STREAMLINE-BASED PARAMETRIZATION OF CENTRIFUGALLY INDUCED SECONDARY FLOWS IN NATURAL STREAMS

1. STREAMLINE-BASED PARAMETRIZATION OF CENTRIFUGALLYINDUCED SECONDARY FLOWS IN NATURAL STREAMS Guido Zolezzi, Ruggero Andreatta, Marco Tubino Department of Civil and Environmental Engineering, University of Trento, Italy Urbana, October 5, 2005

2. z V(z) Context • Increasing use of depth averaged models to predict morphological evolution of river reaches • Need to properly account for 3D effects associated with secondary flows ( = vanishing depth-average) • Intensity and direction of bed shear stress • Bed deformation • Redistribution of longitudinal momentum • Friction losses Urbana, October 5, 2005

3. Secondary flows in curved channelsstate of art • Centrifugal – lateral pressure unbalance (Rozovskij, 1957) • Decomposition of lateral flow in a topographical and a secondary flow components (Kalkwijk & De Vriend, 1980) • Linear closures for regular meandering channels with mobile bed (Johannesson & Parker, 1989, Seminara & Tubino, 1989) • Nonlinear approach (Zolezzi & Seminara, 2001, based on Seminara & Solari, 1998) • Nonlinear closures for sharp bends retaining the effect of longitudinal flow redistribution (Blanckaert & De Vriend, 2004) Urbana, October 5, 2005

4. Streamline Channel axis Problem statement Centrifugal effects in natural rivers are due to the curvature of streamlines rather than to the curvature of channel axis Depth-averaging: need for a new secondary flow parameterization relating centrifugal effects to streamline curvature Urbana, October 5, 2005

5. 2D model 3D model Width variations in a straight channel (Repetto et al., 2002) Pattern of bed deformation The depth-averaged (2D) model does not reproduce lateral bed deformation Why? Urbana, October 5, 2005

6. bed free surface 3D linear model Phase shift between the velocity at the free surface and at the bed • Intrinsic coordinates • Arbitrary ch. curvature and width variations • Nonlinear framework Secondary circulations generate and are the main responsible for lateral bed deformation 2D linear model The modified 2D model reproduces lateral bed deformation Can we formulate a general expression of a depth-averaged model for curved channels with width variations? The lateral bed shear stress is corrected accounting for secondary flows generated by streamline curvature Urbana, October 5, 2005

7. Work plan • Streamline curvature • New secondary flows parameterization • Depth-averaged model for natural, single-thread rivers: curvature and width variations • Validation with test cases • Application: meandering channels with variable width Urbana, October 5, 2005

8. Main notations Relevant dimensionless parameters (axis) curvature ratio width ratio Shields stress relative roughness 3D Governing equations Urbana, October 5, 2005

9. Streamline curvature ratio Streamline curvature Dimensionless streamline curvature Urbana, October 5, 2005

10. Streamlines curvature The closure sub-model for secondary flows Lateral flow decomposition (from Kalkwjik & De Vriend, 1980) Centrifugally induced secondary flow with vanishing depth average Depth average of governing 3D problem Urbana, October 5, 2005

11. Geometrical effect of width variations Effects related to secondary flows Depth averaged model for meandering channels with variable width Geometrical effect of axis curvature Urbana, October 5, 2005

12. Test of the closure submodel Results • Homogeneous solution: alternate bars in a straight channel • Meandering channels with constant width • Straight channel with variable width A first application 2D model of meandering channels with variable width Urbana, October 5, 2005

13. Bar wavenumber Straight channels: homogeneous solution Resonance conditions  morphodynamic influence Resonant width ratio Resonant wavenumber Shields stress Shields stress Linear stability of alternate bars width ratio Urbana, October 5, 2005

14. Migration Stability Meandering channel with constant width Simple model for fluvial bank erosion: Position of near-bank velocity peak Urbana, October 5, 2005

15. Bend migration rate Meander wavenumber Meander migration rate Urbana, October 5, 2005

16. Lateral amplification rate Meander wavenumber Linear bend stability Urbana, October 5, 2005

17. Straight channel with variable width Prediction of lateral bed deformation Urbana, October 5, 2005

18. Lateral bed deformation and longitudinal velocity Correction to the lateral bed shear stress Transverse redistribution of longitudinal momentum and modified inertial effects Urbana, October 5, 2005

19. Lateral bed deformation and longitudinal velocity Planform instability Urbana, October 5, 2005

20. Meandering channels with variable width The present parameterization allows to reproduce the morphodynamic response of channels forced by variations of both channel width and curvature Widest sections correspond to most curved reaches (Seminara & Solari, 2005) Urbana, October 5, 2005

21. Effect on planform stability Width variations may significantly modify the planimetric stability of meandering channels Urbana, October 5, 2005

22. Summary • Closure sub-model for secondary flows based on the curvature of streamlines • Depth-averaged model for meandering channels with variable width • Validation with known test cases • Opened opportunities to study planform stability of natural channels Urbana, October 5, 2005

23. Ongoing developments • Chute cutoffs processes in braided and meandering streams: • planform stability of meandering channels with variable width • how bar migration is affected by a combined forcing effect (channel curvature + width variations)? • Channel shift in braided streams: • how do curvature and width evolve in time? Urbana, October 5, 2005