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Numerical Simulation of Supercooled Large Droplet Icing Phenomenon: A Review

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Abstract

The impingement and ice accretion of supercooled large droplets (SLD) on the aircraft surface is one of the crucial factors threatening flight safety. The movement and impingement of SLD have many unique characteristics that conventional small droplets do not own. Therefore, a large number of experimental and numerical studies about SLD have been carried out to explore its physical properties and simulation method. The distribution and motion characteristics of supercooled large droplet during the process of approaching to the aircraft are first reviewed in this paper. Then the governing equations of SLD under the framework of Lagrangian and Eulerian methods are analyzed and established. The unique phenomena of SLD such as water droplet deformation and breakup, droplet–wall interaction and re-impingement in the literature are analyzed. The research development and results of the droplet–wall interaction phenomenon have been discussed particularly, which is summarized and classified from three aspects: droplet splashing threshold, splashing model and the method of modification of governing equation. Finally, the establishment process and the corresponding modification of the icing model in SLD condition is given, and the related calculation results are exhibited to validate the numerical simulation methods of SLD. Some shortcomings in current research are presented and the aspects needed to be developed further in future studies for the acquisition of more accurate simulated results are also recommended.

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Abbreviations

\(Re\) :

Reynolds number; \(\rho Vd /\mu\) (−)

\(Re_{d}\) :

Droplet Reynolds number (−)

\(Re_{s}\) :

Splashing Reynolds number (−)

\(We\) :

Weber number; \(\rho V^{2} d /\sigma\) (−)

\(We_{b}\) :

Breakup Weber number (−)

\(We_{crit}\) :

Critical Weber number (−)

\(We_{s}\) :

Splashing Weber number (−)

\(Oh\) :

Ohnesorge number; \(\mu /\sqrt {\rho \sigma d} = \sqrt {We} /Re\) (−)

Ga:

Galilean number (−)

Ca:

Capillary number (−)

La :

Laplace number (−)

V :

Velocity vector (m/s)

d :

Diameter of droplet (μm)

g:

Acceleration of gravity (m/s2)

V t :

Terminal speed (m/s)

V slip :

Slip velocity of the droplet (m/s)

C D :

Drag coefficient (−)

C Dsphere :

Drag coefficient of non-deforming spherical droplet (−)

C Dsteady :

Drag coefficient for a solid sphere in steady state flow (−)

u :

Velocity component in the x direction of the coordinate system (m/s)

v :

Velocity component in the y direction of the coordinate system (m/s)

w :

Velocity component in the z direction of the coordinate system (m/s)

m d :

Mass of droplet (kg)

A d :

Cross section area of droplet (m2)

f :

Eccentricity function (−)

F :

External force (N)

x 1 :

Displacement of the droplet from its equilibrium position (m)

y 1 :

Dimensionless displacement of the droplet from its equilibrium position (−)

C b :

Dimensional constants in TAB model (−)

K 1 :

Ratio of the oscillation total energy to fundamental energy (−)

r 32 :

Fragment size denoted by Sauter mean radius (μm)

K 2 :

Liquid to gas density ratio (−)

N :

Liquid-to-gas dynamic viscosity ratio (−)

T :

Breakup time (s)

n :

Number density of droplet (m−3)

R nd :

Dimensionless roughness parameter (−)

R a :

Surface roughness parameter (μm)

k B :

Boltzmann’s constant (J/K)

T a :

Temperature of air (K)

M G :

Molecular weight of the gas (mol)

P :

Pressure of the ambient air (Pa)

f m :

Mass ratio of the splashing droplet to the original droplet (−)

f V :

Velocity ratio of the splashing droplet to the original droplet (−)

\(\Delta T_{s}\) :

Droplet collision contact time (s)

N s :

Number of splashing droplet

E ERE :

Normalized excess rebound energy parameter

D max :

Maximum spread diameter (μm)

F D :

Drag acting on the droplets (N)

F B :

Buoyancy acting on the droplets (N)

F G :

Gravity acting on the droplets (N)

F S :

Force introduced by splashing phenomenon (N)

S k :

Loss coefficients (−)

\(\bar{\varvec{H}}_{c,wall}\) :

Modified flux through the boundary of control volume on the wall

\(\dot{m}\) :

Mass flow rate (kg/s)

Q :

Fluid flux

h :

Thickness of water layer (m)

b :

Thickness of ice layer (m)

k:

Thermal conductivity [W/(m K)]

c pw :

Specific heat of water [J/(kg K)]

T s :

Temperature of solid surface (K)

T f :

Temperature of freezing point (K)

L f :

Specific latent heat of freezing (J/kg)

h cv :

Convective heat transfer coefficient between water layer and air [W/(m2 K)]

e:

Normal restitution coefficient of ice crystal (−)

St c :

Critical Stokes number (−)

St :

Stokes number (−)

μ :

Dynamic viscosity (N s/m2)

ρ :

Density (kg/m3)

\(\bar{\rho }\) :

Apparent density (kg/m3)

σ :

Surface tension (N/m)

δ:

Dimensionaless height of water film (−)

\(\theta_{o}\) :

Impact angle computed from the tangential direction of the surface (°)

\(\theta_{r}\) :

Reflected angle of splashing droplet (°)

\(\lambda\) :

Weighting coefficient (−)

\(\gamma\) :

Adiabatic constant of the gas (−)

n :

Normal direction

a :

Air

d :

Droplet

w :

Water

ice :

Ice

ic :

Ice crystal

e :

Evaporation

es :

Evaporation and sublimation

k :

Kinetic energy

\(\infty\) :

Free flow

o :

Original physical variable

\(disk\) :

Variable of disk droplet

\(sphere\) :

Variable of spherical droplet

stab :

Stable

l :

Vertical height of falling droplet

s :

Splashing

i :

Incident

re-im :

Re-impingement

crit :

Critical value

dry :

Dry surface

wet :

Wet surface

imp :

Impact

flowin :

Flow in

flowout :

Flow out

t :

Tangential component

n :

Normal component

SLD:

Supercooled large droplet

LWC:

Liquid water content

MVD:

Mean volume diameter

CFD:

Computational fluid dynamics

TAB:

Talyor analogy breakup model

DDB:

Droplet deformation and breakup model

AoA:

Angle of attack

VOF:

Volume of fluid

MOF:

Moment of fluid

PDA:

Phase-doppler-anemometer

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    Yihua Cao & Miao Xin

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Cao, Y., Xin, M. Numerical Simulation of Supercooled Large Droplet Icing Phenomenon: A Review. Arch Computat Methods Eng 27, 1231–1265 (2020). https://doi.org/10.1007/s11831-019-09349-5

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