Experiments on the perturbation of a channel flow by a triangular ripple F. Cúñez *, E. Franklin Faculty of Mechanical Engineering, University of Campinas, Brazil * Correspondent author: fernandodcb@fem.unicamp.br Keywords: PIV processing, Turbulence, Turbulent boundary layer ABSTRACT Turbulent boundary layers over flat walls with perturbations are frequently found in environmental and industrial applications. In industry, some examples are the flows over sand ripples and dunes (which have a triangular shape) in closed conduits such as petroleum and air-conditioning pipelines. In this paper, experiments have been performed for the study of the perturbation of a fully-developed channel flow by a two-dimensional hill of triangular shape. Different water flows were imposed over an asymmetric triangular hill fixed on the bottom wall of a closed conduit, and the flow field was measured by PIV (Particle Image Velocimetry). Reynolds numbers based on the channel height varied between 2.75x10 4 and 3.5x10 4 and the regime was hydraulically smooth. From the instantaneous flow fields, the mean velocities and fluctuations were computed, and the shear stress over the ripple and the field of turbulent production were determined. The general behaviors of obtained velocities and stresses are compared to published asymptotic analyses and the surface shear stress is discussed in terms of bed stability. 1. Introduction The perturbation of a channel flow introduces new scales in the problem, changing the velocity and stress distributions along the flow. These new distributions are important to understand the bed instabilities associated with sediment transport, (Franklin, 2011). Many studies have been devoted to perturbed boundary layers over the last few decades, some of them are based on asymptotic methods. Jackson and Hunt (1975) have presented an analytical analysis of the perturbed turbulent boundary layer by a low hill. The unperturbed boundary layer was given by the law of the wall, and they divided the perturbed boundary layer into two regions. The inner region, close to the bed, is in local-equilibrium because the timescale for the dissipation of the energy-containing eddies is much smaller than the timescale for their advection. The localequilibrium condition allows the use of a turbulent model stresses. This region has a small thickness that does not change significantly along the hill, for this reason the perturbations are driven by the pressure field of the outer region.
For the outer region, that is considered far enough from the bed, the timescale for the dissipation of the energy containing eddies is much larger than the timescale for their advection; therefore, the flow is not in local-equilibrium. Hence, the mean flow is almost unaffected by the shear stress perturbations. Recently, Franklin and Charru (2009) studied the isolated three-dimensional dunes, known as barchans, and showed that the symmetry plane of a banchan dune is nearly triangular. Given the three-dimensional shape of the dune, their results are not conclusive with respect to the unstable role of the fluid flow. This study presents some results of an experimental work on a turbulent channel flow perturbed by a two-dimensional hill of triangular shape. A closed-conduit water flow was imposed over an asymmetric triangular hill (for two different flow rates) with aspect ratio O(0.1), and the flow was measured by particle image velocimetry (PIV). From the instantaneous flows fields, the mean velocities and fluctuations were computed, and the shear stress over the ripple could be determined. 2. Experimental setup The experimental device consisted basically of a water reservoir, two centrifugal pumps, a flow meter, a flow straightener, a 5m long transparent channel of rectangular cross section (160mm wide by 50mm high), a settling tank, and a return line, so that the water flowed in a closed loop. The straightener consisted of a divergent-convergent nozzle filled with glass spheres with a diameter of d=3mm; whose function was to homogenize the flow profile, and it was placed at the channel inlet. The test section was 1m long and started at 40 hydraulic diameters (3m) downstream of the channel inlet. There was another 1m long section connecting the test section exit to a settling tank and the return line. Fig. 1 presents a schematic drawing of the experimental device. PVC plates of 7mm thickness were used to cover the entire bottom of the channel. To model the two-dimensional ripple, a small bedform of triangular shape was fixed on a PVC plate in the test section. The triangular bedform had the same scales as the aquatic ripples usually found in nature and industrial applications (Franklin and Charru, 2009). The bedform was made of PVC plates, painted in black in the test section region, and of a triangular perturbation of black polyoxymethylen. The black color was chosen in order to minimize undesirable reflections. Fig. 2 presents the dimensions of the triangular ripple. The employed flow rates were 8m /h and 10m /h. These flow rates corresponded to cross-sectional 3 3 mean velocities U of 0.32m/s and 0.40m/s and to Reynolds numbers Re = U2H '(( /ν of 2.75x10 4
and 3.5x10 4, where Heff is the distance from the surface of the PVC plates to the top wall of the channel. a) b) Fig. 1 Layout of the experimental device (a) Side view. (b) Cross section. To obtain the instantaneous velocity fields of the flow we used PIV. The employed light source was a dual cavity Nd:YAG Q-Switched laser, capable of emitting 2 x 130mJ at 15Hz pulse rate. The power of the laser was fixed at 66% of the maximum power to assure a good balance between the image contrasts and undesirable reflection from the channel walls. 10µm hollow glass beads (S.G.=1.05) were employed as seeding particles. To capture the images we used a 7.4µm x 7.4µm (px ) CCD (charge coupled device) camera with a 2 spatial resolution of 2,048px x 2,048px and acquiring pairs of images at 4Hz. The total field employed was of 140 mm x 140 mm, corresponding to a magnification of 0.1, and the employed interrogation area was of 16px x 16px, corresponding to 1.09mm x 1.09mm. The computations were made with 50% of overlap, corresponding to 256 interrogation areas.
Fig. 2 Bedform of triangular profile employed as a model ripple (in the figure, the flow is downward). The test section was divided in four parts, for each part were acquired 2,000 pairs of images for both flow rates, from which the fields of instantaneous velocities were computed in fixed Cartesian grids by the PIV controller software. Matlab scripts were written to post-process these fields. Fig. 3 presents an example of PIV image (for one of the four parts) for the experiments with the ripple. Fig. 3 Image of a PIV experiment (in the figure, the flow is from right to left). 3. Results 3.1. Unperturbed flow The first measurements were made with the unperturbed flow, corresponding then to a turbulent fully-developed channel flow. For each test, the time-averaged fields and the fluctuation fields (second-order moments) were calculated. As the flow was fully-developed, the space-averaged profile was calculated in the longitudinal direction from the time-averaged fields.
Fig. 4 Vertical coordinate y versus space-averaged longitudinal mean velocity u. Fig. 4 shows the space and time averaged longitudinal component of the mean velocity, the blue symbols corresponding to U=0.32m/s and Re= 2.75x10, and the red symbols corresponding to 4 U=0.40m/s and Re=3.5x10. Note that the longitudinal component of the mean velocity is slightly 4 asymmetric to the vertical coordinate axis. This asymmetry occurs because the experiment s channel is asymmetric. The bottom wall of the channel was covered with PVC plates. Fig. 5 Velocity profiles. The employed symbols are listed in Table 1. For a fully-developed turbulent flow in a two-dimensional channel, the longitudinal component of the mean velocity follows the law of the wall. The shear velocity u,. for each Reynolds number was determined by fitting the experimental data in the logarithmic region (70 < y + < 200).
The corresponding values of u,., B. and the friction factor f., as well as the symbols employed in Fig. 5, are presented in Table 1. The red symbols correspond the mean velocity on the top wall, the blue symbols to the mean velocities on the bottom wall, and the lines correspond to the law of the wall. Table 1 Computed shear velocity u,0, constant B 0 and friction factor f 0 for each water flow rate Q. Q m 7 h U m s Re Symbol B. u,. m s 8 0.32 2.75x10 4 5.46 0.0193 0.0270 8 0.32 2.75x10 4 5.44 0.0185 0.0250 10 0.40 3.5x10 4 * 5.23 0.0236 0.0252 10 0.40 3.5x10 4 5.17 0.0233 0.0245 f. Fig. 6 presents the Reynolds stress profiles in dimensionless form: y/h '(( versus u @ v @ / u C,.. Four different flow conditions are presented, the blue continuous line corresponding on the top wall Re=2.75x10 4, the blue symbols corresponding on the bottom wall Re=2.75x10 4, the red continuous line corresponding on the top wall Re=3.5x10 4 and the red symbols corresponding on the bottom wall Re=3.5x10 4. In summary, the law of the wall and the Blasius correlation are valid for the unperturbed flow section. Fig. 6 Profiles of the xy component of the Reynolds stress in dimensionless form: y/h eff versus u @ v @ / u 2,0.
3.2. Perturbed flow The measurements over the model ripple were made for two water flow rates Q=8m /h and 3 10m /h, that correspond to U=0.32m/s and 0.40m/s and to Re=2.75x10 and Re=3.5x10. In the 3 4 4 following, the perturbed flow is analyzed and compared with the unperturbed flow. Fig. 7 shows some mean velocity profiles (time averaged) V(y) over the triangular ripple. A total of 608 profiles were obtained with spatial resolution of the employed PIV device; however, Fig. 7 presents only 17 of them. The main characteristics of the perturbed flow can be obtained from Fig. 7, which shows that the water stream is deflected by the triangular ripple. Close to the ripple, the vertical component of the mean flow v is no longer negligible. Upstream of the crest, v is directed upward, while downstream of the crest, the water flows detaches and a recirculation region is formed: v is directed upward just downstream of the crest and downward some distance farther. Far from the triangular ripple, the values of v are again negligible. However, although v 0, the longitudinal component u is accelerated in this region, as expected form the mass conservation. Fig. 7 Some profiles of the perturbed mean velocities V over the triangular ripple. The flow is from left to right and Re=2.75x10 4. To analyze a perturbed boundary layer, different velocity profiles must have as reference the solid surface, the vertical position where V=0. In the upper region far from the ripple, the channel walls are suitable references and the vertical coordinate y can be employed. In the region close to the ripple, the ripple surface is the reference and therefore the displaced vertical coordinate:
y K = y h (1) is the better coordinate. Fig. 8 presents the longitudinal component u of some mean velocity profiles for different longitudinal positions. The employed symbols in the figure are indicated in the legends. Fig. 8 a) presents y versus u, being suitable for the analysis in the core flow and top wall regions, called upper region. Fig. 8 b) presents y K versus u, being suitable for the analysis in the lower region. Downstream of the crest, the flow detaches and u has negative values. a) b) Fig. 8 (a) Vertical coordinate y versus longitudinal mean velocity u. (b) Displaced vertical coordinate yd versus longitudinal mean velocity u. Re=2.75x10 4 Fig. 9 presents the displaced vertical coordinate y K versus the vertical mean velocity v. The values of v are one order of magnitude smaller than that of u as expected from the dimensional analysis of the perturbation. Fig. 9 Displaced vertical coordinate yd versus vertical mean velocity v. Re=2:75x10 4
Fig. 10 Some profiles of the Reynolds stress xy component in dimensionless y M d versus u @ v @ / u 2,0, from upstream of the ripple crest. Re=2.75x10 4 Fig. 10 presents some profiles of the xy component of the Reynolds stresses in dimensionless form, M y K versus u @ v @ / u C,., upstream of the ripple crest. To decrease the noise, the obtained u @ v @ profiles were averaged by a sliding window process over the closest nine points. Fig. 10 shows that the Reynolds stress u @ v @ is perturbed in the (50 < y K M < 250) region that corresponds to the overlap sublayer of the unperturbed boundary layer. If the flow is in local equilibrium in the (y K M <250) region, the shear stress on the surface shall scale with u @ v @ and therefore the longitudinal evolution of u @ v @ is of importance. Fig. 10 also shows that, longitudinally, the perturbation of u @ v @ decreases toward the crest. Fig. 11 Maximum normalized Reynolds u @ v @ max/ u,0 2 as a function of the longitudinal position x. The continuous and dashed lines correspond to Re=2.75x10 4 and Re=3.5x10 4, respectively.
Fig. 11 presents the longitudinal evolution of the maximum of the xy Reynolds stress u @ v @ QRS (for each vertical profile) normalized by u C,., for the bottom wall region. Fig. 11 shows (for each case) that u @ v @ QRS increases approximately 30% where the triangular ripple starts at x 0.09m. For Re=2.75x10, u @ v @ 4 QRS decreases in the (0.09m x 0m) region until u @ v @ QRS 0.65u C,.. For Re=3.5x10 4, u @ v @ QRS has a different behavior in the (0.09m x 0m) region, but this cannot be asserted because of the relatively high noise in the data. However, we obtain that u @ v @ QRS C u,. in the region analyzed. This indicates that, for the water flow over a triangular ripple (for each case), the region corresponding to the overlap sublayer of the unperturbed flow is in local equilibrium with the lower sublayers. a) b) Fig. 12 (a) Production of turbulence Re=2.75x10 4. (b) Production of turbulence Re=3.5x10 4. Fig. 12 shows the turbulence production upstream and downstream of the ripple crest Re=2.75x10 4 and Re=3.5x10 4. The maximum value of turbulence production is obtained where the recirculation region starts. That term is calculated by P = u @ v @ du dx. The production term provides the only means by which energy can be passed by the mean flow to the fluctuations. That effect is caused by the vorticity formed in the recirculation region. Fig. 13 a) shows the turbulence production upstream of the ripple crest in adimensional displaced vertical coordinate y K M and Re=2.75x10 4. Fig. 13 b) presents the longitudinal evolution of maximum value of turbulence production and y d + Pmax where y K M ZQRS = y KZQRS u υ. We obtained y K M ZQRS < 60. This region corresponds to the buffer and viscous layers of the unperturbed boundary layer, where the viscous shear stress is very important (Schlichting, 2000). In addition, we obtain P QRS 0.04 m C s 7, but this cannot be asserted because of the relatively noise in the data.
a) b) Fig. 13 (a) Production of turbulence upstream of the ripple crest in adimensional displaced vertical coordinate y d M. (b) Longitudinal evolution of Production of turbulence and y d MPmax. Re=2.75x104. 4. Conclusions This study investigated the perturbation of a turbulent channel flow by a two-dimensional triangular ripple in the hydraulic smooth regime. The present experiments were performed in moderate Reynolds numbers (10,000 < Re < 50,000). The fluid flow was measured by PIV and the obtained mean and turbulent fields were compared with unperturbed flow fields. The xy component of the turbulent stress was computed and u @ v @ profiles were found. The main features of the perturbed turbulent channel flow by a low ripple were confirmed by the experimental results: a recirculation bubble is formed downstream of the ripple crest, and the maximum of the production turbulence was located in the recirculation bubble region. Upstream of the ripple crest, the maxima of the turbulent stresses were of the same order of magnitude of the square of the unperturbed shear velocity u C,.. In addition, the maximum value of u @ v @ was found in the layer corresponding to the overlap layer of the unperturbed flow (y K M 100). Based on these characteristics, it seems that the flow is in local equilibrium in this region and the asymptotic expressions for the boundary layer perturbation based on local equilibrium conditions can be used. The maxima of the production of turbulence upstream of the ripple occurred in the y K M < 60 region that corresponds to the buffer and viscous layers of the unperturbed boundary layer. Acknowledgments Fernando David Cúñez Benalcázar is grateful to SENESCYT (Programa de Becas Convocatoria Abierta 2014 Segunda Fase), Erick de Moraes Franklin is grateful to FAPESP (grant no. 2012/19562-6) and to CNPq (grant no. 471391/2013-1) for the provided financial support.
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