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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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