Reynolds stress | EPFL Graph Search

In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum. Definition The velocity field of a flow can be split into a mean part and a fluctuating part using Reynolds decomposition. We write :u_i = \overline{u_i} + u_i',, with \mathbf{u}(\mathbf{x},t) being the flow velocity vector having components u_i in the x_i coordinate direction (with x_i denoting the components of the coordinate vector \mathbf{x}). The mean velocities \overline{u_i} are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further u'_i denotes the fluctuating (turbulence) part of the velocity. We consider a homogeneous fluid, whose density ρ is taken to be a constant. For such

Experimental study of suspension flow in open channels

The study of the sediment transport in open-channel as well as in river flow is of great importance for fluvial hydraulics. While the transport of sedimentary particles at the bed, as the bed-load, has been the subject of much research, less attention has been paid to the transport of sediments in suspension. This thesis is a contribution to our knowledge of the transport of sediments in suspension. The most important theory on the vertical distribution of the mean sediment concentration in suspension flows is the diffusion-convection theory, given by the Rouse equation. This equation is rather simple and contains few parameters. About one of these parameters, the β-value, there is little known, despite its importance and physical meaning. One of the main aims of this thesis is to experimentally measure the β-values for different suspension flow configurations. The major objective was to investigate suspension flows over moveable beds without bed forms and this in capacity condition. Since flows are at times not in capacity, i.e. the flows are not saturated with sediments, an additional study (see Appendix C) was also performed. Furthermore, two special runs were done (see Appendix D), where the bed was covered with artificial bed forms. In order to evaluate the β-values, the vertical profiles of the mean velocity and its fluctuations as well as the mean concentration and its fluctuations had to be measured. This could only be done by using the new ultrasonic instrument (Acoustic Particle Flux Profiler, APFP), developed and assembled at the Laboratoire de Recherches Hydrauliques (LRH). Suspension flows were investigated focusing on the possible modification of the clear-water turbulence by suspended sediments. The concentration was carefully measured and interpreted by the Rouse equation. In particular, the ratio of the sediment, εs(y) , and momentum, εm(y) , diffusion coefficients, being the β(y) -value, was for the first time, directly measured. The strongest effect of particles on the flow was noticed on the vertical component of the turbulence intensity, , which was considerably suppressed, when compared to clear-water flow. The longitudinal velocity, u(y) , its turbulence intensity component, , and the Reynolds stress, – ρu'v' , were only slightly affected. By using the APFP instrument, the instantaneous sediment concentrations, cs, were, for the first time, directly measured. The calibration of the APFP instrument was achieved by measuring the mean concentration profiles, cs, with the suction method. The largest measured fluctuating concentrations, , and sediment flux, profiles were observed close to the bed, where the mean concentration is also very large. The sediment, εs, and the momentum, εm, diffusion coefficients, which represent the "ability" of sediment and fluid particles to be diffused in the flow by turbulence, were computed from the APFP measurements. For suspension flows over plane-bed, the sediment diffusion-coefficient profiles are always smaller than the momentum ones, εs < εm. This indicates that sediment particles undergo less diffusion than fluid particles; consequently the β-values are less than unity, β < 1. Thus, the usual assumption of εs = εm, that leads to β = 1, is not justified for the present measurements. The Rouse equation gives a better agreement to the concentration distributions measured with the suction method, by using the experimental obtained β-values, , rather than by using β = 1. For practical use, the experimental values and the ones obtained by a best-fit (least-squares method) of the Rouse equation to some concentration distributions taken from the literature are summarized in a plot. It seems that the β-values increase with either scaling parameters, vss/u*, and (vss/u*) · (Cs/csa). The effects of suspended particles on the clear-water turbulence were also investigated in non-capacity suspension flows having increasing concentrations. The measurements show that, the suppression (damping) of the vertical component of the turbulence intensity – caused by suspended particles – increases with the depth-averaged concentration. The tendency of the - values to decrease approaching the capacity condition, i.e. increasing the concentration, was also found. Suspension flows over bed forms were also investigated; in this study special attention was paid to the evolution of the flow structure behind the bed-form crest as well as the effect of bed forms on the β-values. The bed-form crest seems to generate a high-turbulence region with consequent peaks in both the longitudinal and vertical turbulence-intensity profiles as well as in the Reynolds-stress profiles. This high-turbulence region enhances the sediment diffusion coefficient but suppresses the momentum one. As a consequence, we found that for suspension flows over beds with bed forms, εs > εm, leading to β > 1. The most important result of this thesis is a recommendation that the Rouse equation with an improved β-value – itself to be estimated from the experimental plots obtained in this study – can be used to establish the dimensionless concentration profile of suspension flows. For beds without bed forms the β-values are β < l, while for beds with bed forms the β-values are β > 1. The correlation between the velocity fluctuations, associated with coherent structures (burst cycle), and the concentration was also investigated. From this study it becomes evident that the ejection event, being the most important phase of a burst cycle, is responsible of the erosion of sediment on the bed, behaving as an "injector" of sediments into the main flow.

EPFL1998

An experimental study of coherent structures, secondary currents and surface boils and their interrelation in open-channel flow

Ismail Albayrak

River and open-channel flows are free surface boundary layer flows with complex 3D, large-scale, turbulent structures. The study of 2D and 3D large-scale turbulent flow structures is a great challenge for physicists, mathematicians and engineers from such different domains as civil, environmental and mechanic engineering. Different processes can generate 3D, large-scale, turbulent structures which occur at the same time. On the one hand, large scale vortical structures such as secondary currents of Prandtl's second kind play an important role in the understanding of 3D turbulent structures in straight channels and rivers. Secondary currents affect bottom shear stress and longitudinal mean velocity, and contribute to sediment transport and air-water gas exchange by creating upwelling and downwelling motion in the water column. At the free surface, such upwelling and downwelling motion is an important mechanism for the air-water gas exchange and is considered to be responsible for surface boils. On the other hand, experimental work in turbulent boundary layers revealed the existence of bursting resulting in hairpin shaped structures which are responsible for the link between the inner and the outer layer. The interaction between these two layers in turbulent boundary layers is considered in terms of the dynamics of momentum, energy, and Reynolds shear stress transport. In order to advance in the understanding of this fundamental problem in turbulent open-channel flow, recently developed measurement and observation techniques are used in this Ph.D study. A non-intrusive Acoustic Doppler Velocity Profiler (ADVP), Surface Large Scale Particle Image Velocimetry (LSPIV) and a hot-film probe were combined in the investigation of coherent structures, secondary currents, surface boils and their interaction in turbulent rough-bed open-channel flow. The ADVP permits to measure 3D quasi-instantaneous velocity profiles in the entire water depth and to investigate the mean field and the fluctuating field of all three velocity components. The LSPIV system, developed at the LHE, allows visualizing the water surface and obtaining the surface velocity information in relation to instantaneous surface vortical structures. Bottom shear stress was measured with a sensor based on the hot film principle. The instruments provided the mean and instantaneous velocity field in the entire water depth and at the free surface. Six sets of experiments were carried out in turbulent rough bed open-channel for three different width-to-depth ratios (12.25, 15 and 20) at high, moderate and low Reynolds numbers. The results of the ADVP measurements show mean longitudinal velocity patterns undulating across the channel which indicate patterns of secondary currents in the mean flow structure. Upwelling regions can be identified by lower relative mean longitudinal velocities close to the free surface, and downwelling regions can be identified by higher relative mean longitudinal velocities. It is observed that the existence of secondary currents affects the distribution of bed shear stress and Reynolds stress across the channel. Bed shear stress show a cross-channel undulation pattern with bed shear stress in downwelling areas being higher than in upwelling areas. The Reynolds shear stress distribution in the water column has revealed the same undulating pattern. The number of secondary flow cells is determined by the aspect ratio and relative roughness. It is found that the bottom roughness elements of the channel bed make these longitudinal cells stable. The Reynolds number does not affect the spanwise position of the upwelling and downwelling regions of the secondary cells, but it does affect and slightly increase the normalized Reynolds shear stress. Secondary currents with cells whose dimensions are equal to the flow depth are the most stable and dominant pattern. Changes in the vorticity pattern causes changes in turbulence characteristics in upwelling and downwelling regions. Our study and existing investigations demonstrated that one of the most probable mechanisms for the initiation of multi cellular secondary currents is the mutual interaction between the rough bed and the pre-existing secondary currents near the side wall. The occurrence of small and large scale coherent structures, such as hairpin packets, and their relation to secondary currents are investigated through a quantitative analysis of instantaneous flow fields over the entire turbulent boundary layer across the channel. Uniform momentum zones are clearly detected in the instantaneous velocity fields in the longitudinal direction. In the logarithmic layer, the coherent vortex packets originating from the wall layer frequently occur within larger moving zones of uniform momentum, and extend up to the middle of the boundary layer. Good results in terms of dimension and position of large coherent structures relating to zones of uniform streamwise momentum support the concept of a dynamic link of hairpin packets and zonal organization in the outer layer. Secondary currents are large-scale streamwise vortical structures that affect the organization of coherent structures in the outer layer. More hairpin vortex packets could be carried by upwelling, with positive vertical velocity increasing the height of Zone 2. In the downwelling region, the height of Zone 2 and the growth angle of the hairpin packets decrease compared to the upwelling region, because of the negative vertical mean velocity of the secondary currents. This may prevent hairpin vortex packets from reaching the free surface due to the higher gradient of the longitudinal velocity. A quadrant analysis in the upwelling and downwelling regions revealed that in the wall region of downwelling areas, sweep events are dominant, and in the region close to the free surface, ejection events dominate over sweep events. The dominance of ejection events at the free surface explains the occurrence of a large number of surface boils observed in the upwelling regions. The measurements have shown that secondary currents and coherent structures are correlated, thus producing 3D flow structures. The results from LSPIV show a mean multi-cellular pattern of faster and slower primary longitudinal surface velocities. Streaks of faster longitudinal velocity are found in downwelling areas. Upwelling areas are identified with lower velocities. In addition, we observed mean transversal surface currents between upwelling and downwelling zones. We have shown that near the surface, ejections which are part of the large scale burst cycle are more common in upwelling zones between secondary current cells. Measurements reveal that these vortex boils mainly occur in upwelling areas with high vorticity, whereas downwelling areas show lower vorticity. Up- and downwelling zones, as well as surface boils are observed at all Reynolds numbers and aspect ratios. Therefore, they can be considered important processes in river dynamics and affect transport between the surface and the pelagic zone. Based on the combined results from LSPIV, ADVP and hot-film data, this study experimentally demonstrated that secondary currents, surface boils and coherent structures are correlated and produce 3D flow structures. The effect of secondary currents on tracer distribution in open-channel and river flow is to disperse and mix tracers in three dimensions more rapidly than would be the case if turbulent diffusion were acting alone. This has important consequences for pollutant spreading. Together with surface boil vortices, these currents contribute to surface renewal and gas transfer. In downwelling zones, the water masses moved along the surface by the transversal currents are transported downwards faster than by turbulent mixing. Again, the dispersion and mixing due to secondary currents discussed above will then provide for rapid 3D distribution in the entire water column. This thesis is a contribution to understanding of the transport and mixing dynamics in open-channel flow with an emphasis on the effects of coherent structures, secondary currents and surface boils, as well as the interaction between them. The information which was obtained advances the understanding of fine and large scale dynamics in open-channel flow. At the same time it contributes to the improvement of algorithms in numerical predictive water quality models, which in turn improve effective water resources management.

EPFL2008

Secondary Currents and Corresponding Surface Velocity Patterns in a Turbulent Open-Channel Flow over a Rough Bed

Ismail Albayrak, Ulrich Lemmin

River and open-channel flows present complex three-dimensional (3D) structures in the water column and on the surface as a result of the existence of secondary currents driven by either centrifugal force or turbulence. This paper experimentally investigates secondary currents in a straight, turbulent, rough-bed open-channel flow; the effect of different channel width to flow depth ratios (12.25, 15, and 20) on secondary currents; and surface boils associated with secondary currents at moderate Reynolds numbers. Nearly instantaneous profiles of three components of flow velocity and turbulence characteristics in the water column were measured by using an acoustic Doppler velocity profiler (ADVP). Simultaneously, large-scale particle image velocimetry was used to measure the water surface velocities and turbulence structures. Mean velocity patterns in the water undulate across the channel, indicating the presence of secondary currents in the long-term average flow structure. Secondary currents affect the distribution of bed shear stress, Reynolds stress, and turbulence intensities across the channel. The aspect ratio determines the number of secondary cells in the water column. A mean multiband undulating surface velocity pattern in the transverse direction correlates with the secondary cell distribution in the water column below. The instantaneous position of the upwelling and downwelling regions on the surface may deviate from their long-term mean position, indicating a meandering of the surface pattern. Vortex structures are detected from instantaneous surface velocity maps, and vortex boil lines are identified. Boil vortices mainly occur in upwelling areas with high vorticity. The simultaneous detailed velocity measurements in the water column and on the free surface have shown a good agreement between the secondary cell patterns obtained by the two methods. DOI: 10.1061/(ASCE)HY.1943-7900.0000438. (C) 2011 American Society of Civil Engineers.

2011