R. Akahori & M.W. Schmeeckle Numerical Analysis of Secondary-Flow around a Spur Dike using a Three-Dimensional Free Water Surface LES Model R. Akahori & M.W. Schmeeckle Department of Geography, Arizona State University

Rapid Channel Expansion Produces recirculation eddy that may trap sediment -ecological and recreational important Spur dikes are employed to trap sediment and inhibit lateral migration of the channel Natural rapid channel expansions often occur downstream of tributary/ mass wasting inputs or fixed bedrock constrictions

Spur-dike in Toyohira River Example of channel expansion.1 Spur-dike in Toyohira River Prof. S. Ikeda

Channel expansion by Debris fans in the Colorado River Example of channel expansion.2 Channel expansion by Debris fans in the Colorado River

Features of Recirculation Eddy Eddy Fence Reattachment Zone

Features of Recirculation Eddy Eddy Recirculation zone Eddy Fence (Free Shear Layer) Reattachment zone - intense large scale turbulence Eddy Fence Eddy recirculation zone Sediment trap Reattachment zone

Previous Numerical Approaches 2-D depth averaged model -Can’t treat secondary flow -Relies on adjusting unknown lateral diffusion 2-D/ Quasi 3-D models w/ secondary flow based on streamline curvature -Unknown secondary flow structure in recirculation zones w/ complex topography RANS 3-D model (e.g. k-ε) -Cannot capture time variability in the reattachment zone -especially large scale turbulence produced along the free shear layer

Previous Numerical Approaches Some models assume hydrostatic pressure -Flow at the point of separation and reattachment zone have large advective accelerations in the vertical momentum equation Most models assume either a rigid lid or no time-variance of the water surface -Time variance of the water surface is critical to accurately model large scale turbulence -Adequacy of assuming a rigid lid has not been proven

Employed Features Full 3-dimensional equations (non-hydrostatic) Large Eddy Simulation (LES) turbulent model – no time-averaging Body Fitted Coordinates (BFC) system and Moving Grid system – free water surface

Full 3D Equations Continuity Equation Momentum Equations (Navier-Stokes Equation) in which, xi or xj = x, y, z, and ui or uj = u, v, w

Large Eddy Simulation (LES)-1 N-S equations are spatially-filtered, NOT time- or ensemble-averaged. u = u (spatially filtered) + u’ (fluctuation) Eddies, larger than grid scale, are directly calculated by spatially-filtered N-S equations. Eddies, smaller than grid scale (sub-grid scale: SGS), are parameterized.

Large Eddy Simulation (LES)-2 Smagorinsky model is the simplest model for SGS closure. Additional term Spatially-filtered Momentum Equations (Navier-Stokes Equations) Additional terms after Spatial filtering Smagorinsky Model

Body Fitted Coordinates (BFC) Body Fitted Coordinates (BFC) is employed to fit the grid to arbitrarily shaped boundary Cartesian Coordinates System BFC Coordinates System

Moving Grid System Moving Grid system enables the model to trace temporally changing free water surface boundary. Operators for coordinates transformation into a combined system of the BFC and a moving grid system Example of combination of BFC and moving grid system

Existing Spur-dike Experiment Calculation results compared with existing experimental results (Muneta and Shimizu, 1994). Discharge 1870 cm3/sec Slope 1/1000 Channel length 700 cm Channel width 40 cm Downstream depth 7 cm Spur dike length 20 cm Spur dike width 4 cm

Results (particle tracing) Particle tracing method is applied. Recirculation is clearly observed. Reattachment point is varying over time. Jet-like stream can be seen beside a spur-dike, and it has strong secondary-flow. Time: from 100(sec) to 170(sec)

Results (vorticity: z-axis) Intense horizontal vorticity generated in the free-shear layer (eddy fence) Horizonatal eddies are intermittently shed into the reattachment zone x z y Time: from 100(sec) to 170(sec)

Results (vorticity: x-axis) Stable counter-clockwise vorticity exists beside a spur-dike. Large-scale vorticity generated in the reattachment zone x z y Time: from 100(sec) to 170(sec)

Results (vorticity: y-axis) Large scale vorticity generated at the bed and in the reattachment zone Near-bed flow in jet has strong vorticity. x z y Time: from 100(sec) to 170(sec)

Turbulence Intensity (u’u’) Turbulence intensity of downstream velocity Strong turbulence exists in and downstream of the reattachment zone

Turbulence Intensity (v’v’) Turbulence intensity of cross-stream velocity Strong turbulence exists in the middle of the reattachment zone

Turbulence Intensity (w’w’) Turbulence intensity of vertical velocity Turbulence intensity is respectively small also has a peak in the zone of reattachment

Water Surface Fluctuation is intense in front of a spur-dike, and downstream after a recirculation area. Time-averaged calculated water surface level seems reasonable.

Comparison (depth-averaged) Depth-averaged recirculation eddy is observed both in experiment and calculation results. Calculated result shows good agreement.

Comparison (cross-stream velocity) Lateral velocity at cross-stream sections Results show similar tendency near the bottom. But near the surface, calculation results show stronger leftward flow than the experiment. dot: experiment line: calculation

Cross-sectional Vectors Strong secondary-flow is produced as flow is accelerated around the spur-dike, and it remains far downstream.

Conclusion Intense horizontal vorticity (z-axis) generated but confined to the eddy fence (free shear layer) Large-scale intense turbulence generated in the reattachment zone The point of reattachment varies substantially over time

Morphodynamics of Eddies Sediment trapping occurs in the zone of reattachment with generation of large-scale turbulent eddies and temporal changes in the point of reattachment Little sediment can cross the eddy fence because horizontal vorticity confined to narrow shear zone No mean inward secondary flow near the bed means that sediment is not trapped by secondary flow –only large-scale turbulent motion

The END

Other Features Boundary friction from the law of the wall at the bed and bank boundaries Slip condition at water surface. Convection terms are calculated by the Cubic-Interpolated Propagation (CIP) scheme. Water surface level is implicitly calculated by the kinematic boundary condition.

Calculation Conditions Calculation conditions are following. Total calculation time 170 sec Time step 0.001 sec Grid number (streamwise) 175 Grid number (crossstream) 20 Grid number (vertical) 10 At the upstream end, discharge is given. At the downstream end, water depth is given.