On the Nature of Coherent Turbulent Structures in Channel Bends: Burst-Sweep Orientations in Three-Dimensional Flow Fields

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1 On the Nature of Coherent Turbulent Structures in Channel Bends: Burst-Sweep Orientations in Three-Dimensional Flow Fields M. Tilston 1, C. Rennie 2, R.W.C. Arnott 1 and G. Post 3 1 Department of Earth Sciences, Uniersity of Ottawa, Ottawa ON, 2 Department of Ciil Engineering, Uniersity of Ottawa, Ottawa ON, 3 Northwest Hydraulic Consultants Ltd. Abstract It is widely accepted that natural flows are characterized by coherent turbulent structures, known as bursts and sweeps, which transfer fluid momentum across local elocity gradients. While these hae been the subject of extensie laboratory and field inestigations, no studies hae attempted to address their characteristics in channel bends. In part, this is related to the existing analytical frameworks, which are inherently two dimensional. The objecties of this study are to a) deelop a fully three-dimensional scheme for analysing coherent turbulent structures; b) inestigate the orientation of bursts and sweeps in an experimental channel bend. This inoles soling the Reynolds tensor and performing octant analysis in lieu of conentional twodimensional approaches. Results indicate that bursts and sweeps lack a preferred orientation upstream and downstream of the bend. Howeer, flow in the channel bend strongly displays inner-bank oriented bursts and outer-bank oriented sweeps. These findings suggest that burstsweep orientations are controlled by ertical gradient of the cross-stream elocity component and turbulence generated fluctuations in the cross-stream force balance. Introduction All rier systems are characterized by three-dimensional, turbulent flow fields and perhaps no other fluial enironment better illustrates the fundamental role of secondary currents on channel morphology and eolution than meander loops. Their unique forms are a direct result of helical flow fields, and continue to be the subject of intensie study attempting to relate flow structure, sediment transport (Bridge and Jaris, 1982; Dietrich and Whiting, 1989) and channel eolution (Rhoads and Welford, 1991; Darby and Delbono, 2002). Howeer, there is one flow process that has been largely oerlooked in the context of channel bends: coherent turbulent structures. The most basic feature of turbulent flows are rotating eddies, which sere to transfer momentum through the water column. This mixing is accomplished as the leading edge of an eddy brings high momentum fluid from the upper part of the flow towards the bed, whereas the opposite occurs along the tailing half of the eddy (Schidchenko and Pender, 2001). Taken together, these two types of fluid motions are known as sweeps and bursts, respectiely, and represent the physical basis of the Reynolds stress. Historically, these types of motions are studies by using quadrant analysis (Willmarth and Lu, 1971) where bursts correspond to quadrant 2 eents and sweeps correspond with quadrant 4 eents. By definition, this is a two-dimensional analytical framework that is unable to fully capture the dynamics of turbulent fluid motions in highly three-dimensional flow fields. The objecties of this study are; a) deelop a fully three-dimensional scheme for analysing coherent turbulent structures; and b) inestigate the orientation of bursts and sweeps in an experimental channel bend. This inoles soling the Reynolds tensor and using an octant framework in lieu of the principal Reynolds stress and a quadrant framework. 33rd IAHR Congress: Water Engineering for a Sustainable Enironment Copyright c 2009 by International Association of Hydraulic Engineering & Research (IAHR) ISBN:

2 Methodology The measurements used for this research were collected in a model rier bend in the Ciil Engineering Hydraulics Laboratory at the Uniersity of Ottawa. The flume has a centerline length of 18.2 m, consisting of an initial m straight section leading into a 135 bend followed by a second 2.44 m straight section. The channel is 1 meter wide with a radius of curature of 1.5 m; this represents a relatiely low radius of curature to width ratio, thereby allowing for the rapid deelopment of strong secondary currents. Bed sediments consisted of sub-angular particles (d 50 =1.1 mm). The slope was set to 0.044%, the critical alue for entrainment in the straight section required for maximum clear-water scour conditions. Six sections of elocity measurements (2 minute time series) collected using Nortek Vectrino ADVs (sampling rate of 200 Hz) are utilized the present analysis; 4 of these sections were located in the bend proper at 30 interals (Post, 2007). The most standard method for quantifying coherent structures in turbulent elocity signals is to perform quadrant analysis (Nezu and Nakagawa, 1993). Here, a elocity component at time t ( u i (t) ) can be decomposed into a time-aeraged term and an instantaneous fluctuation about the mean using the equation: u t) = u + u ' i ( i i Where ui is the time aeraged term and ui ' is the instantaneous fluctuation. Using this notation, the magnitude of the Reynolds shear tensor (τ ) can be soled by the equation: τ = ij 1 2 ( τ ij ') = 1 ( ρui ' u j ') 2 2 ij 2 Where τ ij ' represents the standard instantaneous Reynolds stress component. Subsequently, an octant diagram can be used to identify burst-sweep orientation. Conceptually, an octant diagram can be thought of as two quadrant diagrams placed side by side, where one quadrant diagram represents all eents with a positie cross-stream elocity fluctuation ( u ) and the other for cases where u ' is negatie (Figure 1). In this framework, an eent with a positie ' Figure 1: Octant framework for identifying burst-sweep orientation. 852

3 u ' (inner bank orientation) corresponds with octants one through four, whereas octants 5 through 8 indicate a negatie alue of u '(outer bank orientation). Results Figure 2 presents the elocity and three components of the Reynolds stress through the study bend. Here, the elocity and Reynolds tensor data are normalized by the bend maxima to illustrate changes through the study bend, whereas the three planes of Reynolds stress are normalized by the sectional maxima to demonstrate tensor composition. As flow traels around the bend, the high elocity core shifts towards the outer bank, decelerates and deelops strong secondary currents. This is in agreement with preious field studies (Bridge and Jaris, 1982). Howeer, the high elocity core moes from the water surface to the lower third of the flow depth between the bend apex and exit and the area aboe the core is characterized by a weakly negatie ertical elocity gradient. Currently the reasons for this are not completely understood, but might be a consequence of the tightness of the bend ( r c / w = 1.5). Interestingly, the Reynolds stress increases through the bend, reaching maximum leels at the 120 section. Yet, the more important result from this study is the tensor composition. Upstream of the bend, the zone of maximum principal Reynolds stress ( u u ' uw' ) is found at the bed across the entire width of the channel and decreases towards the water surface. This is in agreement with preious studies (Song and Chiew, 2001). Although this trend persists through the bend, it is much weaker than the section upstream of the bend entrance and the upper portion of the flow is actually characterized by negatie principal Reynolds stresses downstream of the Figure 2: Velocity and shear stress distributions through the channel bend. 853

4 60 transect. The negatie sign indicates directionality of momentum transfer and is associated with the lowering of the high elocity core. Interestingly, momentum exchange appears to be dominated by the uu ' u ' and u ' uw' planes throughout the bend (Jamieson et al, in reision). Due to the reduction in the ertical gradient of, the increased influence of these planes was anticipated; howeer, their dominance was not expected because preious bend flume studies using similar planform geometries hae demonstrated that the dominant Reynolds stress is associated with the downstream-ertical plane (Blanckaert and Graf, 2001). This is likely related to the weakening of the ertical elocity ( u u ) gradient aboe the near bed region, which dries fluid momentum exchange in this plane. Moreoer, it may influence burst-sweep orientations. Historically, burst-sweep analysis has focused on two key parameters: intensity and frequency (Willmarth and Lu, 1972). This type of analysis has been applied in the present study and extended to include the mean eent impulse for examining burst-sweep orientations, which is the product of these two ariables. In addition to the untreated data, a hole size filter (H=2) based on the cross-sectionally aeraged tensor magnitude was applied to remoe the effects of weaker turbulent motions. Figure 3 and Figure 4 present the patterns of burst and sweep orientations through the study bend. Here, the orientation ratio is defined as the relatie effect of outward oriented bursts (octant 6) and sweeps (octant 8) with respect to the cumulatie effect of all burst and sweep eents. Thus, alues of zero indicate that all bursts and sweeps display inner bank orientations within the time series, whereas alues of 1 indicate an outer bank orientation. In the regions upstream and downstream of the bend, eent intensity and frequency ratios for Figure 3: Burst orientation ratios. Red indicates a strong preference for inner bank orientation and green indicates a strong preference for outer bank orientation. 854

5 Figure 4: Sweep orientation ratios. Red indicates a strong preference for inner bank orientation and green indicates a strong preference for outer bank orientation. bursts and sweeps range between 0.4 and 0.6. This suggests that both the ratio of inner-bank and outer-bank oriented turbulent motions are roughly equal in terms of strength and regularity. Howeer, the situation is ery different within the bend. Here, burst impulse ratios typically range between 0.2 and 0.4, demonstrating a preference for inner-bank orientation (Figure 3). This trend is further highlighted after applying the filter, where impulse ratios ary between 0 and 0.4 and appears to be a product of eleations in both frequency and intensity. Conersely, sweep impulse ratios generally range between 0.4 and 0.8, indicating a preference for outer-bank orientation (Figure 4). Again, impulse ratio patterns are enhanced upon filter application. Yet the trends seen in sweep orientation differ from bursts in two important ways. First, burst orientation appears to be largely controlled by eent frequency rather than intensity. Secondly, impulse patterns are not as consistent as those of the bursts, and begin to show a reersal in orientation through the upper portions of flow in the 90 and 120 sections. Yet it is worth noting that these zones of reersed orientation coincide with those where impulse intensity is at a minimum and thus represent a small fraction of bulk momentum exchange within the section. This is probably an artifact of the weak ertical elocity gradient ( u u ), hence turbulent fluid momentum exchange is being controlled locally by the uu ' u ' and u ' uw' stress planes. Discussion The results demonstrate clearly that turbulent structures display a preferred orientation in channel bends, with bursts being oriented towards the inner bank and sweeps towards the outer bank. Gien the well established links of the burst-sweep cycle with turbulent fluid momentum 855

6 exchange, sediment transport and bedform deelopment (e.g. Best, 1993), the presence of a preferred orientation represents an important finding. In order to understand the mechanisms controlling burst-sweep orientations, it is necessary to take a closer look at the stress distributions within the uu ' u ' and u ' uw' planes. It is well accepted that the nature of turbulent fluid momentum exchange, hence the sign associated with the time-aeraged principal Reynolds stress, is directly related to that of local elocity gradients (Nezu and Nakagawa, 1993). Theoretically, this notion should also apply to the secondary stress planes, which is confirmed along the uu ' u ' plane upstream of the bend (Figure 2). Here, the region between the channel centerline and the inner bank are characterized by negatie Reynolds stresses, whereas the region between the centerline and the outer-bank contain positie Reynolds stresses. This means that the uu ' u ' plane is characterized by high momentum fluid traeling towards the channel boundaries and low momentum fluid being brought towards the high elocity core (Figure 5a). Howeer, the normal patterns of fluid momentum exchange clearly break down upon entering the bend. The 30 and 60 sections indicate that slow moing fluid moes towards the inner-bank and fast moing fluid is oriented towards the outer-bank, and therefore are independent of the cross-stream distribution of u. Similarly, one would expect the u ' uw' plane to be characterized by the transfer of high momentum fluid from the near-surface and near bed regions towards the middle of the water column along the channel centerline; the reerse would be true for low momentum fluid. This process and anticipated stress distribution is presented in Figure 5b, yet results show that there is no reersal in sign associated with the Reynolds stress. Notwithstanding, both the obsered uu ' u ' and u ' uw' planes are similar to their theoretical distributions. Thus it would appear that there is an underlying process acting in channel bends that influences not only burst sweep orientations, but also turbulent momentum exchange along the uu ' u ' and u ' uw' stress planes in channel bends. We propose that burst-sweep orientations in channel bends can be attributed to u Figure 5: Theoretical elocity and stress distributions for the a) uu ' u ' and b) u ' uw' planes. 856

7 Figure 6: Conceptual models illustrating the roles of a) secondary currents and b) planform geometry on burst-sweep orientations in channel bends. two factors: 1) the presence of strong secondary currents; and 2) channel planform geometry. Strong secondary current can affect burst-sweep orientations due to a strong ertical gradient in u (Figure 6a). As a burst brings slow moing up through the water column, mean alues of u become progressiely more oriented towards the outer bank with respect to those at its point of origin. Therefore a burst would appear to be traelling towards the inner bank with respect to the mean alue of u at the point of measurement. The reerse would be true for sweeps. Alternatiely, burst-sweep orientation could also be modifying the cross-stream force balance associated with planform geometry (Figure 6b). While the pressure gradient force would remain relatiely constant, centrifugal force would not since it is entirely dependent on elocity. Gien that bursts are associated with slow moing fluid, they would cause a reduction in centrifugal force and result in inner-bank orientation. Conersely, fast moing sweeps would increase the centrifugal component of the force balance and result in outer-bank orientation. Each of these processes would lead to inner-bank bursting and outer-bank sweeping motions throughout the cross-section. Determining the relatie contribution of these two mechanisms has proen difficult. Results for the 120 section show a strong correlation between ertical gradients of u and a preferred orientation. Howeer, this is also the section where uu is at a minimum, and therefore the section with minimum centrifugal force. Moreoer, the upper portions of flow in the 30 and 60 sections illustrate that sweeps show an outer-bank orientation despite the weak ertical gradient of u. Therefore, it would appear that the dominant mechanism controlling burst-sweep orientation is dependent on the intensity of uu ' with respect to the ertical gradient of u. 857

8 Conclusion In this study we hae deeloped a noel technique for analyzing coherent turbulent structures in three-dimensional flow fields. Results indicate that bursting motions are preferentially oriented towards the inner-bank, whereas sweeping motions are preferentially oriented towards the outerbank. We beliee this phenomenon can be attributed to temporal imbalances in the cross-stream force balance associated with the passage of turbulent structures and changes in mean crossstream elocity through the water column associated with secondary currents. Future work should be conducted in different enironments such as rier confluences to properly asses the influence of secondary currents ersus planform geometry. References Best, J.L. (1993). On the interactions between turbulent flow structure, sediment transport and bedform deelopment: some considerations from recent experimental research. Turbulence: Perspecties on flow and sediment transport. Clifford, N., French, J.R. and Hardisty, J., eds. Wiley and Sons Ltd. New York, NY. Bridge, J.S. And Jaris, J. (1982). The dynamics of a rier bend: a study of flow and sedimentary processes. Sedimentology. 29, Darby, S.E. and Delbono, I. (2002). A model of equilibrium bed topography for meander bends with erodible banks. Earth Surface Processes and Landforms. 27, Dietrich, W.E., and Whiting, P.J. (1989). Boundary shear stress and sediment transport in rier meanders of sand and grael. Rier Meandering, Water Resour. Monogr. Ser. 12. Ikeda, S. And Parker, G. Eds. American Geophysical Union. Washington D.C. Jamieson, E., Post, G., Rennie, C.D. Spatial ariability of three dimensional Reynolds stresses in a deeloping channel bend, Earth Surface Processes and Landforms, in reision. Nezu, I., and Nakagawa, H. (1993). Turbulence in Open Channel Flows. Brookfield, VT.: A.A. Balkema. Post, G. (2007). The Measurement of Reynolds Stresses in a Model Rier Bend using Acoustic Doppler Velocimeters. M.A.Sc.Thesis, Uni. of Ottawa, Ottawa, On. Rhoads, B.L. and Welford, M.R. (1991). Initiation of rier meandering. Progress in Physical Geography. 15(2), Schidchenko, A. B. and Pender, G. (2001). Macroturbulent structure of open-channel flow oer grael beds. Water Resour. Res. 37, Willmarth, W.W. and Lu, S.S. (1972). Structure of the Reynolds stress near the wall. J. Fluid Mech. 55(1)