1. 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

2. 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

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

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

5. Features of Recirculation Eddy
Eddy Fence
Reattachment Zone

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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.

12. 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

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

14. 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

15. Existing Spur-dike Experiment
Calculation results compared with existing experimental results (Muneta and Shimizu, 1994).
Discharge 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

16. 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)

17. 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)

18. 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)

19. 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)

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

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

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

23. 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.

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

25. 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

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

27. 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

28. 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

29. The END

30. 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.

31. Calculation Conditions
Calculation conditions are following.
Total calculation time 170 sec
Time step 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.