On the nature of bend instabilityStefano Lanzoni

University of Padua, Italy

Bianca Federici and Giovanni SeminaraUniversity of Genua, Italy

Meanders wandering in a flat valley

(Alaska -USA)

Meanders evolving in a rocky environment

(Utah- USA)

Tidal meanders within the lagoon of Venice

(Italy)

Field examples of meander featuresFlow direction

Beaver River

Upstream skewed

meanders

Downstream skewed

meandersFlow direction

Fly River

Coexistence of upstream and downstream

skewed meanders

Multiple loops

White River

Flow direction

Flow direction

Pembina River

Scope of the work

• Under which conditions is the planimetric development of meandering rivers downstream/upstream controlled?

• How is downstream/upstream influence related to the nature of bend instability?

• Which are the implications for the boundary conditions to be applied when simulating the planimetric development of natural rivers?

Notations

Planform view Sez. A-A

Formulation of the problem

•Dimensionless planimetric evolution equation (Seminara et al., Jfm 2001)

• Erosion law (Ikeda, Parker & Sawai, Jfm 1981)

  

semilarghezza velocità media

lateral migration speed

long term erosion coefficient

time longitudinal coordinate

depth averaged longitudinal

velocity

lateral coordinate

•Flow field (Zolezzi and Seminara, Jfm 2001)

Characteristic exponents

integration constants

channel axis curvature

um= um(, Cf0, * )

aspectratio

frictioncoefficient

Shieldsparameter

• Dispersion relationship for bend instability (Seminara et al., Jfm 2001)

• Perturbation

Planimetric stability analysis

: complex angular frequency

: complex phase velocity

: complex group velocity

= (, Cf0, * )

Characteristic of bend instability

growth rate

dune covered bedplane bed

phase speed

r

r

r

•

• response excited at resonance

• super-resonance: bend migrate upstream

• sub-resonance: bend migrate downstream

Instability classification

Absoluteinstability

Convectiveinstability

initial impulse perturbation

initial impulse perturbation

Linear analysis of bend instability(Briggs' criterion, 1964)

Absoluteinstability

Convectiveinstability

branch point singularities = 0

> 0•• the spatial branches of dispersion relationship (igiven, rvarying) lie in distinct half -planes for large enough values of the temporal growth rate i

Results of linear theory: First scenario=8, =0.3, d=0.005, dune covered bed

a) i=[i], b) i=1.5[i], c) i=2[i]

Convective instability

=25, =0.7, d=0.005, dune covered bedResults of linear theory: Second scenario

a) i=[i]

b) i=2[i]

c) i=5[i]

Absolute instability

Linear theory

Bend instability is generally convective, but a transition to absolute instability occursfor large values of , dune covered bed and

large values of *

The group velocity ∂r/∂ associated to thewavenumber max characterized by the

maximum growth rate changes sign as resonance is crossed

moreover,

r

Numerical simulations of nonlinear planimetric development

i

= t/E

pi

Boundary Conditions:

i = E (Ui|n=1-Ui|n=-1)

Ui=Ui(*,ds,cmj )

Free B.C.Periodic B.C. cmj j=1,4Forced B.C.

Numerical results: Free boundary conditions

=8, =0.3, d=0.005 dune covered bed

=25, =0.7, d=0.005 dune covered bed

Sub-resonant conditions, Convective instabilitywavegroup migrate downstream

Super-resonant conditions, Convective instabilitywavegroup migrate upstream

Super-resonant conditions, Convective instabilitywavegroup migrate upstream

=15, =0.3, d=0.005 dune covered bed

Numerical results: Periodic boundary conditions

Sub-resonant conditions, Convective instabilitywavegroup migrate downstream

Super-resonant conditions, Convective instabilitywavegroup migrate upstream

=8, =0.3, d=0.005 dune covered bed

=15, =0.3, d=0.005 dune covered bed

Numerical results: Forced boundary conditions

=30, =0.1, d=0.01, dune covered bedsuper-resonant conditions

periodic B.C.

free B.C.

forced B.C.

Numerical results: Free boundary conditions incipient cut off configuration

=15, =0.3, d=0.005 dune covered bed

incipient cut off configuration

planform configurations after several neck cut offs

The length of straight upstream/downstream reaches continues to increase

Cutoff spreads in the direction of morphodynamic influence

Conclusions

• Bend instability is invariably convective• Meanders are typically upstream skewed• Wave groups travel downstream• The upstream reach tends to a straight configuration in absence of a persisting forcing

• The choice of boundary conditions strongly affects numerical simulations of the planimetric development of alluvial rivers

• Sub-resonant conditions ( < r)

• Bend instability may be absolute for a dune covered bed and high enough values of the Shields parameter• Meanders are typically downstream skewed• Wave groups travel downstream• The downstream reach tends to a straight configuration in absence of a persisting forcing

• Super-resonant conditions ( > r)

Open issues

• Systematic field observations are needed to further substantiate the morphodynamic upstream influence exhibited by bend instability under super-resonant conditions

• The role of geological constraints possibly present in nature and their relationships with the features typical of bend instability has to be investigated.

• Which boundary conditions have to be applied when simulating the planimetric development of alluvial rivers?

• Further analyses are required to clarify the effects of chute and neck cut off on river meandering.