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Study of moist air flow through the ludwieg tube

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The time-dependent behavior of unsteady condensation of moist air through the Ludwieg tube is investigated by using a computational fluid dynamics (CFD) work. The twodimensional, compressible, Navier-Stokes equations, fully coupled with the condensate droplet growth equations, are numerically solved by a third-order MUSCL type TVD finite-difference scheme, with a second-order fractional time step. Baldwin-Lomax turbulence model is employed to close the governing equations. The predicted results are compared with the previous experiments using the Ludwieg tube with a diaphragm downstream. The present computations represent the experimental flows well. The time-dependent unsteady condensation charac-teristics are discussed based upon the present predicted results. The results obtained clearly show that for an initial relative humidity below 30% there is no periodic oscillation of the condensation shock wave, but for an initial relative humidity over 40% the periodic excursions of the condensation shock occurs in the Ludwieg tube, and the frequency increases with the initial relative humidity. It is also found that total pressure loss due to unsteady condensation in the Ludwieg tube should not be ignored even for a very low initial relative humidity and it results from the periodic excursions of the condensation shock wave.

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Speed of sound

C p :

Specific heat at constant pressure [J/(kg.K)]

E s :

Total energy per unit volume [J/m3]

E, F :

Numerical flux

g :

Condensate mass fraction [-]


Tube height [mm]

h * :

Characteristic length [mm]

I :

Nucleation rate [l/(m3.s)]

J :


k :

Boltzmann constant [J/K]

L :

Latent heat [J/kg]

M :

Molecular weight [kg/kmol]

m :

Mass [kg]


Prandtl number

p :

Pressure [Pa]

p∞ :

Flat film equilibrium vapour pressure [Pa]


Source term

R, S :

Viscous term

Re :

Reynolds number [-]


Gas constant [J/(kg-K)]

γ :

Droplet radius [m]

γ c :

Critical droplet radius [m]


Initial degree of supersaturation [-]

t :

Time [s] or temperature [°C]

T :

Temperature [K]

U :

Conservation term

u, v :

Cartesian velocity components [m/s]

x, y :

Cartesian coordinates [m]

γ :

Ratio of specific heats [-]

Γ :

Accomodation coefficient of nucleation [-]

ξ :

Coefficient of surface tension [-]

μ :

Dynamic viscosity [Pa.s]

K :

Bulk viscosity

γ :

Coefficient of second viscosity

ο, η :

Generalized coordinates [-]


Condensation coefficient [-] λ

ρ :

Density [kg/m3]

Ő :

Surface tension [N/m]


Surface tension of an infinite flat-film [-]


Stagnation state


Plane surface

l :

Liquid or laminar state

m :


r :

Droplet radius

v :





Turbulent state


Dimensional quantity


Nondimensional quantity


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Correspondence to Seung-Cheol Baek or Soon Bum Kwon or Heuy-Dong Kim or Toshiaki Setoguchi or Sigeru Matsuo or Raghu S. Raghunathan.

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Baek, S., Kwon, S.B., Kim, H. et al. Study of moist air flow through the ludwieg tube. KSME International Journal 17, 2066–2077 (2003). https://doi.org/10.1007/BF02982447

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Key Words

  • Ludwieg Tube
  • Compressible Flow
  • Condensation Shock Wave
  • Periodic Flow
  • Non-Equilibrium Condensation
  • Supersonic Nozzle