a [m]

Assumed bubble length

T [s]

Transit time

c [m/s]

Speed of sound

T0 [s]

Time travelled through the TDX

d [m]

Horizontal bubble length

Tc [s]

Reference transit time for calibration

D ( U i 1 , U j 2 ) [s]

Distance function between trajectories from TDX1 and TDX2

u [-]

Number of points in an interpolated trajectory

E ( U i , j ) [s]

Distance function between pulse-echo and HSC trajectories


Pulse echo trajectory

f0 [s−1]

Basic frequency of the ultrasound

VF [-]

Void fraction

fPRF [s−1]

Pulse repetition frequency

w [s/m]

Scaling factor

Fa [m3/s]

Air flow rate

W' [s/m]

Scaling factor

Fw [m3/s]

Water flow rate

x [m]

Distance between bubble and TDX

h [-]

HSC1 trajectory length

x' [m]

Distance between TDX and the opposite side of the bubble

HSC trajectory

x12 [m]

Distance between TDX surfaces

M [V2]

Signal strength

xc [m]

Reference distance for calibration

N [-]

Number of cycles in a pulse

y [m]

Horizontal distance between the bubble and the TDX axis

n, m [-]

Number of points in a trajectory

ymax [m]

Maximum estimated y

l [-]

HSC2 trajectory length

ymin [m]

Minimum estimated y

R 2 y z [-]

Tilt of bubble trajectory on HSC2

z [m]

Vertical bubble position

t [s]

Measurement time (pulse count)

σ R y z [-]

Standard deviation of bubble tilt

Subscripts and Superscripts

i, j

Trajectory counter

1, 2

TDX or HSC counter



High-speed camera


Ultrasonic transducer