Non-Newtonian Turbulent and Transitional Pipe Flow

Transcription

1 Non-Newtonian Turbulent and Transitional Pipe Flow M. Rudman, L. J. Graham, H. M. Blackburn, 1 L. Pullum 2 1. Thermal & Fluids Engineering group, CSIRO, 2. private consultant This paper presents results from experimental and numerical investigations of non Newtonian fluids at transition to turbulence and in weakly turbulent flows. Experimental results showed flow features similar to turbulent puffs and slugs observed in Newtonian transitional flows. Numerical results showed some quantitative discrepancies with the experimental results but did show turbulence suppression, drag reduction and delayed transition as observed experimentally. The discrepancies may be due to deviations from the ideal power law behaviour in the experimental systems, for example, weak elastic behaviour. 1 INTRODUCTION The motivation for this work was to explore the behaviour of non Newtonian fluids at transition to turbulence or in weakly turbulent pipe flow and to compare this behaviour with Newtonian fluids. In practical terms it is possible that operating in the transitional regime may be advantageous for suspension transport with the intermittency of such flows being useful in re-suspending solids that have settled but still operating at a lower pressure gradient than fully developed turbulent flow. Hence transition is of both academic and practical significance. Banfi et al (1) used a laser-doppler velocimeter to investigate transitional pipe flow, in particular the behaviour of the velocity fluctuations as the Reynolds number was increased from 1500 to It was noted that the velocity fluctuations reached a maximum at a Reynolds number of about 2800 (in the transition region). The turbulent pipe flow of several aqueous polymer solutions was investigated by (2). They noted that the transition to turbulence was practically indiscernible from friction factor versus Reynolds number plots but that the axial turbulence intensity, measured by LDV at about 20% of the radius of the pipe from the wall, was a consistent indicator of transition for the range of

2 fluids tested. These results showed that the axial fluctuation intensity rose to a peak value as the Reynolds number increased before again declining to an asymptote. Park et al (3) used LDV to investigate a slurry which exhibited yield power law behaviour. Their results showed a sharp transition to turbulent flow in contrast to (2). An interesting feature of their results is that as the mean flow velocity was increased the axial velocity fluctuation intensity reached a peak before declining again. This behaviour was observed for both a Newtonian fluid and for the slurry. Escudier and Presti (4) examined the pipe flow of an aqueous solution of laponite, which shows thixotropic behaviour. LDV measurements were made ranging from laminar to turbulent flow conditions. As was the case with the previous two references high axial fluctuation intensities were associated with transitional flow. Flow of non-newtonian fluids in a pipe was investigated by (5). The fluids used were four different concentrations of sodium carboxymethyl cellulose dissolved in water (0.1, 0.2, 0.3 and 0.4% by weight). They defined their Reynolds number in terms of the bulk velocity and the viscosity at the wall (i.e. a generalised Reynolds number). They found that transition to turbulence was a gradual process with the 0.2% solution not achieving turbulence below a Reynolds number of The 0.1% solution reached turbulent flow at a Reynolds number of Drag reduction at higher Reynolds numbers was also noted. A similar set of experiments were performed by (6) who used 0.4%, 0.5% and 0.6% solutions of Tylose (methylhydroxil cellulose). Again wall Reynolds numbers were used. The results showed that transition to turbulence was delayed as measured by the ratio of centreline to bulk velocity but no time traces of the velocity were presented. They noted an increase in the turbulence intensity in the transition regime compared with higher Reynolds number turbulent flows that could be caused by flow intermittency but this was not conclusive since time traces of the velocity were not presented. Drag reduction effects were also exhibited. The flow of a slurry near transitional Reynolds numbers was investigated by (7). Pressure drop measurements and LDV velocity measurements were made in a refractive index matched slurry that consisted of chloroform and silica gel. It was found that the universal logarithmic law for velocity distributions was valid for this slurry with some modification to the von Karman constant. Significant work has been undertaken in the area of non-newtonian transitional and turbulent flows however there are still significant areas that are not well understood, particularly the effect of small amounts of fluid elasticity on transitional behaviour. A further lack in the current literature is the scarcity of direct numerical simulations of transition to turbulence and turbulence for non Newtonian fluids which should also significantly enhance the understanding of the effect that non-newtonian rheology has on flow structures. This paper describes some experimental results from a study into the transitional behaviour of two pure non-newtonian carrier fluids. One fluid is well modelled by a Herschel Bulkley model τ τ y kγ! n = + Equation 1 and the other is modelled by the power law model.

3 n = kγ Equation 2 τ! Here τ is the shear stress, γ! is the strain rate, τ y the yield stress, and k and n are coefficients. Direct numerical simulations for the power law fluid have also been undertaken to provide further insight in to the weakly turbulent flow of non-newtonian fluids. 2 EXPERIMENTAL DETAILS A schematic of the pipe test loop is shown in Figure 1. The test facility consists of several different pipe loops but only data obtained in the 100 mm plastic loop are reported here. Steel pipes 150mm 100mm 50mm Key: C L M N W Temperature transducer Ultrasonic level detector Magnetic flow meter Tachometer Load cells Differential pressure transducer NMR L W LDV PIV PDV nom. 20m N T N T M C N C N T 100mm plastic pipe optical window PIV, PDV Figure 1 Pipe test loop at Highett This pipe loop comprises a fully instrumented mixing tank that feeds a special non-magnetic Warman International 4x3 centrifugal slurry pump. This pump feeds a 40m x 100mmdiameter pipe loop that passes through the MRI imaging facility and then finally returns to the mixing tank. Flow can be diverted to a weigh tank for flow calibration and delivered density measurements. Optical windows are installed at the beginning and near the end of the loop. The MRI and second optical window are positioned at the downstream ends of the loop s straight sections to ensure that established flow conditions are examined. The rig is fully equipped with typical industrial pressure and bulk flow transducers and is operated under computer control via a Labview SCADA system. A two colour TSI laser Doppler velocimeter (LDV) mounted on an industrial robot is used to measure the axial velocity

4 profiles across the vertical diameter of the pipe in the second optical window. Further details of this pipe test loop and associated instrumentation are given in (8). The two fluids used in the present investigation were aqueous solutions of Ultrez 10 (a carbomer polymer supplied by B.F. Goodrich) and sodium carboxymethylcellulose (7HF Aqualon CMC supplied by A.C Hatrick). The Ultrez 10 is a cross linked polymer and hence the molecular weight may be as high as several billion. Here the Ultrez 10 is modelled by the Herschel Bulkley relation. The CMC has a molecular weight of approximately and is modelled as a power law fluid. The concentration of the CMC was adjusted (to approximately 0.6% by weight) so that at the shear rates expected in the pipe flow the apparent viscosity was similar to that of the Ultrez 10. Curve fits to the rheology data for the fluids yielded parameters for the rheology models as shown in Table 1. Table 1: Rheology model parameters τ y k n Ultrez CMC In order to compare the results of different experiments, a generalised Reynolds number is defined in terms of the apparent viscosity at the shear rate at the wall, Hence, µ a τ γ! w = Equation 3. w Re * = ρ VD / µ a Equation 4. It should be noted that this generalised Reynolds number represents the ratio between inertial and viscous forces and does not include an effective viscosity or contribution to the wall shear stress from turbulence effects. 3 NUMERICAL METHOD The spatial discretisation employs a spectral element/fourier formulation, which allows arbitrary geometry in the (x,y) plane, but requires periodicity in the z (out-of-plane) direction. The non-linear terms of the momentum equation are implemented in skewsymmetric form because this has been found to reduce aliasing errors. To allow a semiimplicit treatment of the viscous terms, the non-newtonian viscosity is decomposed into a spatially-constant component, η R, and a spatially-varying component η-η R. The spatially varying component is treated with a second-order explicit formulation and the constant component is treated implicitly, thus enhancing the overall numerical stability of the scheme (see (9) for details). The value of η R is chosen to be approximately equal to the maximum value of η. This value is not known a priori, but can be adjusted during the computation

5 without any adverse effects. In order to drive the flow in the axial (z) direction, a body force equal to the pressure gradient measured in the experiments is applied to the z-momentum equation. This approach allows the pressure to be periodic in the axial direction. The code runs in parallel using the message-passing kernel MPI, and the computations reported here were carried out using 8 processors on an NEC-SX5 supercomputer. The underlying numerical code has been validated for both DNS and LES of pipe and channel flow (see for example (10)). The implementation of the power-law non-newtonian viscosity was validated against laminar pipe flow and axisymmetric Taylor-Couette flow of power-law fluids, both of which have analytic solutions. In all cases, numerical results from the code agreed to within 0.01% of theory. The computational domain consists of th order elements in the pipe cross section (see Figure 2) and 96 Fourier modes in the axial direction (3πD long). Numerical integration was continued until such time as the solution had become statistically steady. Averages were then taken over approximately 5 pipe-length traverse times. In terms of wall units, the nearwall mesh spacing is r + 0.5, Rθ + 8 and z This resolution is marginal in the streamwise direction but sufficient for this initial investigation. Figure 2 Upper part of the 2-D cross-sectional mesh used for the DNS (the nodal mesh is shown on the right side only). A Fourier expansion with 96 modes was used in the axial direction. 4 RESULTS AND DISCUSSION 4.1 Experimental results The transport characteristics of the fluids used are shown in Figure 3. Two experiments were performed with the CMC to determine the effect of changing the upstream geometry on transition to turbulence. It was found that the change to the sharp bend upstream resulted in a small change in the flow curve relative to the long radius bend. The flow curve for the Ultrez was measured with the sharp bend in place. The most interesting result from these tests was that despite the apparent viscosities being similar, the transition to turbulence for the CMC occurred at a significantly higher generalised Reynolds number (approximately 3500 versus approximately 2800) as judged from the change in slope of the curve. Detection of transition by other means will be considered later in this section. It is possible that this delay is associated with elastic effects in the CMC solution even though the level of elasticity was too small to be measured with equipment currently available

6 to the authors, although it is also argued by (5) that such delays in transition can also be affected by in-elastic shear thinning properties alone. It is possible that there were also elastic effects in the Ultrez but since the transitional Reynolds number is closer to the generally accepted value of 2100 they are probably less significant than in the CMC Presure gradient (kpa/m) CMC 12/12/2000 Long rad CMC 12/12/2000 Sharp bend Ultrez 11/8/ Generalised Reynolds number Figure 3. Transport characteristics for Ultrez and CMC. Figure 4 shows the axial velocity time series for the Ultrez 10 solution at a range of generalised Reynolds numbers. The corresponding superficial velocities are also shown. At a Reynolds number of 1209 it can be seen that the flow is essentially laminar with the only fluctuations in the profile being those associated with the noise level in the LDV. As the Reynolds number increases to 1710 a low frequency variation in the velocity trace can be discerned with the frequency being of the order Hz. The waviness becomes more pronounced as the Reynolds number is increased to 2454 and a further increase to 3151 results in some turbulent puffs being present as well as the waviness. This is different to the results that would be expected in a Newtonian transitional flow such as that reported by (11) where the waviness noted here was not present. The time scale of the waves is of the order of 3 seconds or more which indicates that the features are at least 3 metres in length. This is a substantial fraction of the length of the pipe (20 m) and is consistent with (11) reporting that turbulent slugs could be as much as of the same order of length as the pipe. At a Reynolds number of 3885 the velocity trace appears to be turbulent. The CMC velocity traces for the centreline of the pipe are shown in Figure 5. Here the Reynolds numbers are higher than those for the Ultrez 10 as seen previously in the transport characteristic. For a generalised Reynolds number of 3194 the flow is essentially laminar. As the Reynolds number is increased to 3865 and 4356 turbulent slugs or puffs appear in the time history. It can be seen that they last for at least a second implying a length of the order of 3 metres. These are similar to those reported for Newtonian fluids and there does not appear to be the waviness seen in the Ultrez 10 results presented previously. Also the mean velocity in the turbulent region of the time trace is less than that in the laminar region, which is expected

7 given that the turbulent velocity profile is blunter than the laminar profile. At a generalised Reynolds number of 5617 the time history appears to be fully turbulent V e lo city -1 ) (m s R e = 3885 R e = 3151 R e = 2454 R e = 1710 R e = Tim e (s) Figure 4. LDV axial velocity time series at pipe centreline for Ultrez R e = V e lo c ity -1 (m ) s R e = 4356 R e = 3865 R e = Tim e (s) Figure 5. LDV axial velocity time series at pipe centreline for CMC. Escudier et al (2) investigated the pipe flow of fluids similar to those considered here and used the centreline value of the axial velocity fluctuation to provide a clear indication of transition to turbulence. This approach has also been used here with the results shown in Figure 6. These data clearly demonstrate the difference between the Ultrez 10 and the CMC with the

8 transition to turbulence taking place at a much higher generalised Reynolds number for the CMC. Another interesting feature of both sets of data is that the value of the root mean square (rms) velocity reaches a peak through the transitional regime and then declines as the flow becomes fully turbulent. This is also reflected in the time histories presented earlier with the large fluctuations associated with turbulent puffs and slugs in the transitional regime. Escudier et al (2) used similar data from near the pipe wall to clarify the transition to turbulence RMS velocity (m/s) Ultrez CMC Generalised Reynolds Number (-) Figure 6. Axial velocity fluctuation at pipe centreline as a function of generalised Reynolds number.

9 Figure 7 Experimentally measured velocity profiles for CMC compared to DNS results for a Newtonian fluid at Re=5,000. The mean axial velocity profile for the CMC as measured by LDV is presented in Figure 7. For comparison, the DNS profile for a Newtonian flow at Re=5,000 is included. From this data, it is difficult to distinguish between the CMC and Newtonian results. Further consideration of the experimental results will be included in a discussion of the numerical results. 4.2 Numerical Results The results from three simulations are presented here. In Simulation 1, a pressure gradient equal to that measured experimentally is used (resulting in Re G =3964). When the pressure gradient used in the simulation is equivalent to that measured experimentally, a mean flow velocity of 0.73 times that of the experiment is predicted. In Simulation 2, an increased pressure gradient (25% above the experimental value) is applied (giving Re G =5500). The mean flow velocity increases, but only to 0.89 times the experimental value. Both simulation 1 and 2 use the power law rheology model. In simulation 3, the Cross model was used with a higher pressure gradient than experiment (resulting in Re G =4723) and a similar mean flow velocity as simulation 2. Further consideration of the discrepancy is presented later. The wall streaks for all simulations show significant axial extent (Figure 8) especially for Simulation 1. The small disordered patches of red (most prevalent in Simulation 2 Re G =5500) are suggestive of intermittency or bursting, and not of fully developed turbulence.

10 Figure 8 Near wall structure revealed in contours of streamwise velocity at y + =10. From left to right: Simulation 1, (Re=3964), Simulation 3 (Re G =4723) and Simulation 2 (Re G =5500).

11 Figure 9 Velocity profiles in wall units: X Experiment (Re G =4678), Simulation 1 (Re G =3964), Simulation 3 (Re G =4723), Δ Simulation 2 (Re G =5500) and Newtonian (Re=5000). Velocity profiles in wall units are presented in Figure 9 for the experimental results, the three non-newtonian CFD simulations and a DNS of turbulent pipe flow at Re=5000. The nondimensionalisation is undertaken using the wall viscosity given in equation 1. The Newtonian profile is in good agreement with accepted profile for low-reynolds number turbulent pipe flow (shown as the solid line). All profiles have a linear relationship between U + and y + in the near wall region. In the logarithmic region (where the flow is represented by U + = A+B lny + ), the experimentally measured profile for the CMC is very much above the low-re Newtonian profile and has generally similar characteristics to a flow that is not fully developed. The profiles are consistent with results presented in (4) in which the offset A increases with CMC concentration (i.e. decreasing n) and the results of (5) where B increases with increasing CMC concentration as does the buffer layer thickness. The results for Simulation 1 (Re G =3964) fall above the low-re Newtonian profile but significantly below the experimentally measured profile. The results for Simulation 2 lie still closer to the Newtonian profile. As Re G increases, the simulation results appear to approach the Newtonian profile, consistent with more developed turbulence as Re G increases, but in disagreement with the experimental data measured here as well as that presented in (4) and (5) for CMC. A number of different causes for this discrepancy have been investigated but none satisfactorily explain the difference. They include the following: 1. Inconsistencies between the LDV data and the magnetic flow meter were observed that account for approximately 4 of the 25% error in the Re G =3964 simulation.

12 2. As seen in Figure 8, the near wall structures have lengths that are comparable to the domain length (especially for Simulation 1 at Re=3964). This will influence the results, although a similar simulation undertaken on a short domain of length πd resulted in an almost identical under-prediction of the mean velocity. Although domain length is an issue, it is not likely to be a major source of the error in mean flow velocity or profiles. 3. Because the power law model for the CMC solution had been determined from shear rates less than approximately 500 sec -1, and the peak values in the turbulent boundary layer were predicted to be of the order of 5000 sec -1, the suitability of both the power law model parameters and the power law itself were called into question. In particular, the possibility of a high shear rate viscosity plateau (more appropriately modelled using a Cross viscosity model) modifying the turbulent structures was a possibility. Even though Cross model parameter fitting from rheology data suggested that the high shear plateau occurs for this material at significantly higher shear rates than those predicted here, Simulation 3 was undertaken using the Cross model (Re G =4723). Results are seen to be consistent with the power law results (see Figure 8 Figure 11) and is not a source of error. Figure 10 Velocity profiles in wall units: X Experiment (Re G =4678), Simulation 1 (Re G =3964), Simulation 3 (Re G =4723) and Simulation 2 (Re G =5500) It was estimated that the highest shear rates in the turbulent boundary layer had time scales that were approximately two orders of magnitude too long for viscoelastic effects to be important. However rheology measurements and turbulent pipe flow measurements of CMC solutions presented in (12) suggest that first normal stress differences in CMC solutions are significant at concentrations higher than 0.2% and result in drag reduction. This is consistent with the discrepancy between experiment and CFD here and it is believed that the discrepancy between experiment and CFD is due to CMC rheology that is not well modelled as a simple power law fluid once the flow becomes turbulent.

13 Results presented in (13) support this claim. For the turbulent flow of Carbopol 934 (claimed to be well modelled with a power law rheology) over a range of concentrations and flow rates, the logarithmic velocity profile is shown in (13) to be a function of the power law index, n: * U + ln y n n * n 2 n =, where y* = ρ τ K y. Equation 5 W This correlation is plotted along side the simulation and CMC data in Figure 10. As shown, the results for the CMC experiment deviate significantly from Equation 5, but the CFD results are in extremely good agreement and approach the experimental correlation as Re G increases. Additional support for the veracity of the simulation results is found in the results of (14) for turbulent flow of a well-sheared Laponite suspension (a synthetic clay that produces a thixotropic fluid). In (14) the profile for transitional flow appears very much like the experimental profile measured here for CMC, however as Re G increases, the value of A falls from around 8 and approaches the Newtonian value (albeit at much higher Reynolds numbers than simulated here). The value of B does not vary significantly from the Newtonian value once the flow becomes fully developed. On this basis, the simulation results here are believed to give a good representation on the behaviour of weakly turbulent power law fluids. Figure 11 Turbulence intensities and Reynolds stress as a function of r/d. Radial velocity (top left), azimuthal velocity (top right), axial velocity (lower left) and Reynolds stress (lower right). Solid line for Newtonian DNS, symbols as for Figure 10.

14 Figure 12 Contours of axial velocity (red is high, blue low) and in-plane velocity vectors for Newtonian flow at Re=5000 (left) and power law fluid at Re G =5500 (right). Turbulence intensities and Reynolds stresses are presented in Figure 11. From the experimental results, only azimuthal and axial turbulence intensities are available. The radial and azimuthal turbulence intensities are lower by 20-40% for the power law fluid simulations compared to the Newtonian DNS, whereas the axial intensities are marginally higher. Interestingly, the CFD results for Re G =3964 are quite close to the experimental results measured here despite the other discrepancies. Cross-sectional velocities for Re G =5500 are shown in Figure 12 and show a qualitatively similar picture to low Reynolds number turbulence in a Newtonian fluid. 5 DISCUSSION AND CONCLUSIONS The experimental measurements reported here are in qualitative and quantitative agreement with previously published experimental results for CMC (2), (5) and are believed to be correct. They also share some features of transitional flow in shear thinning thixotropic fluids (12), however differ qualitatively from experimental results for power fluids reported in (15). The simulation results for a power law fluid show some agreement as well as some significant differences with the experimental results. Importantly, the superficial velocity and velocity profiles for a given pressure drop differ significantly. The results are consistent with turbulence suppression, drag reduction and delayed transition in the experiment. These observations, coupled with the inability to match the DNS results to the experiment or data previously reported in (2) all point to a 0.5% CMC solution having more complex rheological properties than a simple power law fluid. Despite this, it is certainly true that CMC is generally well described by this model in the macro sense and that the laminar flow profiles measured in the pipe were in good agreement with power law predictions. The simulation results have a different form to the experimental results for CMC (Figure 9, Figure 10). The mean flow profiles have some qualitative agreement with experimental results for the shear thinning turbulent flow of Laponite (4), although this is probably

15 fortuitous given that shear history effects that are not included in the DNS are important in Laponite. Importantly, the DNS results are in good agreement with the experimental data of (15) for a power law fluid, with results close to the log law profile measured experimentally there. Additional simulations for different power law indices (n) and higher Re G need to be undertaken to confirm this result, although it is believed the simulations are correctly predicting turbulence of a power law fluid. This paper has presented results from an investigation into the transitional behaviour of two non-newtonian fluids. The apparent viscosities of these fluids were matched as closely as possible to allow a good basis for comparison. The transport characteristics, velocity measurements and pressure gradient measurements showed that the CMC underwent transition to turbulence at a higher generalised Reynolds number than the Ultrez. This may be due to elastic effects in the CMC which had a much higher concentration than the Ultrez. Sa Pereira and Pinho (6) observed similar delays in transition to turbulence with polymer solutions with elasticities that were of the order of the minimum detectable by their instrumentation. Their results and the present results indicate that even small levels of fluid elasticity may have a significant effect on the transitional behaviour. Both fluids tested here also showed evidence of flow features similar to puffs and slugs noted in Newtonian fluids undergoing transition to turbulence. Some of these features were at least 3 metres long (30 pipe diameters) which is a substantial fraction of the 20 meter pipe length (200 diameters). Operating in the transitional flow regime may prove advantageous in practical pipe flow applications because the large velocity fluctuations available can be used to stir up sedimented particles. 6 ACKNOWLEDGEMENTS This work is partially sponsored by AMIRA project P599 High Concentration Suspension Pumping. The authors are grateful to AMIRA, BHP Cannington, Rio Tinto, De Beers, Warman International and WMC Resources for their support. The authors also gratefully acknowledge the assistance of Mr Ross Hamilton with some of the experiments. 7 REFERENCES 1. G. P. Banfi, R. De Micheli, A. Henin, Velocity fluctuation enhancement in the transition to turbulence in a pipe., Journal of Physics D: Applied Physics 14, (1981). 2. M. P. Escudier, F. Presti, S. Smith, Drag reduction in turbulent pipe flow of polymers, Journal of Non-Newtonian Fluid Mechanics 81, (1999). 3. J. T. Park, R. J. Mannheimer, T. A. Grimley, T. B. Morrow, Pipe Flow Measurements of a Transparent Non-Newtonian Slurry, Journal of Fluids Engineering Vol. 111, (1989). 4. M. P. Escudier, F. Presti, Pipe flow of a thixotropic liquid, J. Non-Newtonian Fluid Mech. 62, (1996). 5. F. T. Pinho, J. H. Whitelaw, Flow of non-newtonian fluids in a pipe, Journal of Non- Newtonian Fluid Mechanics 34, (1990). 6. A. Sa Pereira, F. T. Pinho, Turbulent pipe flow characteristics of low molecular weight polymer solutions, Journal of non-newtonian Fluid Mechanics 55, (1994).

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