time-resolved 3d-piv measurements of traveling waves in a...

16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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Time-resolved 3D-PIV measurements of traveling waves in a torus

Jakob Kühnen1,*, Markus Holzner2, Björn Hof3, Hendrik Kuhlmann1

1: Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Austria

2: Institute of Environmental Engineering, ETH Zürich, Switzerland 3: Max Planck Institute for Dynamics and Self Organization, Göttingen, Germany

* corresponding author: [email protected] Abstract Transition to turbulence in curved pipes is considerably different compared to transition in straight pipes, because the onset of turbulence is significantly delayed to higher Reynolds numbers even for small pipe curvature. Moreover, the transition scenario will be modified, since the curvature-induced secondary flow may become unstable to stable travelling waves, a process that is not yet understood in detail. The aim of our present work is to accurately investigate the transition to turbulence in curved pipe flow experimentally with special attention to the detection and investigation of travelling waves and coherent structures. To that end a novel experiment was set up that realizes a flow in a toroidal pipe. The flow is driven by a sphere that is moved from outside the closed toroidal cavity using a magnet. The ratio of the coiling to the tube diameter is 20.26. Laser Doppler Velocimetry and a high-speed stereoscopic Particle Image Velocimetry (S-PIV) system has been used to investigate and capture the appearance and development of the transitional flow. Using S-PIV we were able to reconstruct all three components of the velocity vectors in the measurement plane covering the entire cross-section of the tube. For a Reynolds number below 4080 we find a steady laminar flow which is axisymmetric except with respect to the vertical midline of the cross section of the tube. Above Re=4080 we find a supercritical instability of the steady basic flow to a flow which is periodically modulated in the streamwise (toroidal) direction. Contours of the instantaneous streamwise flow field during one full wavelength are presented. 1. Introduction The physics of transition to turbulence in straight circular pipes has recently experienced significant progress due to the application of modern optical measurement techniques and computationally-based theoretical modeling (e.g. Hof et al. 2004, Eckhardt et al. 2007, Mullin 2011). Based on these advances it is now possible to also tackle the transition to turbulence in curved pipes using these new methods and concepts. Transition to turbulence in curved pipes is theoretically interesting, because the onset of turbulence is significantly delayed to higher Reynolds numbers even for small pipe curvature. Moreover, the transition scenario will be modified, since the curvature-induced secondary flow becomes unstable to stable traveling waves (Sreenivasan and Strykowski 1983, Webster and Humphrey 1993, Webster and Humphrey 1997). Very recently Piazza (2011) investigated incompressible flow in toroidal pipes of circular cross-section by three-dimensional, time-dependent numerical simulations using a finite-volume method. They found consecutive transitions between different flow regimes from stationary to periodic, quasi-periodic and chaotic flow. A torus can be seen as a generalized model for curved pipes. For this reason up to now most theoretical and numerical studies on curved pipe flow have been conducted for toroidal models, while, to our knowledge, so far no study has achieved to attain a steady flow in a real torus for experimental investigations. Madden (1994) and del Pino (2008) used a torus model to experimentally investigate spin-up from rest and spin-down from a certain velocity, i.e. transient flow. The aim of our present work is to accurately investigate the transition to turbulence in curved pipe


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flow experimentally with particular attention to the detection and investigation of traveling waves and coherent structures. To that end a novel experiment with a sphere in a torus driving a flow was set up. Among the quantities to be experimentally investigated is the full three-dimensional and time-resolved structure of the transitional flow including all three velocity-components over the entire cross-section of the pipe. 2. Experimental setup The main challenge was to steadily drive the flow in a closed toroidal cavity. The rather simple mechanism used to realize a precisely adjustable stationary flow in the torus was by placing an adequate actuator, realized as an iron sphere or a custom-made plunger, into it. The actuator is then controlled and moved from outside the cavity via a magnet. Figure 1 provides a sketch of the driving mechanism and the main parts of the experimental setup. Two highly transparent and polished plexiglas disks into each of which a concentric notch of half-circular cross section has been machined were flush mounted such that they form a toroidal cavity. The toroidal cavity is filled with the working fluid. The steel sphere is driven by a magnet which is mounted on a rotating arm. To achieve a constant, precisely adjustable flow rate in the torus the boom is rotated by an electric gear motor, thereby driving the sphere and, hence, the flow. A reed switch is used as a sensor to measure the angular velocity (motor and sensor are not shown in fig. 1).

Figure 1: Illustration of the mechanism used to realize a constant flow rate in a torus: two disks of plexiglas into each of which a concentric notch of half-circular cross sections has been machined, and a rotating boom driving a sphere by means of a magnet. When measuring flow details with PIV a large tube diameter d=2a is desirable. Van Doorne and Westerweel (2004) and Hof et al. (2004), e.g., used tube diameters of 40mm for their PIV measurements. To machine such a diameter or radius respectively as a concentric notch into the plexiglas disks would require the use of a computer numerically controlled machine which was not available at our workshop. For our workshop the maximum producible radius for the tube with custom-made circular-profiled form-cutter blades was about 30mm. Since we had to permit rolling motion for a commercially available ferromagnetic sphere which was used as the actuator, a diameter of 30.3mm was chosen for the tube and a diameter of 30mm for the sphere. A small sickle-shaped gap of max. 0.3mm would hence remain between tube and sphere, resulting in a minor leak flow of approx. 2% of the bulk flow past the sphere. Since small radius ratios are in the focus of our work, a large coiling diameter of the torus was desirable as well. As our workshop is limited to overall diameters of turning workpieces of maximum of 700mm due to the existing turning lathe, the plexiglas disks were first roughly cut to 700mm and then machined to an outer diameter of 680mm. Some supporting area with space for bolting the two disks together and also gasket contact face is needed. Therefore, a diameter of


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D=614mm was chosen for the torus, yielding a diameter ratio of d/D=0.049. The manufacturing tolerance of the torus dimensions is D=614+/-0.1mm and d=30.3+/-0.03mm respectively. Based on the measured revolution times of the rotating boom and the temperature in the torus, the bulk velocity in axial direction U, the kinematic viscosity ν and hence the Reynolds number Re=Ud/ν of the flow can be calculated. The setup provided the possibility to investigate the flow in the torus for adjustable Reynolds numbers ranging from 0 to 14.000 with an accuracy of +/-2%. To experimentally investigate the flow in the torus by means of optical visualization and measurement techniques such as Laser Doppler Velocimetry and Particle Image Velocimetry, the index of refraction of water (nw=1.33) was changed by adding ammonium thiocyanate (e.g. Budwig 1994, Hopkins et al. 2000) to match the refractive index of the plexiglas plates (np=1.492). Since the index of refraction is temperature-dependent, matching indices of refraction was done accurately only to the second decimal point, yielding a tolerance of nw=1.492+/-0.004 measured with an Abbe refractometer. The two photographs in fig. 2 show a grid as seen through the torus model from above when filled with air (left) and when using the index-matched fluid (right). For the matching mixture, the distortion disappeared. The temperature-dependence of the refractive index under variations of the ambient temperature could not be recognized by the naked eye and thus was assumed negligible. For PIV measurements silver-coated hollow glass spheres (S-HGS-10, mean diameter 10µm, Dantec Dynamics) were used, since polyamide seeding particles or hollow glass spheres, both having a refractive index close to 1.5, would not be visible in the index-matched fluid.

Figure 2: Two photographs of a short section of the torus from above. To illustrate the effect of refractive index matching, a checkered pattern was printed on a sheet of paper and put below the plexiglas disks. The picture on the left was made with an empty torus, while on the right side the torus is filled with index matched fluid (distilled water with ammonium thiocyanate). As the index of refraction varies with the wavelength of the (ambient) light, the rainbow colors in the right picture at the torus wall are due to chromatic dispersion. For the monochromatic light of the laser there is no impact though. A prism, also made from plexiglas, acted as a triangular window for the stereoscopic PIV measurement setup (see fig. 3). In absence of the prism the light would be reflected at the wall of the plexiglas plate, which would reduce the effective viewing angle of the cameras. A few drops of glycerin were used to assure seamless connection between the prism and the plexiglas plate. For preliminary investigations, we visualized the flow by adding suspensions of anisotropic light-reflecting particles, allowing visual identification of pertinent flow structures. Still photographs as well as video recordings were made. For investigations with Laser Doppler Velocimetry (LDV), a 2D-LDV system (Dantec Dynamics, Argon-ion-laser Model 5500A, 750 mW, class IV) was used to make velocity time-series measurements of the streamwise velocity component at single points within the flow. A high-speed stereo PIV system has been used to measure the three components of the velocity vectors over a cross section perpendicular to the flow and, hence, to investigate and capture the appearance and development of transitional flow. It consisted of a pulsed Quantronix Darwin Duo laser (diode pumped Nd:YLF laser, wavelength 527nm, 60mJ total energy) and two Phantom high-speed cameras with a full resolution of 2400x1900px. The actual resolution used was 1456x1704px.


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The temporal resolution was set to 100Hz. The calibration target, needed for the calibration of the 3D-PIV measurements, was a custom made (by Die Signmaker GmbH, Göttingen), 5mm spaced lattice of black dots with a diameter of 1mm printed on both sides of a 1.5mm plexi-glass plate. The plate had a cross-sectional diameter of 30mm, i.e. practically the diameter of the tube. The plate was kept in position (congruent with the light sheet of the laser) for the calibration procedure by a tiny locking screw drilled into the torus wall. After the calibration images were shot the screw and hence also the plate could be released. Since the experimental setup was too sensitive to be disassembled after calibration, the calibration plate was just left in the torus and went with the flow in close proximity of the actuator.

Figure 3: Sketch (side view) of the stereoscopic PIV system. To reduce optical reflections, a prism is placed on the upper plexiglas disk of the torus. (1) lower plexiglas disk, (2) upper plexiglas disk, (3) screws, (4) prism, (5) light sheet, (6) cameras, (d) pipe diameter. A major disadvantage of the custom made calibration target was its relatively low precision. As the lattice of black dots was printed on both sides the accuracy achievable is lower compared to calibration targets which are produced by means of CNC-machines. Furthermore the printed shapes of the dots are not as clearly contoured and sharp-cut, but tend to frazzle at the borders. Nevertheless, with the additional use of a self-calibration step we achieved a high accuracy that is comparable to the one achievable with high precision targets. The evaluation of the 3D-vector fields from the PIV images was performed with commercial PIV-software (DaVis 8, LaVision). The interrogation area was 32x32px with an overlap of 50%. The maximum particle displacement (between two frames) was approx. 11px in horizontal and 8px in vertical direction. Subsequently the acquired data were analyzed and evaluated with Mathworks Matlab R2011b. 3. Results For the diameter ratio of d/D=0.049 investigated and for Reynolds numbers up to ~ 4000 the flow is laminar, i.e. stationary, toroidally symmetric and symmetric with respect to the vertical midplane of the cross section. Only in the direct proximity of the actuator the flow is disturbed. The extent of the region of perturbed flow increases with increasing Reynolds number. To exclude any influence of the actuator, measurements were only taken in the half of the torus opposing the actuator. I.e., if the actuator is considered to be located at the toroidal angle φ=0, we examined only the range [π/2, 3π/2] or less (see also fig. 6).


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Figure 4: Coordinate system and denominations of the characteristic areas within the cross section as used in the text and the figures. r denotes the distance from the center of the cross section of the pipe, α the poloidal angle in the cross section, and φ the toroidal angle, i.e. the angular distance between two cross sections. As a characteristic of curved pipe flow, the location of the maximum streamwise velocity is considerably shifted towards the outer wall and two secondary counter-rotating Dean vortices are created (Dean 1927, Dean 1928). At low Reynolds numbers the Dean vortices, which are vortical secondary flow structures, are steady and located symmetrically with respect to the equatorial plane of symmetry. As sketched in fig. 4, the core region in the bulk of the flow is centrifugally driven toward the outer wall. Fluid with high streamwise velocity in the outer half of the cross section is convected toward the inner wall by a cross-stream flow leading to elongated contour shapes of the streamwise flow. With increasing Reynolds number and/or increasing curvature the pressure-driven wall layers returning the cross-stream flow to the inner wall become thinner and confined to a distance close to the wall. Region 1 indicates the regions near the top and bottom walls in the inner half of the pipe cross section, where Webster and Humphrey (1997) observed the fluctuations of the streamwise velocity to be the largest.

Figure 5: Left: Mean streamwise velocity (color contours) and in-plane velocity (vector field) at Re=3600, both normalized with the bulk velocity U. The outside of the torus is at the left. The flow field is an average of 400 PIV-images. Right: Three-dimensional view of the characteristic streamwise velocity profile. As the maximum velocity is strongly skewed toward the outer wall of the tube and the gently s-shaped 'back' of the core region is encircled by distinctly superior 'arm rests' of the wall layer, the appearance reminds of a padded armchair. Figure 5 depicts a representative flow field (for Re=3600) in the regime where the bulk flow is steady and axisymmetric. The position of the maximum streamwise velocity and, therewith, the


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whole profile is severely skewed to the outer wall, causing a sharp peak about 0.12d off the outer tube wall with a maximum at about 1.6 of the normalized velocity. Fig. 5 hence shows a steep gradient of the streamwise velocity along the horizontal axis near the outer wall of order 1.3U/0.1d and a substantially more snugly inclined gradient towards the inner wall. The 'back' of this snugly inclined slope along the horizontal midplane is gently wavy or s-shaped, yielding an inflection point in the inner half of the tube, about 0.25d from the tube centerline. In the upper and lower cross-stream wall layer, i.e. in the regions of maximum in-plane velocity, the streamwise velocity profile develops two characteristic local maxima near the wall. These local maxima are best seen in the 3D view provided in fig. 5 (right). Even though the cross-stream (r and α) velocity components are about an order of magnitude smaller than the streamwise velocity, the vector field in fig. 5a representing the secondary flow is clearly visible and well resolved. The secondary flow field exhibits the expected two-vortex structure with velocity maxima close to the upper and lower wall. The inward secondary flow arises in thin boundary layers close to the outer, upper and lower pipe wall, while the flow directed radially outward is quite evenly distributed over the rest of the cross section. Region 1 is, furthermore, characterized by a rather sharp turn of the cross stream velocity and could be regarded as the apparent center of the Dean vortices. For a Reynolds number of 4080 we found a supercritical transition from the basic toroidally symmetric flow to a laminar flow which is periodically modulated in the streamwise (toroidal) direction. The emerging pattern is almost stationary in a frame of reference moving with the actuator, exhibiting a wave celerity slightly above (~12%) the mean bulk velocity of the flow depending on the Reynolds number. Slightly above the critical Reynolds number the amplitude of the streamwise modulation is very weak. With increasing Reynolds number, the modulation amplitude of the streamwise velocity grows and the associated flow pattern becomes more distinct. Figure 6 shows a time trace of the streamwise velocity at the center of region 1 (see fig. 4) for Re=4300. A very clear periodic modulation of the streamwise velocity can be observed. In the vicinity of the actuator the modulation is disturbed. The frequency of the modulation is visible already in the closer vicinity of the actuator, but the amplitude is lower than in the bulk. In the section marked as 'used range' the amplitude of the modulation is constant within ~5%. Thus, the flow is considered fully developed and not influenced by the details of the driving mechanism. All data used for the subsequent analysis are taken from this range.

Figure 6: Time trace of the streamwise velocity at the center of region 1 for Re=4300. A typical PIV run of 12s, comprising approximately one full revolution of the actuator, is displayed. The streamwise velocity is periodically modulated in the toroidal direction. The transit of the actuator through the measurement plane at ~8s is clear-cut. In the vicinity of the actuator the flow is considerably disturbed. In the section marked as 'used range' the flow is considered fully developed. Only this section has been employed for the analysis of the instability and the flow field. The duration of 2.25s of the 'used range' corresponds to 0.5m or ~26% of one full revolution.


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Figure 7: Contours of the instantaneous streamwise velocity during one full wavelength λ at Re=4300. a) λ=0, b) λ =1/4, c) λ =1/2 and d) λ =3/4. Each image is an average out of 8 periods. The velocity has been normalized with the bulk velocity U and is shown in increments of 0.1. The outside of the bend is at the left. The first instability, arising at Re=4080, is mainly characterized by a regular periodic modulation of the instantaneous streamwise flow field. Figure 7 illustrates the incipient modulation of the


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streamwise velocity by means of a sequential arrangement of the contours of the instantaneous streamwise velocity during one full wavelength λ at Re=4300. As the cross-stream velocities show very little modulation being practically imperceptible on the scale of the streamwise velocity, the figure displays only contour lines for the streamwise velocity. Re=4300 was selected for the purpose of illustration as the modulation is well established at this Reynolds number. Looking at the streamwise velocity profile, the core region of high streamwise velocity at the left half, i.e. the outer half of the tube, is subject to minor 'swell' during one wavelength. Only an almost negligible change between a slightly more or less sharply cut peak considering a vertical profile and a slight oscillation of the position of the maximum streamwise velocity can be recognized. Also the modulation in the low speed core region on the right seems insignificant. The modulation is most notable for the contour-line value 1U, especially at the 'end' of the ‘arm rests’ in region 1. The strongest alteration of the flow field is, hence, realized by periodic speeding up and slowing down of the streamwise velocity in the area located in region 1, i.e. at the end of the cross-stream wall layer. The frequency of the modulation is 3.25Hz at Re=4300, yielding a nondimensional frequency f’=fd/U of 0.485. As the basic flow in toroidal direction is characterized by a strong shear gradient, small perturbations of the flow in poloidal direction lead to large fluctuations of the streamwise velocity. These fluctuations can be regarded as periodic streaks. 4. Summary We have considered the pressure-driven flow in a toroid with d/D=0.049. By detailed stereoscopic PIV measurements the time-dependent flow structure was determined quantitatively. In particular, we have reconstructed all three velocity components on the entire cross-stream plane. We have precisely determined the critical Reynolds number of Re=4080 +/-2% for the onset of toroidal waves with wavelength 2.1d and celerity 1.12Ubulk. The bifurcation was found to be supercritical. The amplitude of the wave takes its maximum in region 1.


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References Budwig, R. (1994), ‘Refractive index matching methods for liquid flow investigations’, Experiments in Fluids 17, 350–355. Dean, W. R. (1927), ‘Note on the motion of fluid in a curved pipe’, Philos. Mag. 4, 208. Dean, W. R. (1928), ‘The streamline motion of fluid in a curved pipe’, Philos. Mag. 7, 673. del Pino, C., Hewitt, R. E., Clarke, R. J., Mullin, T. & Denier, J. P. (2008), ‘Unsteady fronts in the spin-down of a fluid-filled torus’, Physics of Fluids 20(12), 124104. Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. (2007), ‘Turbulent transition in pipe flow’, Annual Review of Fluid Mechanics 39, 447. Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M. & Faisst, H. (2004), ‘Experimental observation of nonlinear traveling waves in turbulent pipe flow’, Science 305, 1594–1598. Hopkins, L. M., Kelly, J. T., Wexler, A. S. & Prasad, A. K. (2000), ‘Particle image velocimetry measurements in complex geometries’, Experiments in Fluids 29, 91–95. Madden, F. N. & Mullin, T. (1994), ‘The spin-up from rest of a fluid-filled torus’, Journal of Fluid Mechanics 265, 211. Mullin, T. (2011), ‘Experimental studies of transition to turbulence in a pipe’, Annual Review of Fluid Mechanics 43(1), 1–24. Piazza, I. D. & Ciofalo, M. (2011), ‘Transition to turbulence in toroidal pipes’, Journal of Fluid Mechanics 687, 72–117. Sreenivasan, K. R. & Strykowski, P. J. (1983), ‘Stabilization effects in flow through helically coiled pipes’, Experiments in Fluids 1, 31–36. van Doorne, C.W. H. (2004), Stereoscopic PIV on transition in pipe flow, PhD thesis, Technische Universiteit Delft. van Doorne, C. W. H. & Westerweel, J. (2007), ‘Measurement of laminar, transitional and turbulent pipe flow using stereoscopic-PIV’, Experiments in Fluids 42, 259–279. Webster, D. R. & Humphrey, J. A. C. (1993), ‘Experimental observation of flow instability in a helical coil’, Journal of Fluids Engineering 115(3), 436–443. Webster, D. R. & Humphrey, J. A. C. (1997), ‘Traveling wave instability in helical coil flow’, Physics of Fluids 9, 407–418.

time-resolved 3d-piv measurements of traveling waves in a...

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