Quadrant method application to the study of the beginning of sediment motion of sedimentary particles

Transcription

1 V Conferência Nacional de Mecânica dos Fluidos, Termodinâmica e Energia MEFTE 214, etembro 214, Porto, Portugal APMTAC, 214 Quadrant method application to the study of the beginning of sediment motion of sedimentary particles E Carvalho 1, R Aleixo 2 1 Departmento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal 2 National Center for Computational ydroscience Engineering, The University of Mississippi, Mississipi, UA elsac@fe.up.pt, rui.aleixo@ncche.olemiss.edu ABTRACT: In the present paper the quadrant analysis method is used to analyse the turbulence over a spherical particle of a uniform spheres bed. Two components velocity fluctuations, (u, v ), were measured with a Laser Doppler Anemometer and divided by four quadrants according to their signal. For each quadrant an event is associated and it is possible to study the contribution of each quadrant to the total Reynolds stress. Each event is associated with a different physical phenomenon, namely ejections and sweeps that can be related with the beginning of the sediment motion. In order to analyse the influence of different variables such as the particle s diameter, D, the flow rate, Q, the channel slope, i, and the distance to the top of the particle s top, y, different experimental conditions on supercritical flows over rough bed were measured. KEY-WORD: Quadrant analysis; upercritical flows; Ejection; weep. 1 INTRODUCTION The study and project of stable channels is an important topic on fluvial hydraulics. It is necessary that the conditions for the initiation of sediment motion be known and understood. To study the stability of fluvial beds it is necessary to characterize the fluid-bed interaction where the flow turbulence and it characteristics play an important role. The quadrant analysis is one method for turbulence study. This method consists in plotting the velocity fluctuations in a four quadrant Cartesian plot, according to the signal of the two components [1]. It allowing also the evaluation of the contribution of each event to the mean value of the Reynolds stress, u v [2]. Each quadrant is then associated to a turbulent event (Figure 1): 1 st Quadrant: for u > and v >; it is associated to outward interactions; 2 nd Quadrant: for u < and v >; it describes the ejection events; 3 rd Quadrant: for u < and v <; it is related to inward interactions; 4 th Quadrant: for u > and v <; it refers to sweep events. v 2ºQ Ejeção Ejection u < e v > 1ºQ Interação Exterior Outward Interaction u > e v > 3ºQ Interação Interior Inward Interaction u < e v < 4ºQ Varrimento weep u > e v < u Figure 1: Velocity fluctuations Cartesian plot for quadrants analysis and events. MEFTE 214, et 214, Porto, Portugal 189

2 With the quadrant method it is also possible to make a conditional analysis by defining a hyperbolic region, referred as the hole, delimited by a curve u v = constant (as depicted in Figure 3). The dimension of the hole,, represents the threshold [2] and it is defined by: where.5 and.5 u '.5 v '. 5 (1) are the rms of the velocity components u and v, respectively Figure 2: Definition of the hyperbolic region. With this method it is possible to eliminate the smaller contributions inside the hyperbolic region, considering just the larger contributions of the events associated to each quadrant. The contributions for the total Reynolds stress, u v, of the quadrant i, out of the hyperbolic region with dimension, is given by [2, 3]: where I i,, t is the detection function, defined by: The fraction contribution of each event to T 1 lim i ( t) ( t) Ii,, t (, ) dt (2), T T if, Qi and Ii,, t (, ) (3) otherwise, i,, is defined by: i, i, (4) The sum of the fraction contributions of all events in a measuring point, for =, is equal to one. 4 i, 1 i (5) As an example, Figure 3 shows: (a) the representation of the two velocity components fluctuations, allowing for the identification of the pairs of values associated to events of each quadrant and (b) the correspondent conditional analysis ( = 1) for which the values with that respect the condition: are represented. u '.5. 5 (6) 19 MEFTE 214, et 214, Porto, Portugal

3 =1 (b) v (m/s) v (m/s) = (a) u (m/s) - u (m/s) Figure 3: (a) Fluctuations contributions for = ; (b) Condition analysis for = 1. 2 EXPERIMENTAL ETUP The experiments were carried out in the channel of FEUP s ydraulics Laboratory (Figure 4). This channel is 17 m long, has a section of.4.6 m2 and its slope is adjustable. Both sides of the channel have glass windows to allow the optical access to the interior. The flow rate is controlled by means of a valve and it is measured by an electromagnetic flowmeter, both located upstream of the channel. At the channel outlet a sluice gate allows controlling the water level. Figure 4: Experimental setup (FEUP s ydraulic Laboratory channel). A double bottom with 8 m length was placed inside the channel in order to allow the use of different roughness beds. For this study, a uniform spheres bed with triangular arrangement was considered. Two beds made of particles with diameters of 4 mm and 5 mm respectively (Figure 5) were tested. The spheres were glued in an area of.4 1 m2 and test section was located at 6.5 m from the beginning of the double bed, to ensure the development of the turbulent boundary layer, which was confirmed by the measured velocity profiles. Figure 5: Bed spheres triangular arrangement. Velocity profiles measurements were made by means of a two components Laser Doppler Anemometer system and the signal was processed by a DANTEC burst spectrum analyzer. The LDA is an established and well known technique and a complete description can be found in [4, 5], for example. Table 1 presents the main characteristics of the LDA system used in this study. MEFTE 214, et 214, Porto, Portugal 191

4 Table 1: LDA main characteristics. LDA1 LDA2 Wavelength nm 488 nm Control Volume Dimensions Major axis mm mm Minor axis.8189 mm.7767 mm Different flow conditions were considered for the measurement of the two velocity components over a loose particle of the bed. This allowed the analysis of the different parameters influence on the contributions for the Reynolds stress mean value. A total of 72 test conditions were measured for 6 flow rates, Q, two channel slopes, i, two diameters, D, and 3 distances from the top of the sphere, y (see Table 2). For each condition 1 velocity samples were considered. Table 2: Tested conditions. i 1.3 %.8 % y 2.5 mm 4.25 mm 1 mm 2.5 mm 4.25 mm 1 mm D = 4 mm D = 5 mm Q (L/s) 3 EXPERIMENTAL REULT AND CONCLUION The quadrant analysis was applied to supercritical flows for the aforementioned flow conditions. The two velocity components were measured for each condition allowing the application of the quadrant method. Figures 3 presents (a) the fraction contributions of each quadrant to the Reynolds stress, for different flow rates, and (b) the difference between the contributions of the ejections and sweep events Q=1L/s Q=17.5L/s Q=2L/s Q=1L/s.8.6 Q=17.5L/s.4 Q=2L/s.. D = y=2.5mm y=4.25mm y=1mm i=1.3% i=.8%.4.6 Q=1L/s Q=1L/s Q=17.5L/s Q=2L/s Q=17.5L/s.8 Q=2L/s Figure 6: (a) contributions of each quadrant to Reynolds stress; (b) difference between the ejections and sweeps contributions. The analysis of the results allowed to conclude that ejection and sweep events have higher contributions to Reynolds stresses. Ejections and sweeps have similar contributions values. The sum of the contributions of the second and fourth quadrants, positive contribution to Reynolds stress, is around 16% for =. The interaction events (corresponding to the first and third quadrants) have a total contribution 192 MEFTE 214, et 214, Porto, Portugal

5 of 6%. From the study of the contributions for different distances from the top of the spheres it was possible to verify that the sweep events have higher contributions near the wall. With the increase of the distance it is possible to observe that the contributions of the ejections increase. These experiments have shown that the beginning of sediment motion is either triggered by an ejection or sweep event. More experiments have to be made with imaging techniques to capture the turbulent event capable of triggering the sediment motion. ACKOWLEDGEMENT This work was funded by a PhD scholarship ref. FR/BD/19575/24 from Fundação para a Ciência e a Tecnologia, Portugal. REFERENCE [1] W Willmarth, Lu (1972). tructure of the Reynolds stress near the wall. Journal of Fluid Mechanics 55(1): [2] I Nezu, Nakagawa (1993). Turbulence in Open Channel Flows, AA Balkema, Rotterdam. [3] Dey, A Papanicolaou (28). ediment threshold under stream flow: A state-of-the-art review. KCE Journal of Civil Engineering 12(1):45 6. [4] F Durst, A Melling, J Whitelaw (1976). Principles and Practice of Laser Doppler Anemometry, Academic Press, London. [5] C Tropea, A Yarin, J Foss, Eds. (26). pringer andbook of Experimental Fluid Mechanics, pringer-verlag Berlin eidelberg. MEFTE 214, et 214, Porto, Portugal 193

6 This page was intentionally left blank 194 MEFTE 214, et 214, Porto, Portugal