Abstract
The interaction of a two-phase flow with a wedge where a stationary shock wave is initially settled is studied in a two-dimensional configuration. Before the introduction of the dispersed phase, the flow around the wedge is a supersonic one phase flow such as an attached stationary shock wave is present. Then, the dispersed phase is introduced upstream the initial position of the stationary shock wave. The purpose of this study is to point out two-phase and droplets break-up effects on the oblique shock wave. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using Riemann solver for the gas phase equations and also for the dispersed phase equations wich is an original approach due to the authors (Saurel et al. 1994). In addition to drag forces and heat and mass transfers, the process of droplets fragmentation based on the particle oscillation is considered.
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Abbreviations
- C vs :
-
constant-volume specific heat of dispersed phase [J\kg−1\k−1]
- D :
-
droplet diameter [m]
- C d :
-
drag coefficient
- F dx,y :
-
drag force per unit volume [N·m−3]
- h :
-
convection coefficient around the droplet [W·m−2·k−1]
- L v :
-
specific latent heat vaporisation [J·kg−1]
- \(\dot n_{br} \) :
-
term of droplets production [m−3 st−1]
- N:
-
number of droplet per unit volume [m−3]
- Nu:
-
Nusselt number
- P g :
-
gas pressure [Pa]
- Pr:
-
Prandlt number
- Oh:
-
Ohnesorge number
- Q :
-
convective heat transfer [W·m−3]
- R:
-
universal gas constant [J·kg−1·k−1]
- Re:
-
Reynolds number
- t :
-
time [s]
- T g,s :
-
steam temperature, drop temperature [K]
- U :
-
vector of conservative quantities
- u g,s :
-
x component of the velocity [m·s−1]
- v g,s :
-
y component of the velocity [m·s−1]
- We:
-
Weber number
- Wec:
-
critical number Weber
- W R :
-
solution of the Exact Riemann Problem
- x, y :
-
space coordinates
- α:
-
volume fraction
- δT j :
-
contour of elementT j
- δT j∩δT jk :
-
interface between elementsT j andT jk
- d nj :
-
slopes alongx at time t of conservative variables for elementT j
- δ nj :
-
slopes alongy at timet of conservative variables for elementT j
- Δt :
-
time step [s]
- ⎔τ br :
-
fragmentation time [s]
- Γ:
-
mass transfer [kg·m−3s−1]
- ΓEC:
-
kinetic energy transfer [kg·m−1s−3]
- Γ V:
-
momentum transfer due to mass transfer [kg·m−2·s−2]
- μ:
-
gas viscosity [kg·m−1s−1]
- π:
-
density [kg·m−3]
- ω i :
-
production rate of speciesi
- ω j :
-
center of gravity of elementT j
- λ g :
-
steam thermal conductivity [w·m−1k−1]
- α s :
-
droplet surface-tension [Nm−1]
- inj:
-
relative to the injection
- s:
-
particles
- g:
-
gas
- sx:
-
relative to the particles after break-up
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Utheza, F., Saurel, R., Daniel, E. et al. Droplet break-up through an oblique shock wave. Shock Waves 5, 265–273 (1996). https://doi.org/10.1007/BF02425219
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DOI: https://doi.org/10.1007/BF02425219