Abstract
Intense turbulence develops in the twophase flow region of hydraulic jump, with a broad range of turbulent length and time scales. Detailed air–water flow measurements using intrusive phasedetection probes enabled turbulence characterisation of the bubbly flow, although the phenomenon is not a truly random process because of the existence of lowfrequency, pseudoperiodic fluctuating motion in the jump roller. This paper presents new measurements of turbulent properties in hydraulic jumps, including turbulence intensity, longitudinal and transverse integral length and time scales. The results characterised very high turbulent levels and reflected a combination of both fast and slow turbulent components. The respective contributions of the fast and slow motions were quantified using a triple decomposition technique. The decomposition of air–water detection signal revealed “true” turbulent characteristics linked with the fast, microscopic velocity turbulence of hydraulic jumps. The highfrequency turbulence intensities were between 0.5 and 1.5 close to the jump toe, and maximum integral turbulent length scales were found next to the bottom. Both decreased in the flow direction with longitudinal turbulence dissipation. The results highlighted the considerable influence of hydrodynamic instabilities of the flow on the turbulence characterisation. The successful application of triple decomposition technique provided the means for the true turbulence properties of hydraulic jumps.
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Abbreviations
 C :

Timeaveraged void fraction
 \(\overline{C}\) :

Decomposed timeaveraged void fraction of average signal component
 C′:

Decomposed timeaveraged void fraction of lowfrequency signal component
 C″:

Decomposed timeaveraged void fraction of highfrequency signal component
 C _{max} :

Local maximum timeaveraged void fraction in the shear flow region
 c :

Instantaneous void fraction
 \(\overline{c}\) :

Decomposed instantaneous void fraction of average signal component
 c′:

Decomposed instantaneous void fraction of lowfrequency signal component
 c″:

Decomposed instantaneous void fraction of highfrequency signal component
 d _{1} :

Inflow water depth immediately upstream of the jump toe (m)
 d _{2} :

Downstream water depth (m)
 F :

Bubble count rate (Hz)
 \(\overline{F}\) :

Decomposed bubble count rate of average signal component (Hz)
 F′:

Decomposed bubble count rate of lowfrequency signal component (Hz)
 F″:

Decomposed bubble count rate of highfrequency signal component (Hz)
 F _{max} :

Maximum bubble count rate in the shear flow region (Hz)
 Fr _{1} :

Inflow Froude number, \({Fr}_{ 1} {\, =\, }{{V_{ 1} } \mathord{\left/ {\vphantom {{V_{ 1} } {\sqrt {g \times d_{ 1} } }}} \right. \kern0pt} {\sqrt {g \times d_{ 1} } }}\)
 g :

Gravity acceleration (m/s^{2})
 h :

Upstream gate opening (m)
 L _{r} :

Length of jump roller (m), defined as the distance over which the freesurface level increased monotonically
 L _{X} :

Longitudinal integral turbulent length scale (m)
 L _{X}′:

Decomposed longitudinal integral turbulent length scale of lowfrequency signal component (m)
 L _{X}″:

Decomposed longitudinal integral turbulent length scale of highfrequency signal component (m)
 (L _{X}″)_{max} :

Maximum decomposed longitudinal integral turbulent length scale of highfrequency signal component (m)
 L _{xx} :

Advection length scale (m)
 L _{xx}′:

Decomposed advection length scale of lowfrequency signal component (m)
 L _{xx}″:

Decomposed advection length scale of highfrequency signal component (m)
 L _{Z} :

Transverse integral turbulent length scale (m)
 Q :

Flow rate (m^{3}/s)
 R _{xx} :

Normalised autocorrelation function
 R _{xx′} :

Normalised crosscorrelation function between leading and trailing phasedetection probe signals
 R _{xx′}″:

Decomposed crosscorrelation function between highfrequency signal component
 R _{xz} :

Normalised crosscorrelation function between sidebyside phasedetection probe signals
 Re :

Reynolds number, \(Re{ = }{{\rho \times V_{ 1} \times d_{ 1} } \mathord{\left/ {\vphantom {{\rho \times V_{ 1} \times d_{ 1} } \mu }} \right. \kern0pt} \mu }\)
 T :

Time lag for maximum crosscorrelation coefficient (s)
 T′:

Time lag for maximum decomposed crosscorrelation function of lowfrequency signal component (s)
 T″:

Time lag for maximum decomposed crosscorrelation function of highfrequency signal component (s)
 T _{X} :

Longitudinal integral turbulent time scale (s)
 T _{X}′:

Decomposed longitudinal integral turbulent time scale of lowfrequency signal component (s)
 T _{X}″:

Decomposed longitudinal integral turbulent time scale of highfrequency signal component (s)
 (T _{X}″)_{max} :

Maximum longitudinal integral turbulent time scale of highfrequency signal component (s)
 (T _{X}″)_{mean} :

Depthaveraged longitudinal integral turbulent time scale of highfrequency signal component (s)
 T _{xx} :

Autocorrelation time scale (s)
 T _{xx}′:

Decomposed autocorrelation time scale of lowfrequency signal component (s)
 T _{xx}″:

Decomposed autocorrelation time scale of highfrequency signal component (s)
 T _{xx′} :

Longitudinal crosscorrelation time scale (s)
 T _{xx′}′:

Decomposed longitudinal crosscorrelation time scale of lowfrequency signal component (s)
 T _{xx′}″:

Decomposed longitudinal crosscorrelation time scale of highfrequency signal component (s)
 T _{xz} :

Transverse crosscorrelation time scale (s)
 T _{Z} :

Transverse integral turbulent time scale (s)
 T _{0.5} :

Time lag for maximum autocorrelation coefficient (s)
 Tu :

Turbulence intensity
 Tu′:

Decomposed turbulence intensity of lowfrequency signal component
 Tu″:

Decomposed turbulence intensity of highfrequency signal component
 V :

Average air–water interfacial velocity (m/s)
 V′:

Decomposed interfacial velocity of lowfrequency signal component (m/s)
 V″:

Decomposed interfacial velocity of highfrequency signal component (m/s)
 V _{1} :

Average inflow velocity (m/s)
 v′:

Standard deviation of interfacial velocity (m/s)
 W :

Channel width (m)
 x :

(1) Longitudinal distance from the upstream gate (m)
(2) Signal of leading sensor of phasedetection probe
 x′:

Signal of trailing sensor of phasedetection probe
 x _{1} :

Longitudinal position of jump toe (m)
 Y _{90} :

Characteristic elevation where C = 0.9 (m)
 y :

Vertical distance from the channel bed (m)
 Δx :

Longitudinal separation distance between two phasedetection probe sensors (m)
 Δz :

Transverse separation distance between two phasedetection probe sensors (m)
 μ:

Dynamic viscosity (Pa × s)
 ρ:

Density (kg/m^{3})
 τ:

Time lag (s)
 τ_{0.5} :

Time lag between maximum and half maximum crosscorrelation coefficient (s)
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Acknowledgments
The authors thank Jason Van Der Gevel (The University of Queensland) for manufacturing the phasedetection probes. The financial support of the Australian Research Council is acknowledged.
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Wang, H., Felder, S. & Chanson, H. An experimental study of turbulent twophase flow in hydraulic jumps and application of a triple decomposition technique. Exp Fluids 55, 1775 (2014) doi:10.1007/s0034801417758
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Keywords
 Turbulence Intensity
 Void Fraction
 Hydraulic Jump
 Bubble Count Rate
 Integral Turbulent Length