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An experimental study of turbulent two-phase flow in hydraulic jumps and application of a triple decomposition technique

Abstract

Intense turbulence develops in the two-phase flow region of hydraulic jump, with a broad range of turbulent length and time scales. Detailed air–water flow measurements using intrusive phase-detection probes enabled turbulence characterisation of the bubbly flow, although the phenomenon is not a truly random process because of the existence of low-frequency, pseudo-periodic 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 high-frequency 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 :

Time-averaged void fraction

\(\overline{C}\) :

Decomposed time-averaged void fraction of average signal component

C′:

Decomposed time-averaged void fraction of low-frequency signal component

C″:

Decomposed time-averaged void fraction of high-frequency signal component

C max :

Local maximum time-averaged 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 low-frequency signal component

c″:

Decomposed instantaneous void fraction of high-frequency 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 low-frequency signal component (Hz)

F″:

Decomposed bubble count rate of high-frequency signal component (Hz)

F max :

Maximum bubble count rate in the shear flow region (Hz)

Fr 1 :

Inflow Froude number, \({Fr}_{ 1} {\, =\, }{{V_{ 1} } \mathord{\left/ {\vphantom {{V_{ 1} } {\sqrt {g \times d_{ 1} } }}} \right. \kern-0pt} {\sqrt {g \times d_{ 1} } }}\)

g :

Gravity acceleration (m/s2)

h :

Upstream gate opening (m)

L r :

Length of jump roller (m), defined as the distance over which the free-surface level increased monotonically

L X :

Longitudinal integral turbulent length scale (m)

L X′:

Decomposed longitudinal integral turbulent length scale of low-frequency signal component (m)

L X″:

Decomposed longitudinal integral turbulent length scale of high-frequency signal component (m)

(L X″)max :

Maximum decomposed longitudinal integral turbulent length scale of high-frequency signal component (m)

L xx :

Advection length scale (m)

L xx′:

Decomposed advection length scale of low-frequency signal component (m)

L xx″:

Decomposed advection length scale of high-frequency signal component (m)

L Z :

Transverse integral turbulent length scale (m)

Q :

Flow rate (m3/s)

R xx :

Normalised auto-correlation function

R xx′ :

Normalised cross-correlation function between leading and trailing phase-detection probe signals

R xx′″:

Decomposed cross-correlation function between high-frequency signal component

R xz :

Normalised cross-correlation function between side-by-side phase-detection probe signals

Re :

Reynolds number, \(Re{ = }{{\rho \times V_{ 1} \times d_{ 1} } \mathord{\left/ {\vphantom {{\rho \times V_{ 1} \times d_{ 1} } \mu }} \right. \kern-0pt} \mu }\)

T :

Time lag for maximum cross-correlation coefficient (s)

T′:

Time lag for maximum decomposed cross-correlation function of low-frequency signal component (s)

T″:

Time lag for maximum decomposed cross-correlation function of high-frequency signal component (s)

T X :

Longitudinal integral turbulent time scale (s)

T X′:

Decomposed longitudinal integral turbulent time scale of low-frequency signal component (s)

T X″:

Decomposed longitudinal integral turbulent time scale of high-frequency signal component (s)

(T X″)max :

Maximum longitudinal integral turbulent time scale of high-frequency signal component (s)

(T X″)mean :

Depth-averaged longitudinal integral turbulent time scale of high-frequency signal component (s)

T xx :

Auto-correlation time scale (s)

T xx′:

Decomposed auto-correlation time scale of low-frequency signal component (s)

T xx″:

Decomposed auto-correlation time scale of high-frequency signal component (s)

T xx′ :

Longitudinal cross-correlation time scale (s)

T xx′′:

Decomposed longitudinal cross-correlation time scale of low-frequency signal component (s)

T xx′″:

Decomposed longitudinal cross-correlation time scale of high-frequency signal component (s)

T xz :

Transverse cross-correlation time scale (s)

T Z :

Transverse integral turbulent time scale (s)

T 0.5 :

Time lag for maximum auto-correlation coefficient (s)

Tu :

Turbulence intensity

Tu′:

Decomposed turbulence intensity of low-frequency signal component

Tu″:

Decomposed turbulence intensity of high-frequency signal component

V :

Average air–water interfacial velocity (m/s)

V′:

Decomposed interfacial velocity of low-frequency signal component (m/s)

V″:

Decomposed interfacial velocity of high-frequency 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 phase-detection probe

x′:

Signal of trailing sensor of phase-detection 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 phase-detection probe sensors (m)

Δz :

Transverse separation distance between two phase-detection probe sensors (m)

μ:

Dynamic viscosity (Pa × s)

ρ:

Density (kg/m3)

τ:

Time lag (s)

τ0.5 :

Time lag between maximum and half maximum cross-correlation coefficient (s)

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Acknowledgments

The authors thank Jason Van Der Gevel (The University of Queensland) for manufacturing the phase-detection probes. The financial support of the Australian Research Council is acknowledged.

Author information

Correspondence to Hang Wang.

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Wang, H., Felder, S. & Chanson, H. An experimental study of turbulent two-phase flow in hydraulic jumps and application of a triple decomposition technique. Exp Fluids 55, 1775 (2014) doi:10.1007/s00348-014-1775-8

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Keywords

  • Turbulence Intensity
  • Void Fraction
  • Hydraulic Jump
  • Bubble Count Rate
  • Integral Turbulent Length