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 3.8E+5 to 7.1E+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 > 0.95–0.97 in which the distributions of droplet chord sizes and integral advection scales presented some marked differences with the rest of the flow.
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Abbreviations
- 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 \({d = {\int\nolimits_{C = 0}^{C = 0.90} {(1 - C)\,\hbox{d}y}}}\)
- d c :
-
critical flow depth (m): \({d_{\rm c} = \root 3\of{{\it{Q}_{\it w} ^{2} /(gW^{2})}}}\)
- 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/s2 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 = VT xx
- L xy :
-
transverse/streamwise air–water integral turbulent length scale (m): \({L_{{\rm xy}} = {\int\limits_{Y = 0}^{Y_{{\rm max}}} {(R_{{\rm xy}})_{{\rm max}}\,\hbox{d}Y}}}\)
- (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 (m3/s)
- Re :
-
Reynolds number defined in terms of the hydraulic diameter
- R xx :
-
normalised auto-correlation function
- 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 = sin θ
- 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): \({T_{{\rm xx}} = {\int\nolimits_{\tau = 0}^{\tau = \tau (R_{{\rm xx}} = 0)} {R_{{\rm xx}} (\tau)\,\hbox{d}\tau}}}\)
- T xy :
-
cross-correlation time scale (s): \({T_{{\rm xy}} = {\int\nolimits_{\tau = \tau (R_{{\rm xy}} = (R_{{\rm xy}})_{{\rm max}})}^{\tau = \tau (R_{{\rm xy}} = 0)} {R_{{\rm xy}} (\tau)\,\hbox{d}\tau}}}\)
- 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 /(dW)
- 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
- Δx :
-
streamwise separation distance (m) between sensor
- Δz :
-
transverse separation distance (m) between sensor
- μ:
-
dynamic viscosity (Pa s)
- μw :
-
water dynamic viscosity (Pa s)
- θ:
-
angle between the pseudo-bottom formed by the step edges and the horizontal
- ρ:
-
density (kg/m3)
- ρw :
-
water density (kg/m3)
- 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′/(2D o) + (y′ − 1/3)3 /(3D o)
- Ø:
-
diameter (m)
- w:
-
water flow
- xx:
-
auto-correlation of reference probe signal
- xy:
-
cross-correlation
- 90:
-
flow conditions where C = 0.90
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Acknowledgments
The writers acknowledges the technical assistance of Graham ILLIDGE and Clive BOOTH.
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Chanson, H., Carosi, G. Turbulent time and length scale measurements in high-velocity open channel flows. Exp Fluids 42, 385–401 (2007). https://doi.org/10.1007/s00348-006-0246-2
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DOI: https://doi.org/10.1007/s00348-006-0246-2