Diapositiva 1

1
Nonparallel spatial stability of shallow water
flow down an inclined plane of arbitrary slope P.
Bohorquez R. Fernandez-FeriaE. T. S.
Ingenieros Industriales, Universidad de Málaga,
Spain
Introduction
FIGURE 1depicts the flow after the sudden release
of a dam over an inclined plane of arbitrary
slope (Bohorquez Fernandez-Feria 2006). We have
found for longer times (see Fig. 2), when the
kinematic wave approximation is broadly valid,
the unexpected development of roll waves the
flow suddenly changes from a quasi-uniform and
quasi-steady state to a strong non-uniform and
unsteady state. However, there is no previous
evidence in experimental or theoretical research
(i.e., Chanson 2006) of these hydrodynamic
instabilities. What is the stability threshold in
this nonparallel flow? Is the Jeffreys (1925)
instability condition Froude larger than 2
still valid? ... And what happens to the
asymptotic solution of Hunt (1982)?
THESE ARE ROLL WAVES!!!
FIGURE 1. Sketch of the dam-break problem over an
inclined plane and water depth profiles for
several times, h010m, q20 and ks1mm. Note the
longer time the more uniform profile. Figure
extracted from Bohorquez Fernandez-Feria (2006)
FIGURE 2. Velocity profi-les at different
instants of time for the same parame-ters as
Fig.1. We observe the spontaneous formation of
roll waves. Figure extracted from Bohorquez
Fernandez-Feria (2006)
Statement of the problem
We consider here the one-dimensional flow over a
constant slope bed. In the shallow water
approximation, the dimensionless equations for
the mass conservation and momentum in the
direction of the flow can be written as (see Fig.
3)
FIGURE 3. Sketch of coordinates and variables q
is the angle between the bed and the horizontal,
t is the time, X is the coordinate along the bed,
h is the depth of water measured along the
coordinate Y perpendicular to the bed, U is the
depth-averaged velocity component along X and f
is the Darcy-Weisbach friction factor.
All the magnitudes in these equations have been
non-dimensionalized with respect to a length
scale h0, corresponding to some initial depth,
and a velocity scale U0(gh0)1/2, where g is the
acceleration due to the gravity. We analyse the
linear, spatial stability of the basic flow
including the first order effects of the
streamwise gradient of both the velocity V(X,t)
and the water depth H(X,t). After performing a
local spatial stability analysis (Fernandez-Feria
2000) we obtain a set of four homogeneous linear
equations
yielding a dispersion relation of the form det(F)
? D(a, w ?, f, V, VX, H, HX) 0 that determines
the complex wavenumber a(X)?g(X)ia(X) for a
given real frequency w. The real part g(X) is the
local exponential growth rate, and the imaginary
part a(X) is the local wavenumber.
Results
When the kinematic wave approximation is
satisfied the number of parameters in the
dispersion relation is subtantially reduced with
the change of variable that follows
Parallel flow roll waves The flow is stable
(slt0) for FrltFrc2, while it is convectively
unstable (sgt0) for any value of the frequency ?
if FrgtFrc.
Nonparallel flow kinematic wave
approximation FIGURE 6 shows the neutral curves
in the plane Fr,? for several values of ?gt0,
while Fig. 7 depicts contour lines of constant
growth rate for a particular value of ?gt0. There
are marked differences with the neutral curves
for the parallel case (Fig. 4). Firstly, the flow
is always stable independently of the Froude
number for very small frequencies i.e., for
?lt??(?). Secondly, the minimum, or critical,
Froude number for instability, Frc1(?),
corresponding to the frequency ?c1(?), is always
less than 2 when ? gt 0. This critical Froude
number tends to zero as ? decreases, though the
frequency ?c1 also vanishes as ? ? 0. As in the
parallel case, the flow is unstable for almost
any frequency when Frgt2 (exceptfor very small
frequencies, as commented on above). For high
frequencies the stability region shrinks to
disappear for very high ?. The very high
frequencies, like the very small ones ?lt??, are
too extreme to be physically meaningful.
FIGURE 6. Neutral curves (s0) for different
values of ?gt0 (as indicated) in the plane Fr,?.
The critical points (Frc1,?c1) and (Frc2,?c2) are
shown for one of the curves.
FIGURE 4. Neutral curves (s0) and contour lines
for a constant growth rate s in the plane
Fr,?.Red lines s.001,01,.2,.4,.6,.8,
1,1.2,2,3.Blue lines s-.001,-.01,-.05,-.1,-.15,
-.2,-.25.
FIGURE 7. Contour lines for a constant growth
rate s in the plane Fr,? for ?10-4. Black
line s0. Red lines s.001,.01,.2,.4,.6,.8,1,1.2
,2,3. Blue lines s-.001,-.01,-.05,-.1,-.15,-.2,
-.25.
We have found the spontaneous (i.e., with the
only forcing of the round-off numerical noise)
formation of roll wavesin our numerical
simulations of thedam-break problem in an
inclinedchannel for Froude numbers largerthan
2.5 (Fig. 8), and for Froudelarger than 2.1 when
disturbanceswith much higher amplitude thanthe
numerical noise are introducedupstream. However,
we have notfound the formation of roll wavesfor
Froude numbers less than 2.
FIGURE 8. Velocity profiles U(X) for different
instants of time in the dam-break problem for
Fr2.5, ?01m and ?1. Comparation of the
numerical simulations (line) with Hunts solution
(Hunt 1982, circles)
FIGURE 5. Spatial evolution of a perturbation
(510-5) in a background parallel flow with
Fr2.5, V1 and ?1 at several times.
For further information
References
Bohorquez, P. Fernandez-Feria, R. 2006.
Transport of suspended sediment under the
dam-break flow on an inclined bed of arbitrary
slope. J. Hydr. Res. (submitted) Chanson H. 2006.
Analytical solutions of laminar and turbulent
dam-break waves. Riverflow 2006, Lisbon,
Portugal. Jeffreys, H. 1925. The flow of water in
an inclined channel of rectangular section. Phil.
Mag., 49, 793-807. Hunt B. 1982. Asymptotic
solution for dam-break problem. J. Hydr. Div.
ASCE, 108, 115-126. Fernandez-Feria R. 2000.
Axisymmetric instabilities of Bödewadt flow.
Phys. Fluids, 12, 1730-1739.
Please contact pbohorquez_at_uma.es or
ramon.fernandez_at_uma.es More information on this
and related projects can be obtained at
www.fluidmal.uma.es

Acknowledgments
This research has been supported by a fellowship
(PB) from the Ministerio de Educación y Ciencia
of Spain, and by the COPT of the Junta de
Andalucía (Spain Ref. 807/31.2116)