Fluid Acceleration in the Bulk Turbulence of Dilute Polymer Solutions

We report experimental measurements of Lagrangian accelerations in the bulk of intense turbulent flows of dilute polymer solutions by following tracer particles with a high-speed optical tracking system. We observed a significant decrease in the acceleration variance in dilute polymer solutions. The shape of the normalized acceleration probability density functions, however, remain the same as in Newtonian water flows. We also observed an increase in the turbulent Lagrangian acceleration autocorrelation time in these dilute polymer solutions.

It has been known for many years that the addition of a very small amount of long-chain polymers to a fluid can have a dramatic effect on the turbulence near solid boundaries and result in significantly reduced flow resistance [1]. This "drag-reduction" phenomenon has been the subject of extensive studies [2,3,4,5,6,7,8,9]. On the other hand, the effect of polymers on the bulk turbulence, away from boundaries, has received less attention. Tabor & De Gennes [4,10] and Lebedev & coworkers [11,12,13] proposed theories for homogeneous isotropic turbulence in dilute polymer solutions. Experiments [7,14,15] and numerical simulations [16,17] have provided evidences that in bulk turbulence the small scales Eulerian quantities are damped by the polymer additives. Validation of these theories, however, has been very difficult due to the lack of detailed experimental data.
Recently, there has been growing interest in studying turbulence from a Lagrangian viewpoint, i.e., by following the motion of the fluid particles [18]. It has been shown, in particular, that the dynamics of turbulent flows with polymer additives is closely related to the correlation of velocity gradient following Lagrangian trajectories [19].
In this Letter, we report experimental measurements of fluid acceleration and its Lagrangian correlations in intense turbulent flows of dilute polymer solutions with various concentrations. The Taylor microscale Reynolds number of the turbulent flows before adding polymers was in the range of 140 ≤ R λ ≤ 485. We measured fluid accelerations by optically following tracer particles with a high-speed three-dimensional particle tracking system. We observed strong decreases of turbulent Lagrangian accelerations and an increase in acceleration anisotropy in these flows, similar effects on Eulerian small scale quantities have been reported previously [14,15]. Moreover, we observed that both effects increase with polymer concentration. The probability density function (PDF) of acceleration normalized by the standard deviation, however, is not affected by the polymer additives. The measured auto-correlation time of Lagrangian acceleration components increases with polymer concentration. These measurements provide detailed data of the effect of polymers on small scale Lagrangian turbulence. In addition, we observed an increase of the correlation time of Lagrangian acceleration magnitude, which suggests that larger scales of turbulence are also affected.
We studied a turbulent water flow generated between two counter-rotating coaxial baffled disks with blades that force the flow inertially rather than through a boundary layer, as shown in earlier experiments in a similar apparatus [20]. The flow was seeded with 50µm or 26µm neutrally buoyant polystyrene particles. These particles have previously been shown to faithfully follow the turbulent flow in pure water at Reynolds numbers up to O(10 3 ) [21]. In this study, no difference was seen between measurements made with the two different sized particles in the water or in the polymer solutions. In addition, within the range of polymer concentrations studied in this work (0-10ppm), we observed only small changes in fluctuation velocity with polymer concentration, comparable to the estimated uncertainties of velocity measurement.
A 35W pulsed frequency-doubled Nd:YAG laser was used to illuminate the tracer particles. A 4.1 × 4.1 × 2.05 mm 3 volume was imaged onto four silicon strip detectors, each of which measured one coordinate of the particle position with spatial resolution 8µm/pixel. The silicon strip detectors can record up to 70,000 frames per second. The recording frequencies were chosen such that the time resolution of the measurement is at least 100 frames per τ η , sufficient for accurate acceleration measurement. The details of the flow and imaging system have been described in [21,22].
The measured particle positions were connected to form three dimensional particle trajectories using the algorithms described in Ref. [21]. Velocities and accelerations were computed by convolution of the trajectories with an appropriate kernel based on successive derivatives of a Guassian filter of width σ k , which performs simultaneously a low-pass filtering and the differentiation [23]. The variances of acceleration components, a 2 i , for both water and the polymer solutions, were estimated as in [21], i.e., a 2 i is computed as a function of σ k and then is extrapolated to σ k going to zero, in order to remove the effect of measurement noise on acceleration variance.
We developed a protocol to prepare the polymer solutions such that the experiments are robust and reproducible. First a high concentration stock solution was prepared by mixing the dry polymer powder with water. A magnetic stir was used to mix the solution at slow speed for 16 hours. Then the stock solution stood for several hours and was depressurized to remove soluble gases. Finally it was gently added to the turbulence chamber to form a dilute solution at the desired concentration. We typically waited 10 to 15 minutes before starting the measurement so that the solution was wellmixed. In this Letter, we present measurements on solutions of polyacrylamide (PAM) 18522 from Polysciences. All initial stock solutions were 400 ppm except one experiment with a final concentration of 1ppm, which had a stock solution of only 20 ppm.
According to the manufacturer, the weight average molecular weight, M w , for the 18522 PAM is M w = 18 × 10 6 amu. The manufacturer also gave a measure of the polydispersity of the sample M w /M n = 3 − 4, where M n is the number average molecular weight. Using the Zimm model [24] we calculated that the radius of gyration R g = 0.5µm, the maximum extension R max = 77µm and the polymer relaxation time τ p = 43ms for this polymer. As it is widely accepted, the most important dimensionless number that characterizes the polymer dynamics in turbulence is the Weissenberg number W i ≡ τ p /τ η , which measures the ratio of the polymer relaxation time to the Kolmogorov time τ η , the smallest time scale of turbulence. Only when W i 1, the polymer molecules can be stretched by the turbulent flow and may affect the turbulence. The corresponding Weissenberg number for each Reynolds number that we have measured is shown in Table I. Some experiments have shown that the effect of polymer on turbulence is sensitive to the polydispersity of the polymer sample with the longer polymers in the sample having a greater effect on the turbulence [25]. In view of this, the τ p and W i calculated from the M w may be treated as a lower bound. Figure 1 shows the component acceleration PDF for water and the polymer solution of 5ppm by weight at R λ = 285. The unnormalized PDFs for the dilute polymer solution are strongly suppressed, which indicates damping of dissipation scale fluctuations by the presence of polymers. It is interesting to note that the PDF 's of the normalized acceleration, a i / a 2 i 1/2 , for water and for the dilute polymer solution overlap with each other, as shown in Figures 1 (c) and (d), which suggests that the accelerations are suppressed only by a scaling factor.  fp is the rotation speed of the baffled disks. R λ is the Taylor microscale Reynolds number. η and τη are the Kolmogorov length and time scales, respectively. R λ , η, and τη refer to the turbulent water flows before adding polymer solutions, and are measured as in Ref. [21]. N f is the temporal resolution of the recording system, in frames per τη. W i ≡ τp/τη is the Weissenberg number.
The rms acceleration, a 2 i 1/2 , in the polymer solutions is dramatically lower than in water. Figure 2 shows the ratio of the rms acceleration in polymer solutions to that of water for the same rotation rate of the propellers. In the range of polymer concentrations investigated, the dependence on concentration is roughly linear. There is no clear dependence on R λ or W i.
It should be pointed out that the decrease in acceleration variance can not be explained by a decrease in energy dissipation rate ε. In earlier experiments carried out in similar geometry, it has been shown that the total energy dissipation rate remains unchanged when polymers were injected into the flow if baffled disks were used to provide the driving [20]. The inertial forcing by the vanes was not altered by the presence of polymers.
The anisotropy of the small scales is also increasing  upon addition of the polymer. The insert in Figure 2 shows the ratio of the horizontal acceleration variance to the axial acceleration variance as a function of polymer concentration. If the polymers were causing the energy cascade to be truncated at a smaller wave number, then the separation between large and small scales would be reduced and the anisotropy of the large scales would affect the small scales more. This is similar to seeing the anisotropy increase with decreasing R λ . The increase of anisotropy might also come from the alignment of polymers in particular directions due to the large scale anisotropic flow. Since the polymer responses in longitudinal and transverse directions (relative to the alignment of polymers) are different [26], this difference could appear as anisotropy in acceleration.
Since the dynamics of polymer solutions depends on the velocity field following Lagrangian trajectories, we studied the Lagrangian acceleration autocovariance c ij (τ ) = a i (t)a j (t + τ ) and the Lagrangian autocorrelation C ij (τ ) = ai(t)aj (t+τ ) ( ai(t) 2 aj (t+τ ) 2 ) 1/2 . To account for the effect of the finite volume bias, we calculated the modified autocovariancec ij (τ ) in which t&τ indicates that only measurements of acceleration where a measurement at both t and t + τ existed on a particle track were averaged over and t indicates that all measurements of acceleration were averaged over. The autocorrelation is theñ where a 2 i is determined from the procedure described above. Figure 3 showsc ij (τ ) andC ij (τ ) for water and polymer. The height of the autocovariance is depressed because, as we have discussed above, a 2 i is dramatically decreased in the polymer solutions. It has been shown that the integral of the acceleration correlation vanishes [27] and the zero-crossing time of the correlation curve is usually used as a measure of the correlation time [28,29]. The curve at the zero-crossing has a shallow slope. Small measurement errors in the acceleration correlation can cause large deviations in the zero-crossing time. A more robust measure is the exponential decay time τ 1/e , defined such thatC(τ 1/e ) = 1/e. In pure water, the τ 1/e times are approximately 0.7-0.8 times the Kolmogorov times. As shown in Figure 3, in the range of polymer concentration used, the τ 1/e for polymer is about 3-4 times the τ 1/e for water and increases with polymer concentration.
The width of the acceleration correlation is proportional to the Kolmogorov time, τ η . In pure water we observe that component acceleration correlation functions for R λ = 285 to 970 collapse when the time is normalized by τ η [30]. The Kolmogorov time in turn is proportional to ν 1/2 . The acceleration variance is proportional to ν −1/2 . If the polymer solutions were to behave simply as if they had an increased effective viscosity, then we would expect the decrease in the acceleration variance to match the increase in the correlation time. Instead we find that the decrease in a 2 i is much larger than the increase in the autocorrelation time. The acceleration variance for a 3.5 ppm solution is more than 4 times higher than for a 10 ppm solution. However the τ 1/e for the 10 ppm solutions is not even twice as much as that of the 3.5 ppm solution. The backreaction of the polymer is obviously more complex than just a change in the bulk viscosity and is most likely anisotropic and dependent on the local velocity gradients [19] as for example the elongational viscosity is known to change by orders of magnitude but not the shear viscosity [9].
We also measured the autocorrelation of the magnitude of Lagrangian acceleration, defined as C |a| = (|a(t)|− |a| )(|a(t+τ )|− |a| ) (|a(t)|− |a| ) 2 and obtained using an estimator equivalent to Eq. (2). The results are shown in Fig. 4. The time scale given by the acceleration magnitude correlation increases with polymer concentration. This may indicate that the large scales of the flow are affected by the polymer as well, since the acceleration magnitude is correlated over much longer time scales of the flow than the acceleration components. Due to the finite size of the measurement volume, we only measured the correlation of the acceleration magnitude over a few τ η . It is possible that, at longer times, the curve for the polymer solution rejoins that for water.
In summary, we studied the effect of polymers on the small scales of turbulence by measuring the fluid acceleration in turbulent flows of dilute polymer solutions. We observed strong decreases of acceleration variance in polymer solutions while the shape of the acceleration PDF remains the same as in water. The effect on acceleration variance was insensitive to the Reynolds number or the Weissenberg number. The Lagrangian correlation . C |a| (τ ) for water and polymer. • water R λ = 485, σ k = 0.35τη. • water R λ = 285, σ k = 0.31τη . 5ppm, R λ = 485, σ k = 0.35τη. △ 5ppm, R λ = 285, σ k = 0.31τη.
10ppm, R λ = 285, σ k = 0.31τη. The polymer statistics for R λ = 485 were taken from the first 3.3 hours of the run. time of acceleration components increases with polymer concentration. The observed effect of polymer additives cannot be explained by a change of viscosity alone. In addition, the measured Lagrangian correlation time of acceleration magnitude increases with concentration, which implies that polymers may affect larger scales.