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Time-resolved two-dimensional X-ray densitometry of a two-phase flow downstream of a ventilated cavity

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Abstract

To measure the void fraction distribution in gas–liquid flows, a two-dimensional X-ray densitometry system was developed. This system is capable of acquiring a two-dimensional projection with a 225 cm2 area of measurement through 21 cm of water. The images can be acquired at rates on the order of 1 kHz. Common sources of error in X-ray imaging, such as X-ray scatter, image distortion, veiling glare, and beam hardening, were considered and mitigated. The measured average void fraction was compared successfully to that of a phantom target and found to be within 1 %. To evaluate the performance of the new system, the flow in and downstream of a ventilated nominally two-dimensional partial cavity was investigated and compared to measurements from dual-tip fiber optical probes and high-speed video. The measurements were found to have satisfactory agreement for void fractions above 5 % of the selected void fraction measurement range.

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Abbreviations

BFS:

Backward-facing step

ESF:

Edge spread function

FFT:

Fast Fourier transform

II:

Image intensifier

LSF:

Line spread function

TTL:

Transistor–transistor logic

PSF:

Point spread function

RMSD:

Root-mean-square deviation

SNR:

Signal-to-noise ratio

2D:

Two-dimensional

a i :

Fraction of light scatter contributed by term i

b i :

Measure of propagation distance of scattered light by term i

d :

Bubble diameter (m)

D :

The detected image

E :

The un-degraded image

f e :

Bubble probe sampling frequency (1/s)

H :

Height of the backward-facing step (127 mm)

\(\widehat{H}\) :

Fourier transform of h

h :

Point spread function (1/m2)

I :

Intensity of photon flux, number of photons per unit area and time (1/(m2s))

G :

Gray scale value of light intensity (1)

K :

Number of X-ray image frames

L :

Length (m)

l 12 :

Streamwise distance between the optical probe’s tips (m)

l L :

Latency length of the optical probe’s tips (m)

l c :

Chord length of the gas measured by the upstream tip of the optical probe (m)

N :

Total number of materials (1)

n b :

Number of gas structures detected on the upstream tip of optical probe

Q :

Injected air volume flow rate (m3/s)

Q det :

Image intensifier detection efficiency

q :

Non-dimensional gas flow rate, Q/UHW (1)

r :

Radial distance (m)

Re :

Reynolds number, UL/ν (1)

t :

Time (s)

t a :

Transit time of a gas structure between the two tips of the optical probe(s)

t aP :

Most probable transit time of gas structures (s)

T :

Measurement time of the bubble probe (s)

U :

Free-stream velocity at the BFS (m/s)

U a :

Gas velocity (m/s)

\(\overline{U}_{a}\) :

Statistical mean value of the gas velocity (m/s)

U aP :

Most probable velocity of the gas (m/s)

U w :

Water velocity (m/s)

W :

Width of the model (209.6 mm)

v :

Bubble probe signal voltage (V)

V :

Photon energy (eV)

x n :

Mass thickness of material, n (kg/m2)

x :

Coordinate along the test surface, zero at the BFS (m)

y :

Coordinate normal to the test surface (m)

z :

Spanwise coordinate (m)

α :

Void fraction (1)

δ :

Boundary layer thickness (m)

Δ:

Linear dimension of a square on imager

ε α :

Error in void fraction (1)

ρ :

Density (kg/m3)

θ :

Momentum thickness (m)

σ :

Surface tension (N/m)

ν :

Kinematic viscosity (m2/s)

μ n /ρ n :

Mass attenuation coefficient of material n (m2/kg)

a :

Air

i, j :

Number of term

inj:

Injection

k :

Frame number

m :

Mixture

min:

Minimum

max:

Maximum

n :

Material designator (air, water, mix, etc.)

OP:

Optical probe

w :

Water

0:

Initial

2:

Two-dimensional

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Acknowledgments

We acknowledge the assistance of Prof. Michael Flynn and Dr. Alexander Mychkovsky during the design of the X-ray densitometry system, and Mr. Christopher Haddad for his contribution in processing of the high-speed video images. The authors are grateful for the support of the Office of Naval Research under Grant N00014-10-1-0974 with Dr. L. P. Purtell, Program Manager.

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, MI, USA

    Simo A. Mäkiharju, Marc Perlin & Steven L. Ceccio

  2. Research Institute of the French Naval Academy, École Navale, Brest, France

    Celine Gabillet

  3. Korea Institute of Ocean Science and Technology, Daejeon, Korea

    Bu-Geun Paik

  4. Naval Surface Warfare Center, Carderock Division, Bethesda, MD, USA

    Natasha A. Chang

Authors
  1. Simo A. Mäkiharju
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  2. Celine Gabillet
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  3. Bu-Geun Paik
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  4. Natasha A. Chang
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  5. Marc Perlin
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  6. Steven L. Ceccio
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Corresponding author

Correspondence to Simo A. Mäkiharju.

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Mäkiharju, S.A., Gabillet, C., Paik, BG. et al. Time-resolved two-dimensional X-ray densitometry of a two-phase flow downstream of a ventilated cavity. Exp Fluids 54, 1561 (2013). https://doi.org/10.1007/s00348-013-1561-z

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