velocity gradients (both laterally and vertically) and ... the vegetation zone, and (c) the vertical distribution .... free surface was set as a frictionless rigid lid and.
Large Eddy Simulation of Fully-Developed Turbulent Flow Through Submerged Vegetation T. Stoesser, C. Liang, W. Rodi, G.H. Jirka Institute for Hydromechanics (IfH), University of Karlsruhe, Germany
ABSTRACT: Large Eddy Simulations (LES) are performed for an open channel flow through submerged vegetation with a water depth (h) to plant height (hp) ratio of h/hp=1.5 according to the experimental configuration of Fairbanks and Diplas (1998). Fairbanks and Diplas measured longitudinal and vertical velocities as well as turbulence intensities along several verticals in the flow and the data are used for the validation of the simulations. The code MGLET is used to solve the filtered Navier Stokes equations on a Cartesian non-uniform grid. In order to represent solid objects in the flow, the immersed boundary method is employed. The computational domain is idealized with a box containing four submerged circular cylinders and periodic boundary conditions are applied in both longitudinal and transverse directions. The predicted streamwise as well as vertical mean velocities are in good agreement with the LDA measurements. Furthermore, good agreement is found between calculated and measured streamwise r.m.s velocities. Largescale flow structures of different shape are present in the form of vortex rolls above the vegetation tops as well as von Karman type vortices generated by flow separation at the cylinders. In this paper the mean and instantaneous flow field is analyzed and further insight into the complex nature of flow through vegetation is provided based on Large Eddy Simulation.
1 INTRODUCTION Both aquatic and riparian vegetation have become central to river and coastal restoration schemes, the creation of flood retention space and coastal protection projects. Aquatic and riparian plants obstruct the flow and reduce the mean flow velocities relative to non-vegetated regions, and the additional drag exerted by plants influences strongly the mean and instantaneous flow field as well as transport processes and system morphology. The flow through partially-vegetated channels or submerged vegetation is characterized by significant velocity gradients (both laterally and vertically) and strong secondary currents, which result in shear layer formation between the canopy and the flow region above the vegetation. A greater understanding of the interaction of flow with vegetation will improve our ability to accurately model flow and sediment transport through vegetation. Hence, both conceptually- and physically-based representations of vegetation-flow interactions are sought with the objective to reduce modelling uncertainty in engineering practice. Most research activities on vegetation effects in the benthic and riparian environment have been devoted to laboratory flumes of simple cross-section, and many researchers (for example: Kouwen et al., 1969; Tsujimoto et al. 1992; Dunn et al., 1996; Nepf
and Vivoni, 1999 to name only a few) have observed that the mean streamwise velocity profile within an emergent or submerged vegetated layer (irrespective of whether the vegetation is rigid or flexible) no longer follows the universal logarithmic law. In the surface flow layer above a submerged vegetation zone, researchers (for example Kouwen et al. 1969) have generally assumed that the logarithmic law prevails. However, detailed studies of flowvegetation interaction and the consequences on the instantaneous flow field are rare. Presently, steady Reynolds Averaged Navier Stokes (RANS) models are the most practical approaches for high-Reynolds-number-fluvial hydraulics applications despite the rapid advancements in computational power and numerical algorithm development. These steady RANS models allow the resolution of the timeaveraged turbulent flow field by adding a subgrid force to the RANS and turbulence transport equations to account for vegetative drag effects. For multi-dimensional flow problems, such methods were used by Shimizu and Tsujimoto (1994), Lopez and Garcia (1997), and Fischer-Antze et al. (2001) together with a k-ε turbulence closure approach to study the mean flow through submerged vegetation. Neary (2003) employed the k-ω model as turbulence closure and reported results very similar to the studies using a k-ε model. Naot et al. (1996) and
Choi and Kang (2001) used a Reynold's Stress model (RSM) accounting for the anisotropy of turbulence, to simulate the flow through rigid submerged vegetation elements. However, within the surface-layer region there was only minor improvement in the computed mean velocity, turbulent intensity and Reynolds stress profiles for the RSM relative to the k-ε or k-ω model. Mean flow features resolved by the steady RANS models include: (a) the suppression of the streamwise velocity profile in the vegetated zone, (b) the inflection of the velocity profile at the top of the vegetation zone, and (c) the vertical distribution of the turbulent shear stress, with its maximum value at the top of the vegetation zone. However, although mean velocities were predicted with satisfying accuracy, RANS models have been less successful at correctly predicting streamwise and vertical turbulence intensities, because these models cannot account for organized large-scale unsteadiness and asymmetries (coherent structures) resulting from turbulent flow instabilities. These coherent structures include: (a) the transverse and secondary vortices in the form of rolls and ribs (Finnegan, 2000), which occur at the top of the vegetation layer as a result of a Kelvin-Helmholtz instability due to the inflection of the streamwise velocity profile (Figure 1), and (b) 3D vortices produced by the complex interaction of the approach flow with the stem (e.g trailing or necklace vortices) and the enforced vortex shedding in the wake of the stem due to flow separation (Figure 2). Recently, modeling techniques that directly resolve large-scale, organized, unsteady structures in the flow and advanced numerical techniques for simulating flows around multiple flexible bodies were introduced for the simulation of such or similar flow problems, e.g. URANS simulations by Paik et al. (2003) or Large Eddy Simulations by Cui and Neary (2002). Such techniques elucidate the largescale coherent structures described above, their important role in vegetative resistance, and the interaction and feedback between the region within and outside the vegetation layer. The Large Eddy Simulation of flow through and above vegetation is not new and finds its origins in boundary-layer meteorology. To our knowledge, the first LES for flow over vegetation were presented by Deardoff (1972) in which the atmospheric boundary-layer over a wheat field was simulated. Further LES in boundary-layer meteorology were for the flow and the turbulent structures above forests, a flow problem that has been studied extensively with LES (e.g. Moeng, 1984, Shaw and Shumann, 1992, Kanda and Hino, 1993, Dwyer et al., 1997). The advantage of LES lies in the fact that a highly resolved temporal and spatial picture of the flow field can be obtained. The vertical distribution of Reynolds stresses and turbulent fluctuations as
calculated with LES were found to be in good agreement with laboratory and field observations. These simulations show the enormous potential of LES in accurately predicting the flow and its associated time-dependent structures. In this paper we present Large-Eddy simulations of turbulent channel flow through a matrix of cylinders. The flow around the individual cylinders is fully resolved by a high resolution grid and the cylinder-matrix can be regarded as an idealized vegetation layer. The time-averaged velocity field as well as turbulence quantities are presented and compared with laboratory measurements. Moreover, large-scale structures are shown to occur above and within the vegetation layer.
Figure 1: Kelvin-Helmholtz instabilities resulting in ribs and rolls above vegetation layers (from Finnegan 2000).
Figure 2: Vortices originating at tall and short cylinders (from Kawamura et al, 1984).
2 NUMERICAL METHOD The LES code MGLET, originally developed at the Institute for Fluid Mechanics at the Technical University of Munich (Tremblay et al., 2000), was used to perform the Large Eddy Simulations. The code solves the filtered Navier-Stokes equations discretised with the finite-volume method and is based on a staggered Cartesian grid. Convective and diffusive fluxes are approximated with central differences of second order accuracy and time advancement is achieved by a second order, explicit Leapfrog scheme. The Poisson equation for coupling the pressure to the velocity field is solved iteratively with the SIP method of Stone (1968). The subgridscale stresses appearing in the filtered Navier-Stokes equations are computed using the Lagrangian dynamic approach of Meneveau et al. (1996). The no-slip boundary condition is applied on the surface of the cylinders and the immersed boundary method (e.g. Verzicco, 2000) is employed. This method is a combination of applying body forces in order to block the cells that are fully inside the cylinder geometry with a Lagrangian interpolation scheme of third order accuracy, which is used for the cells that are intersected by the cylinder surfaces to maintain the no-slip condition.
Reynolds number based on the bulk velocity ubulk and the cylinder diameter D, ReD, is approximately 800. The ratio of cylinder height hp to water depth is hD/h=2/3 and the computational box, which is the shaded area indicated in Figure 2, contained four submerged cylinders. The domain spans 20D in streamwise, 10D in spanwise and 18D (which is the full water depth) in vertical direction, respectively. Several simulations with different grid resolutions were carried out, however, in this paper we will only present the results from the finest grid. The grid consisted of 612 x 308 x 141 grid points in streamwise, spanwise and vertical directions, respectively, which sums to a total of approximately 27 Million grid points. The grid spacings in terms of wall units in streamwise and spanwise direction were ∆x+ = ∆y+ ≈ 40-60 in the areas between the cylinders and ∆x+ = ∆y+ ≈ 1 near the cylinder surface. In the vertical direction the grid was stretched likewise and values of ∆z+≈ 60 near the water surface and ∆z+ ≈ 1 near the top of the cylinders and near the channel bed were obtained. Figure 3 shows the grid in a horizontal plane indicating the refinement of the grid towards the cylinders (note that only every 4th grid line is plotted). Periodic boundary conditions were applied in the streamwise and spanwise directions. At the channel bed the no-slip condition is used and the free surface was set as a frictionless rigid lid and was treated with the symmetry condition. The flow was established over a period of approximately 50 flow through time and circa 80 flow throughs was taken as averaging period. The calculations are still ongoing at the time of the writing and the latest results will be presented at the conference.
4
2
Y/D
Figure 2: Flow Configuration indicating the positions of the 6 measurement verticals and the computational box
0
-2
3 FLOW CONFIGURATION The geometry and the Re number of 15000, based on the channel depth and bulk velocity, are chosen analogous to experimental laboratory investigations undertaken by Fairbanks and Diplas (1998). This allows a comparison between LES and experiments and hence a validation of the LES code as well as the model approximations for the subgrid scale stresses and the treatment of the individual cylinders with the immersed boundary method. In order to investigate the flow through vegetation, Fairbanks and Diplas placed rigid cylinders in a staggered arrangement into a rectangular flume and carried out detailed LDA measurements at the six verticals within the flow as indicated in Figure 2. The
-4 0
5
10
15
20
X/D
Figure 3: Computationalth grid for the Large Eddy Simulations (only every 4 grid line is shown) 4 RESULTS AND DISCUSSIONS
4.1 Time-averaged flow field In Figures 4 and 5, the time-averaged flow field in two selected longitudinal planes (position is indicated in the sketch in the upper right corner) is
presented. Due to flow separation at several locations on the cylinder walls, e.g. on the top and the rear of the cylinder, the flow develops a number of recirculation zones, due,. The retardation of the flow in the vegetation layer is clearly visible and prevails over the whole width of the channel. Above the cylinders the flow accelerates, causing a strong shear layer. The mean flow field exhibits a wavy shape from the top of the cylinders to the free surface which can be attributed to standing waves forming above the recirculation bubble on the top of the cylinder. In the vegetation layer the velocities are almost constant, though slightly higher in planes where no cylinders are present, except in the rear of the cylinders were recirculation over the whole cylinder height occurs.
Figure 4: Distribution of time-averaged streamwise velocity in a plane through cylinders (see sketch in the upper right corner)
Figure 5: Distribution of time-averaged streamwise velocity in a longitudinal plane midway between cylinders (see sketch in the upper right corner)
Figure 6: Distribution of mean streamwise velocity along the six measurement positions #1 to #6. Figure 6 presents the comparison of LES with the measurements along the six selected verticals in the flow for time-averaged streamwise velocities. Overall, a good match between the computed and measured quantities could be achieved. Especially the well known vertical distribution of streamwise velocities of the flow through vegetation was reproduced and exists irrespective of the position of the profile. At position #1, which is directly behind the cylinder, the LES underestimates somewhat the streamwise velocities in the vegetation layer. However, this is the region where the highest turbulence occurs, and a longer averaging period is required to obtain a smooth vertical distribution of velocity. Along the Position #6, which is also behind the cylinder but further downstream, the match between measured and computed velocities is very good. Equally good is the agreement at the positions in front of the cylinder (#3 and #4) and along the positions between the cylinders (#2 and #5). Also noticeable is the velocity bulge near the channel bed, resulting from the near bed necklace vortex around the cylinder, which is predicted to a very satisfying degree (see e.g. Positions #1 and #6).
vegetation layer and interacts with the von Karman vortices (see Figure 11) that are shed from the cylinders. In longitudinal planes between the cylinders (Figure 9) the above mentioned shear layer between the vegetation layer and the region above it with elevated levels of kinetic energy is clearly visible and extents between Z/D=8 – Z/D=15.
Figure 8: Distribution of time-averaged kinetic energy in a longitudinal plane through cylinders (see sketch in the upper right corner) Figure 7: Distribution of streamwise turbulence intensities Urms along the six measurement positions #1 to #6. The above mentioned coherent flow structures affect significantly the turbulence intensity distributions along the six verticals. In Figure 7 measured and computed streamwise turbulence intensities are compared. Similar to the comparison of mean velocities, the match between computed and measured r.m.s values is very good. As can be seen clearly, the peak in the turbulence intensities appears in the vicinity of the vegetation tops (indicated by the green line). This is the area of highest vertical shear where most of the turbulence production occurs. This region is regarded as the interface between the upper part of the flow and the vegetation layer where momentum transfer between these layers occurs. It should be noted that for a smooth distribution of r.m.s velocities a longer averaging time is needed. Figures 8 and 9 present the distribution of the kinetic energy in two selected longitudinal planes (indicated in the sketch in the upper right corner). In the plane where the cylinders are present (Figure 8), two significant regions of high turbulence can be identified. The highest values of kinetic energy occur in the small recirculation bubble on the top of the cylinder. A second region of high turbulence is the area were the trailing vortex penetrates into the
Figure 9: Distribution of time-averaged kinetic energy in a longitudinal plane midway between cylinders (see sketch in the upper right corner) 4.2 Instantaneous Flowfield Figure 10 presents instantaneous streaklines originating at different positions in the flow. All typical features of the flow through and above a vegetation layer can be identified. Near the channel bed the necklace vortices cause a more or less strong
disturbance of the near bed flow. Near the bottom of the second cylinder (from the left) a necklace vortex, also known as the horseshoe vortex, is visualized and strong vertical accelerations are present. In the middle part of the vegetation layer the von Karman vortices dominate the flow and instantaneous flow separation occurs. However, the vertical acceleration is fairly small. Near the top of the vegetation layer, the trailing vortices spinning from the top of the cylinder and penetrating the flow within the vegetation layer, is visible. At this instant in time a trailing vortex is present at each of the cylinder tops. Further above the cylinders the flow is characterized by the organized structures in forms of ribs and rolls, which causes up and downward movement of the flow. In Figure 11 two dominant flow structures present in the vegetation layer and above the cylinders are visualized. The red isosurfaces of instantaneous pressure fluctuation p’ illustrate von Karman vortices shed from the cylinders as a result of alternating flow separation. The vortices travel downstream until they are destroyed by either the trailing vortex that penetrates from the upper region or by interaction with the following cylinder. The upper flow structure, here also visualized by the blue isosurfaces of instantaneous pressure fluctuation, can be identified as vortex rolls that are generated by the shear layer between the vegetation layer and the region above. As can be seen clearly, these two different structures are independent of each other due to the fact that their origin is different. As was shown previously, another flow structure that is responsible for flow interaction is the trailing vortex that disturbs the flow within the vegetation layer transporting momentum from the upper layer of the channel into the vegetation layer and is also responsible for momentum exchange between these two layers as the large-scale ribs and rolls.
responsible for the formation of coherent flow structures being present in the form of vortex rolls. Within the vegetation layer, vortex shedding from the cylinders occurs and von Karman vortices are present. Near the channel bed, the presence of the necklace vortex was confirmed. Also confirmed was the trailing vortex originating at the cylinder tops. It was found that the latter vortex as well as large-scale flow structures in form of ribs and rolls are the main mechanism for flow interaction between the upper part and the vegetation layer.
Figure 10: Instantaneous streamtraces colour-coded with instantaneous streamwise velocity
5 CONCLUSIONS In this paper we have presented the results of a Large Eddy Simulation of open channel flow through and above a matrix of cylinders. The time averaged data of the simulations were compared to LDA measurements from laboratory experiments of the same flow. The well known distribution of streamwise velocities was reproduced and the comparison along the six measurement verticals showed good agreement of calculated values with measured data. Furthermore, turbulence intensities were compared and again a good match between computed and measured quantities was found. In addition to the mean flow field the instantaneous flow field was investigated. A strong shear layer develops above the vegetation layer that is
Figure 11: Instantaneous isosurfaces of pressure fluctuations p’ visualizing vortex rolls (blue) and von Karman vortices (red).
6 ACKNOWLEDGEMENTS This work is part of the research project funded by the German Research Foundation (DFG) under project number Ro 558/29-1. The computations were carried out on the high performance computer HP XC6000 of the computing center at the University of Karlsruhe.
7 REFERENCES Choi S-U, Kang H. (2001). Reynolds Stress Modeling of Vegetated Open Channel Flows. XXIX IAHR Congress Conference Proceedings. Beijing, China. Theme D. Vol 1. 264-269. Cui, J. and V.S. Neary (2002). Large eddy simulation (LES) of fully developed flow through vegetation. IAHR's 5th International Conference on Hydroinformatics, Cardiff, UK, July 1-5, 2002. Dunn, C., Lopez F., Garcia, M. H.. (1996). Mean flow and turbulence in a laboratory channel with simulated vegetation. Hydraul. Engr. Series No. 51, UILU-ENG-96-2009, UIUC. Dwyer, Michael J, Patton, Edward G, Shaw, Roger H. (1997). Turbulent Kinetic Energy Budgets from a Large-Eddy Simulation of Airflow Above and Within a Forest Canopy. Boundary Layer Meteorology., No. 1: 23-44. Fairbanks, J. D. and Diplas, P. (1998): Turbulence characteristics of flows through partially and fully submerged vegetation, Wetlands Engineering and River Restoration Conf., March 22-27, Denver, Colorado, USA. Finnigan, J. (2000). Turbulence in Plant Canopies. Annual Review of Fluid Mechanics, Vol. 32, 519-571. Fischer Antze, T, Stoesser, T, Bates, PB, Olsen, NR. (2001). 3D Numerical Modelling of OpenChannel Flow with Submerged Vegetation. IAHR Journal of Hydraulic Research 39: 303 - 310. Kanda, M., Hino, M. (1994). Organized Structures in Developing Turbulent Flow within and above a Plant Canopy, Using a LES. Boundary-Layer Meteor., 68, 237 – 257. Kawamura, T., Hiwada, M., Hibino, T., Mabuchi, I., Kamuda, M. (1984). Flow around a finite circular cylinder on a flat plate. Bulletin of the JSME 27 (232), 2142–2151. Kouwen N, Unny TE, Hill HM. (1969). Flow Retardance in Vegetated Channels. Journal of Irrigation and Drainage Division, ASCE 95: 329342
Kouwen, N. & T.E. Unny (1973). Flexible roughness in open channel. J. of Hydraulic Div., ASCE, Vol. 99, HY5, pp. 713-728. Lopez, F. & M. Garcia (2001). Mean flow and turbulence structure of open-channel flow through non-emergent vegetation. ASCE J. Hydraulic Engrg., 127(5), 392-402. Meneveau, C., Lund, T. S. and Cabot, W. H. (1996): A Lagrangian dynamic subgrid-scale model of turbulence, Journal of Fluid Mechanics, 319, pp. 353-385. Moeng, C. H. (1984). A large-eddy-simulation model for the study of planetary boundary-layer turbulence. Journal of the Atmospheric Sciences, Vol. 41, pp 2052 – 2062. Naot, D, Nezu, I, Nakagawa, H. 1996. Hydrodynamic Behavior of partly vegetated open-channels. ASCE J. Hydraulic Engrg., 122 (11): 625 – 633. Neary, V.S. (2003). Numerical solution of fullydeveloped flow with vegetative resistance. Journal of Engineering Mechanics. 129(5), 558563. Nepf, H.M. and Vivoni, E.R. (1998). Drag and Diffusivity in Emergent Vegetation. ASCE Wetlands Engineering and River Restoration Conference. Denver, CO. Paik, J., Jones, S. C., and Sotiropoulos, F. (2003). DES and URANS of Turbulent Boundary Layer on a Concave Wall. The proceedings of the 3rd International Symposium on Turbulent and Shear Flow Phenomena, Sendai, Japan, June 2003. Shaw, R. H., Schumann, U. (1992). Large-Eddy Simulation of Turbulent Flow above and within a Forest. Boundary-Layer Meteor., 61 (1-2), 4764. Shimizu, Y. & T. Tsujimoto (1994). Numerical analysis of turbulent open-channel flow over vegetation layer using a k-turbulence model. J. of Hydroscience and Hydraulic Engrg., JSCE, Vol. 11, No. 2, pp. 57-67. Stone, H.L. (1968). Iterative Solution of Implicit Approximation of Multi-dimensional Partial Differential Equations. SIAM J. Numerical Analysis. No. 3. Tremblay, F., Manhart, M. and Friedrich, R. (2000): LES of flow around a circular cylinder at a subcritical Reynolds number, EUROMECH Colloquium 412, October, 4-6, Munich, Germany. Verzicco, R., Mohd-Yusof, J., Orlandi, P., Haworth, D. (2000). Large Eddy Simulation in Complex Geometric Configurations Using Boundary Body Forces. AIAA , Vol. 38 (3), pp. 427-433.
