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Weakly nonlinear analysis of dunes
by the use of a sediment transport formula
incorporating the pressure gradient
Satomi Yamaguchi (Port and airport Institute, Japan)
Norihiro Izumi (Tohoku University, Japan)
Transition of dune formation
Water depth variation during flood
(Simons and Richardson, 1961)
Hysteresis in the transition
Weakly nonlinear stability analysis (Yamaguchi & Izumi, 2003)
- The subcritical bifurcation appears in the transition
Transition of dune formation
Weakly nonlinear analysis (Yamaguchi & Izumi, 2003)
- the subcritical bifurcation in the transition
- effect of bed inclination in sediment transport formulas
[ the gravity effect ]
- Fredsoe’s formula
The subcritical bifurcation appears
when the effect of bed inclination is small.
- Kovacs-Paker’s formula The subcritical bifurcation does not appear.
|
The gravity effect on inclined bed
is estimated to be large.
Effect of pressure gradient
Pressure distribution on dune form (Raudkivi, 1963)
Effect of bed inclination
[the gravity]
[resistance due to the pressure gradient]
In this study,
a bedload transport formula incorporating the pressure gradient
on the basis of Kovacs-Paker’s formula
for the weakly nonlinear analysis of dunes.
Effect of pressure gradient on bedload transport
The coordinate system
- on the basis of Kovacs-paker’s formula
- 2D (bed inclination
in the streamwise direction)
Effect of pressure gradient on bedload transport
Force balance
on a sediment particle
drag force
gravity force
resistance due to pressure gradient
dynamic Coulomb friction force
the force balance in the tangential direction to the bed
Sediment particle velocity
vp
Effect of pressure gradient on bedload transport
Force balance
on the bedload layer
bed shear stress
gravity force
the sum of the pressure p 1 - p 4
grain stress
fluid shear stress at the bottom
(
)
the force balance in the tangential direction to the bed
Volume of particle in the bedload layer x
Effect of pressure gradient on bedload transport
Bedload sediment transport rate
volume of particle in the bedload layer
the coordinate
system
critical Shields shear stress
particle velocity
- Effect of pressure gradient on bedload sediment transport
: decrease
: increase
- Balance between the gravity and pressure gradient
tangential to the bed
Application to analysis of dunes
- Formulations
flow (2D Reynolds equations )
time variation of bed elevation
- Linear stability analysis
- Weakly nonlinear stability analysis
the coordinate system
Application to analysis of dunes
- Linear stability analysis
W : growth rate, k : wave number
- Weakly nonlinear stability analysis
the growth rate expansion method
Landau equation
Landau constant
a 1 < 0 : supercritical bifurcation
a
a 1 > 0 : subcritical bifurcation
|
hysteresis in the transition
Fc
b
Subcritical bifurcation
Results of the linear stability analysis
Instability diagram
S = 0.002 (average bed slope)
stable
mc = 0.84 (dynamic Coulomb
friction coefficient)
t*co= 0.047 (the critical Shields
shear stress for
a horizontal bed)
unstable
neutral curve
Kovacs-Parker’s formula
the present formula incorporating pressure gradient
Results of the weakly nonlinear stability analysis
The present formula
incorporating pressure gradient
Kovacs-Parker’s formula
a1 < 0
Landau constant
supercritical bifurcation
a1 > 0
subcritical bifurcation
|
hysteresis in the transition
Balance between the gravity and pressure gradient
phase difference (from the phase of bed form)
out of phase
: 0.500p
|
The effect of the gravity
is reduced by
the pressure gradient
gravity
pressure gradient
: -0.546p
the phase of linear solutions
Conclusions
- We propose a bedload formula incorporating pressure gradient for the
analysis of dunes.
- The weakly nonlinear analysis with the use of the proposed formula
shows that the subcritical bifurcation occurs in the transition between
dune-covered and flat beds even if the effect of bed inclination is
reasonably large.
- It is found that the pressure gradient reduces the gravity effect in the
bedload formula.