CHANSON, H., and CAROSI, G. (2007). "Turbulent Time and Length Scale Measurements in High-Velocity Open
Channel Flows." Experiments in Fluids, 42 (3), pp. 385-401 (DOI 10.1007/s00348-006-0246-2) (ISSN 0723-4864).
Page 1
TURBULENT TIME AND LENGTH SCALE MEASUREMENTS IN HIGH-VELOCITY
OPEN CHANNEL FLOWS
by H. Chanson and G. Carosi
Division of Civil Engineering, The University of Queensland, Brisbane QLD 4072, Australia.
Address :
Hubert Chanson, Professor
Giovanna Carosi, Research student
Division of Civil Engineering, The University of Queensland
Brisbane QLD 4072, Australia
Fax: (61 7) 33 65 45 99 Email: h.chanson@uq.edu.au
Abstract :
In high-velocity open channel flows, the measurements of air-water flow properties are complicated by the strong
interactions between the flow turbulence and the entrained air. In the present study, an advanced signal processing of
traditional single- and dual-tip conductivity probe signals is developed to provide further details on the air-water
turbulent level, time and length scales. The technique is applied to turbulent open channel flows on a stepped chute
conducted in a large-size facility with flow Reynolds numbers ranging from 3.8 E+5 to 7.1 E+5. The air water flow
properties presented some basic characteristics that were qualitatively and quantitatively similar to previous skimming
flow studies. Some self-similar relationships were observed systematically at both macroscopic and microscopic levels.
These included the distributions of void fraction, bubble count rate, interfacial velocity and turbulence level at a
macroscopic scale, and the auto- and cross-correlation functions at the microscopic level. New correlation analyses
yielded a characterisation of the large eddies advecting the bubbles. Basic results included the integral turbulent length
and time scales. The turbulent length scales characterised some measure of the size of large vortical structures
advecting air bubbles in the skimming flows, and the data were closely related to the characteristic air-water depth Y
90
.
In the spray region, present results highlighted the existence of an upper spray region for C > 0.95 to 0.97 in which the
distributions of droplet chord sizes and integral advection scales presented some marked differences with the rest of the
flow.
CHANSON, H., and CAROSI, G. (2007). "Turbulent Time and Length Scale Measurements in High-Velocity Open
Channel Flows." Experiments in Fluids, 42 (3), pp. 385-401 (DOI 10.1007/s00348-006-0246-2) (ISSN 0723-4864).
Page 2
.
Keywords : air-water flow measurement, turbulent time and length scales, open channel, upper spray region.
LIST OF SYMBOLS
C void fraction defined as the volume of air per unit volume of air and water; it is also called air concentration
or local air content;
C
mean
depth-average void fraction defined in terms of Y
90
: C
mean
= 1 - d/Y
90
;
D
H
hydraulic diameter (m) also called equivalent pipe diameter;
D
o
dimensionless constant
d equivalent clear water flow depth defined as: d =
⌡
⌠
C=0
C=0.90
(1 - C) *dy ;
d
c
critical flow depth (m) : d
c
=
3
Q
w
2
/(g*W
2
);
F air bubble count rate (Hz) or bubble frequency defined as the number of detected air bubbles per unit time;
F
max
maximum bubble count rate (Hz) at a cross-section;
g gravity constant: g = 9.80 m/s
2
in Brisbane, Australia;
h vertical step height (m);
K' dimensionless integration constant;
K
*
dimensionless constant;
L
xx
air-water advection integral length scale (m) :
L
xx
= V * T
xx
L
xy
transverse/streamwise air-water integral turbulent length scale (m) :
L
xy
=
⌡
⌠
Y=0
Y
max
(R
xy
)
max
* dY
(L
xx
)
max
maximum advection air-water length scale (m) in a cross-section;
(L
xy
)
max
maximum air-water integral length scale (m) in a cross-section;
l horizontal step length (m);
N power law exponent;
Q
w
water discharge (m
3
/s);
Re Reynolds number defined in terms of the hydraulic diameter;
R
xx
normalised auto-correlation function;
CHANSON, H., and CAROSI, G. (2007). "Turbulent Time and Length Scale Measurements in High-Velocity Open
Channel Flows." Experiments in Fluids, 42 (3), pp. 385-401 (DOI 10.1007/s00348-006-0246-2) (ISSN 0723-4864).
Page 3
R
xy
normalised cross-correlation function between two probe output signals;
(R
xy)max
maximum cross-correlation between two probe output signals;
S
o
bed slope : S
o
= sinθ;
T time lag (s) for which R
xy
= (R
xy
)
max
;
T integral turbulent time scale (s) characterising large eddies advecting the air bubbles;
Tu turbulence intensity defined as: Tu = u'/V;
T
xx
auto-correlation time scale (s) :
T
xx
=
⌡
⌠
τ=0
τ=τ(R
xx
=0)
R
xx
(τ) * dτ
T
xy
cross-correlation time scale (s) :
T
xy
=
⌡
⌠
τ=τ(R
xy
=(R
xy
)
max
)
τ=τ(R
xy
=0)
R
xy
(τ) * dτ
T
0.5
characteristic time lag (s) for which R
xx
= 0.5;
T
max
maximum integral time scale (s) in a cross-section;
(T
xy
) maximum cross-correlation time scale (s) in a cross-section;
U
w
flow velocity (m/s) : U
w
= Q
w
/(d*W);
u' root mean square of longitudinal component of turbulent velocity (m/s);
V interfacial velocity (m/s);
V
c
critical flow velocity (m/s);
V
90
characteristic interfacial velocity (m/s) where C = 0.90;
W channel width (m);
x distance along the channel bottom (m);
Y separation distance (m) between two phase-detection probe sensors;
Y
90
characteristic depth (m) where the void fraction is 90%;
y distance (m) measured normal to the invert (or channel bed);
y' dimensionless distance (m) normal to the invert (or channel bed) : y' = y/Y
90
;
z transverse distance (m) from the channel centreline;
CHANSON, H., and CAROSI, G. (2007). "Turbulent Time and Length Scale Measurements in High-Velocity Open
Channel Flows." Experiments in Fluids, 42 (3), pp. 385-401 (DOI 10.1007/s00348-006-0246-2) (ISSN 0723-4864).
Page 4
Greek symbols
∆x streamwise separation distance (m) between sensor;
∆z transverse separation distance (m) between sensor;
µ dynamic viscosity (Pa.s);
µ
w
water dynamic viscosity (Pa.s);
θ angle between the pseudo-bottom formed by the step edges and the horizontal;
ρ density (kg/m
3
);
ρ
w
water density (kg/m
3
);
τ time lag (s);
τ
0.5
characteristic time lag τ for which R
xy
= 0.5 * (R
xy
)
max
;
χ dimensionless parameter: χ = K' - y'/(2*D
o
) + (y'-1/3)
3
/(3*D
o
)
∅ diameter (m);
Subscript
w water flow;
xx auto-correlation of reference probe signal;
xy cross-correlation;
90 flow conditions where C = 0.90.
1. INTRODUCTION
In high-velocity open channel flows, the strong interactions between the turbulent waters and the atmosphere lead often
to some air bubble entrainment. The entrained air is advected within the bulk of the flow and the air-water mixture has
a whitish appearance (Fig. 1A). In civil engineering applications, the flow velocity exceeds typically 5 to 10 m/s, and
the flow Reynolds number ranges from 1 E+7 to over 1 E+9 in large dam spillways. The void fraction ranges from
100% above the "free-surface" to some small, often non-zero value close to the invert (e.g. Cain and Wood 1981b).
These high-velocity, highly-aerated flows cannot be studied analytically nor numerically because of the large number
of relevant equations and parameters. Present knowledge relies upon physical modelling and experimental
measurements. Accurate measurement systems for air-water flow measurements include intrusive phase-detection
probes, hot-film probes, and LDA/PDA systems. Authoritative reviews include Jones and Delhaye (1976), Cain and
CHANSON, H., and CAROSI, G. (2007). "Turbulent Time and Length Scale Measurements in High-Velocity Open
Channel Flows." Experiments in Fluids, 42 (3), pp. 385-401 (DOI 10.1007/s00348-006-0246-2) (ISSN 0723-4864).
Page 5
Wood (1981a), Chanson (1997a,2002) and Chang et al. (2003). The processing of these measurement techniques yield
basically the void fraction, bubble count rate, interfacial velocity and turbulence intensity. Further information requires
more advanced instrumentation : e.g., 4- or 5-sensor probes (Kim et al. 2000, Euh et al. 2006).
In the present study, it is shown that an advanced signal processing of traditional single- and dual-tip conductivity
probes may provide further information on the air-water turbulent time and length scales. The technique was applied to
turbulent open channel flows on a stepped chute. The measurements were conducted in a large-size facility (θ = 22º, h
= 0.1 m) in which detailed air-water flow properties were recorded systematically for several flow rates including
turbulence levels and turbulent time and length scales.
2. EXPERIMENTAL APPARATUS AND PROCEDURES
2.1 Experimental flume
New experiments were performed a 3.2 m long 1 m wide flume with an average bed slope S
o
≈ 0.37 (θ = 21.8º) and a
stepped invert (Fig. 1B). Previous experiments were conducted in the same channel by Chanson and Toombes
(2001,2002) and Gonzalez (2005). Waters were supplied from a large feeding basin leading to a sidewall convergent
with a 4.8:1 contraction ratio. The test section consisted of a broad-crested weir (1 m wide, 0.6 m long, with upstream
rounded corner) followed by ten identical steps (h = 0.1 m, l = 0.25 m) made of marine ply. The stepped chute was 1 m
wide with perspex sidewalls followed by a horizontal concrete canal ending in a sump pit.
The water was delivered by a pump controlled with an adjustable frequency AC motor drive, enabling an accurate
discharge adjustment in a closed-circuit system. Further details and the full data set were reported by Carosi and
Chanson (2006).
2.2 Instrumentation
Clear-water flow depths were measured with a point gauge. The discharge was measured from the upstream head above
the crest with an accuracy of about 2%. The discharge measurements were derived from Gonzalez' (2005) detailed
velocity distribution measurements on the broad-crested weir.
The air-water flow properties were measured with two types of conductivity probes : single-tip and dual-tip probes
(Fig. 2). Basic air-water flow measurements were performed with the single-tip conductivity probes (needle probe
design). Figure 2A shows two single-tip conductivity probes side-by-side. Each probe consisted of a sharpened rod (∅
= 0.35 mm) coated with non-conductive epoxy set into a stainless steel surgical needle acting as the second electrode.
