Erik Bollaert, Anton Schleiss
Laboratory of Hydraulic Constructions, LCH, Swiss Federal Institute
of Technology Lausanne
EPFL-LCH, CH-1015 Lausanne, Switzerland
Phone:
++41 21 693 23 85
Fax:
++41 21 693 22 64
E-mail:
secretariat.lch@epfl.ch
internet:
http://lchww.epfl.ch
Abstract:
High velocity plunging
jets, appearing for example at the downstream end of spillways of dams, create
local erosion of the rock bed. The appropriate
prediction of these scours is important for the safety evaluation of the dam and
its abutments. Traditionally, scour formation is estimated with empirical or
semi-empirical formulae, mostly developed from physical models. Nevertheless,
these approaches are often unreliable and not representative because neglecting
some of the basic physics involved. The full understanding of the scour process
and the interaction of the various parameters is still incomplete and a more
reliable theoretical approach to the problem is actually missing. Above all, the
relationship between dynamic water pressures in rock fissures and jet
characteristics in the plunge pool is unknown
for prototype jet velocities. Experimental
pressure measurements in simulated rock fissures have been performed which
revealed the appearance of violent transient flow phenomena inside the fissures,
such as oscillations, resonance conditions, e.g. These phenomena propagate at
specific celerities, depending on the air content of the two-phase air-water
mixture inside the fissures. It was found that pressure fluctuations in rock
fissures are highly dependent on the mean pressure value inside and on the air
concentration of the two-phase mixture. Furthermore, the air content in the
fissure can be related to jet and plunge pool characteristics. Numerical
simulation of the unsteady two-phase flow conditions inside rock fissures will
finally provide the framework for an improved scouring evaluation method.
Keywords:
high velocity jets, scour in fissured rock mass, transient pressurized air-water
flow, air concentration, pressure wave celerity, dam spillways
In
the following paper, parts of the experimental results of a research study,
which concerns the pressure fluctuations inside rock fissures due to high
velocity jet impact, are presented. A clearer understanding of these dynamic
pressures is required to better assess the physical processes of rock mass
break-up by hydraulic jacking and uplift. Pressure measurements inside
artificially created 1D and 2D rock fissures were performed which highlighted
the formation of violent transient flow phenomena in the form of standing waves
and resonance conditions. Maximum pressures inside the fissures attained values
up to several times the jet stagnation pressure. The influence of air bubbles in
the pressurized flow on this wave propagation process is discussed in detail,
because it appears to be of crucial significance. The final goal is the
development of an improved scouring evaluation method that accounts for the
processes mentioned before. The project’s objective and actual
state-of-the-art can be visualized in function of the three governing phases:
the liquid phase (water), the gas phase (air) and finally the solid phase
(rock). The 3D-cube presented at Figure 1 summarizes the most important existing
evaluation methods and compares them with the objective of the research project.
In the following, special attention will be drawn on aeration aspects.
Fig. 1 3D-visualization of the actual state-of-the-art in rock scour research and of the project’s objective
The experimental set-up is shown at Figure 2 and has two main parts: the upper part consists of a 3m-diameter cylindrical basin in steel reinforced plastic, while the lower part incorporates a 1 mm thin steel strip that is pre-stressed like a sandwich between two 100 mm thick, 1 ton heavy steel plates. The jet has diameters up to 72 mm and mean velocities up to 35 m/s. Pressure sensors simultaneously register at the pool bottom and inside the simulated rock fissure, for acquisition rates up to 20 kHz, in order to trace any violent transients. The water depth in the plunge pool varies between 0 and 1 m.
Fig. 2 Perspective/side view
of experimental facility: (1) cylindrical jet outlet, (2) reinforced plastic
cylindrical bowl, (3) pre-stressed two-plate steel structure, (4) PC-DAQ and
pressure sensors, (5) restitution system, (6) thin steel strips (1D and 2D
fissures).
Table 1 summarizes the main jet characteristics for a 0.072 m diameter cylindrical nozzle:
Table 1 Main characteristics of vertical plunging jets investigated
Special
attention has to be paid to the ratio of pool depth to jet diameter Y/Dj.
Two types of jet impact in plunge pools can be distinguished: core
impact, for Y/Dj < 4-5 (Figure 3a), and developed jet impact, for
Y/Dj > 4-5 (Figure 3b).
Fig. 3 (a) jet
core impact for Y/Dj < 4-5, (b) developed jet impact for Y/Dj
> 4-5 Furthermore,
parameters that influence the aeration rate at pool impact b,
defined by the ratio of the air discharge (Qa) over the water
discharge (Qw), are the Froude, Reynolds and Weber numbers, the
initial turbulence intensity Tu and the fall length over jet diameter
ratio L/Dj (Table 1). As can be seen
in Figures 4a-c, the bottom pressure patterns generated by core and developed
jet impact are completely different. While core impact is related to high mean
pressures with small fluctuations (low RMS) and a negatively skewed probability
density function (PDF), developed jet impact provides much lower mean pressures
but higher RMS values, for a positively skewed PDF. In other terms, core impact
involves high pressures and developed jet impact stands for lower mean pressures
but with considerable fluctuations. The plunge point
aeration rate is highly dependent on the initial jet turbulence intensity (Tu)
and is mostly expressed in function of jet velocity (Vj) and the
ratio of fall depth to jet diameter (L/Dj). Most of the existing
relations for b have been
established at low velocity range (< 10 m/s). However, based on a comparison
of the expressions valid for circular plunging jets (by Bin 1984, Van de Sande
& Smith 1973, Ervine & al. 1987 and Ervine 1998), a reasonable extension
towards higher velocities is set up and indicates a plunge point air
concentration (ai)
almost growing linearly with jet velocity. Depending on the L/Dj
ratio, values for ai
of 25-40 % for Vj = 10 m/s up to 45-60 % for Vj = 35 m/s
are obtained (Fig 4d).
(a)
(b)
(c)
(d) Fig.
4 Pool bottom pressures for core impact (+) and developed
jet impact (¨):
(a) mean dynamic pressure coefficient Cp; (b) RMS-coefficient C’p;
(c) probability density function (PDF); (d) air content aI
(mean value according to existing expressions). The air
concentration at the pool bottom, close to the fissure entry, can be determined
by expressing the diffusion-advection bubble transport process from the point of
impact up to the rock-water interface (Chanson 1998). However, the considerable
influence of the pool bottom and the very small decay of the air concentration
through the small pool depths justify the use of the above determined plunge
point air concentrations as values close to the fissure entry. The impact of a high velocity
air-water jet onto a rock fissure principally contains all the characteristic
elements of a resonator system: the jet provides the necessary excitation, while
the open or closed-end rock fissure plays the role of a resonance chamber.
Application of transient pressures to fissure lengths Lf @
1-10 m, with celerities c @
1’400 m/s, can create oscillatory conditions inside the fissures when the jet
has considerable energy in a frequency range beyond 35-70 Hz (fundamental mode fr
= c/(4Lf) or c/(2Lf) for closed or open-end resonator
systems). Plunge pool macro-turbulent flow has turbulent energy mainly at low
frequencies (less than 25 Hz). However, experiments performed by the authors
indicate that a high velocity jet possesses sufficient turbulent energy beyond
this macro-turbulent frequency range to create a resonant excitation inside rock
fissures with open or closed ends. Figures 5a-b present simultaneously
recorded pressures at the plunge pool bottom and inside a 1D fissure, for Y/Dj
= 9.3, thus in the developed jet region according to Figure 3. Figure 5b shows
the non-dimensionalized spectral density in the Strouhal (Strouhal number of
fissure Sh,f = fr.Lf/c) domain, for the
pressure waves measured inside the fissure. The Strouhal number of 0.25,
obtained for maximum energy, corresponds to a resonance frequency of fr
= c/(4Lf), according to the closed-end resonator theory (Wylie &
Streeter 1978). Figure 5a shows values in the time domain. The pressures
recorded at the pool bottom (next to the fissure entry) were used as input
condition for a numerical calculation (by linear method of characteristics, thus
constant wave celerity) of the transient pressure waves inside the 1D fissure.
These calculated transient waves were finally compared with the measured
transient pressure waves. The time domain calculated values show good agreement
with the measured ones, but only for compressive waves (upper peak values). On
the other hand, rarefaction waves calculated by the numerical model are
completely different from the recorded ones. This can be explained by alternating
air bubble release and re-solution effects of the air-water mixture propagating
inside the fissure. In fact, if a liquid with a certain gas content in solution
undergoes a sudden pressure drop, supersaturation and thus gas release occurs.
The amount of released gas depends in a linear manner on the pressure drop below
the governing saturation pressure. The variation of the gas content a
with pressure is governed by the equation of state for gases, expressed under
isothermal conditions as
While the mass density of the mixture is hardly
modified, a very slight change in free gas content however drastically changes
the mixture’s compressibility and thus pressure wave celerity. Application
to high velocity jet impact onto rock fissures accounts for air bubble
entrainment through the pool into the fissures by a combined mechanism of
re-solution and release of the bubbles. Following the equation of state for
gases, it is obvious that the free air content will strongly depend on the
pressure situation inside the fissure. Figure 6 presents the calculated air
contents based on dynamic pressure measurements inside 1D and 2D simulated rock
fissures. These contents are derived from the relationship between the free air
content of a gas-liquid mixture and its compressibility, thus wave celerity. The
experiments allow measuring these celerities inside the fissures by three
different means: cross-correlations, power spectral density peaks and time
domain pressure pulse propagation speed evaluation. (a)
Fig.
5 Pressure values inside 1D rock fissure: (a) time domain:
measured versus calculated values, (b) Strouhal domain: spectral density
(log-log). The results are
presented at Figure 6 for the two distinct impact conditions that exist in
plunge pools. For core impact, the measured celerities progressively grow with
the mean absolute pressure inside the fissures. As a result, the corresponding
mean free air content stays more or less constant between 0.5 and 2%. This is
explained by the fact that the pressure conditions at the pool bottom (Figure
4), and thus also inside the fissures, are characterized by a high mean value
and relatively small fluctuations around this value: the related pressure drops
inside, and thus air release, is negligible. However, developed jet impact has a
different pressure pattern, with important fluctuations and thus pressure drops
inside the fissures. This allows considerable air bubble release and thus very
low wave celerities, even for high mean pressures. Celerities less than 100 m/s
and free air contents beyond 10 % have been noticed. Therefore, the derived
relationship allows defining a mean free air content in 1D and 2D rock fissures,
in function of the falling jet and plunge pool parameters. Appropriate two-phase
wave celerities of transient pressures inside rock fissures can so be accounted
for. (a)
Fig.
6 Air bubble effects inside rock fissures for core impact (+)
and developed jet impact (¨):
(a) measured mean wave celerity, (b) calculated mean free air content inside
rock Experimental
study and analytical analysis of the unsteady character of the propagation of
transient two-phase pressure waves in 1D and 2D artificially created rock
fissures due to high velocity jet impact is performed. In particular, the
importance of air bubbles in the plunge pool and inside the rock fissures is
outlined and its influence evaluated in a combined empirical-analytical
approach. It is found that the amount of free air in water inside rock fissures
is controlled by a mechanism of air release and re-solution, depending on the
governing pressure conditions. This two-phase
transient approach will allow a better physical understanding of the processes
of scour like rock break-up i.e. hydraulic jacking, and rock block uplift.
Numerical simulation of transient pressure propagation in fissured media, based
on two-phase compressible flow equations, will form the theoretical bases for an
improved scouring evaluation method taking into account the basic physical
processes involved. References Bin
A.K. (1984): Air entrainment by plunging liquid jets. IAHR Symposium on Scale
Effects in Modelling Hydraulic Structures, Esslingen, Germany. Chanson
H. (1998): Air bubble entrainment in free surface shear flows. Academic Press. Ervine
A. (1998): Air entrainment in hydraulic structures: a review. Proceedings Instn
Civ. Engrs. Wat., Marit. & Energy, Vol. 130, 142-153. Ervine
A. , Elsawy E.M. (1975): The effect of a falling nappe on river aeration.
Proceedings of the XVI IAHR Congress, Sao Paulo, Brazil, paper C45. Ervine
A., Falvey H. (1987): Behaviour of turbulent water jets in the atmosphere and in
plunge pools. Proceedings of the Institution of Civil Engineers, Part 2, Vol.
83, 295-314. Wylie
B., Streeter V.L. (1978): Fluid Transients. Mc Graw-Hill Inc. 
4
PLUNGE POOL BOTTOM PRESSURES
5
DYNAMIC PRESSURES IN ROCK FISSURES
(i = initial conditions).
(b)
(b)
6
CONCLUSIONS