1. Guido Zolezzi, Ruggero Andreatta, Marco Tubino
STREAMLINE-BASED PARAMETRIZATION OF CENTRIFUGALLY INDUCED 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. 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) z
V(z)
Intensity and direction of bed shear stress
Bed deformation
Redistribution of longitudinal momentum
Friction losses
Urbana, October 5, 2005

3. Secondary flows in curved channels state 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. to streamline curvature
Problem statement
Streamline
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
Channel axis
Urbana, October 5, 2005

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

6. The modified 2D model reproduces lateral bed deformation
3D linear model
bed
free surface
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
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?
2D linear model
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. Relevant dimensionless
Main notations
Relevant dimensionless
parameters
(axis) curvature ratio
width ratio
Shields stress
relative roughness
3D Governing equations
Urbana, October 5, 2005

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

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

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

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

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

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

16. Linear bend stability Lateral amplification rate Meander wavenumber
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