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TRANSCRIPT
ORIGINAL
Ice accretion simulation on multi-element airfoils using extendedMessinger model
S. Ozgen Æ M. Canıbek
Received: 25 October 2007 / Accepted: 22 July 2008
� Springer-Verlag 2008
Abstract In the current article, the problem of in-flight
ice accumulation on multi-element airfoils is studied
numerically. The analysis starts with flow field computa-
tion using the Hess-Smith panel method. The second step is
the calculation of droplet trajectories and droplet collection
efficiencies. In the next step, convective heat transfer
coefficient distributions around the airfoil elements are
calculated using the Integral Boundary-Layer Method. The
formulation accounts for the surface roughness due to ice
accretion. The fourth step consists of establishing the
thermodynamic balance and computing ice accretion rates
using the Extended Messinger Model. At low temperatures
and low liquid water contents, rime ice occurs for which
the ice shape is determined by a simple mass balance. At
warmer temperatures and high liquid water contents, glaze
ice forms for which the energy and mass conservation
equations are combined to yield a single first order ordinary
differential equation, solved numerically. Predicted ice
shapes are compared with experimental shapes reported in
the literature and good agreement is observed both for rime
and glaze ice. Ice shapes and masses are also computed for
realistic flight scenarios. The results indicate that the
smaller elements in multielement configurations accumu-
late comparable and often greater amount of ice compared
to larger elements. The results also indicate that the multi-
layer approach yields more accurate results compared to
the one-layer approach, especially for glaze ice conditions.
1 Introduction
Ice accumulation on parts of the airframe is one of the
fundamental problems of aviation. Ice growth on wings,
tail surfaces, fuselage and other items like the engine
intakes and pitot tubes result in severe performance deg-
radation, thus threatening flight safety. For example,
modification of the wing shape due to accumulated ice
results in reduced lift together with increased drag and
weight. Ice formation on control surfaces results in serious
and often unpredictable degradations in the controllability
of aircraft. If an airplane is to fly in icing conditions, it
must demonstrate that it can operate safely in conditions
prescribed by Certification Authorities, like those defined
in Federal Aviation Regulations, Part 25, Sect. 25.1419.
Certification process may involve flight and/or laboratory
testing and numerical simulation. Numerical ice accretion
simulation reduces (but never totally replaces) the demand
for flight and laboratory testing.
Efforts towards understanding the effects of ice on
performance and flight mechanics started in the 1940s.
These were mainly based on experiments and in-flight
testing. Among the pioneering works, the published work
of Messinger [6] represents an important foundation and a
milestone in numerical ice accretion simulation. With the
advent of digital computers in the 1970s, theoretical
research was directed towards representative geometries
such as airfoils, wings and helicopter rotor blades. The
major contributors to the aircraft icing simulations are
S. Ozgen (&) � M. Canıbek
Turkish Aerospace Industries, Flight Sciences Department,
Middle East Technical University Technopolis,
06531 Ankara, Turkey
e-mail: [email protected]; [email protected]
M. Canıbek
e-mail: [email protected]
S. Ozgen
Department of Aerospace Engineering,
Middle East Technical University, 06531 Ankara, Turkey
123
Heat Mass Transfer
DOI 10.1007/s00231-008-0430-4
NASA Lewis Research Center (USA), Defence Research
Agency (DRA-UK), Office National d’Etudes et des
Recherches Aerospatiales (ONERA-France), Anti-Icing
Materials International Laboratory (AMIL-Canada) and
Italian Aerospace Research Center (CIRA-Italy), each
having developed an ice accretion simulation code.
Cebeci et al. [1] describe a numerical method for com-
puting ice shapes on airfoils and their effects on lift and
drag coefficients. The Interactive Boundary Layer Method
developed by Cebeci has been incorporated into the
LEWICE code of NASA to improve the accuracy of ice
shape predictions and to compute the performance char-
acteristics of airfoils.
A NASA Report [13] summarizes the results of a ten-
year collaborative research on ice accretion simulation
between NASA, DRA and ONERA. The report includes
the descriptions of the codes developed by these institu-
tions and the results obtained. The report also presents
comparisons with ice shapes obtained experimentally in the
NASA Lewis Icing Research Tunnel with a 21’’ chord
NACA 0012 airfoil.
Mingione and Brandi [7] present results on ice shape
simulation over multi-element airfoils. They describe and
compare different ways to solve the transient ice accretion
problem, i.e., single-step, multi-step and predictor-correc-
tor methods.
In a review paper, Gent et al. [3] present the background
and the status of analyses addressing aircraft icing problem.
Methods for water droplet trajectory calculation, ice
accretion prediction and aerodynamic performance degra-
dation are discussed and recommendations for further
research are made.
Myers [8] presents a one-dimensional mathematical
model, extending the original Messinger Model describing
ice growth. It is demonstrated that the model can also be
extended to two and three-dimensions. A modified version
of the two-dimensional extension proposed by Myers is
employed in the current study.
Myers et al. [9] discuss a mathematical model for water
flow in glaze ice conditions. Water flow can significantly
complicate the problem and can have a major impact on
final ice shapes. It has been pointed out that previous codes
cannot deal adequately with this issue. The model is
applied to ice accretion problem and results are presented
for ice growth and water flow driven by gravity, surface
tension and constant air shear.
Fortin et al. [2] propose an improved roughness model,
in which the water state on the surface is represented in the
form of beads, film or rivulets. The model is tested for
severe icing conditions at six different temperatures cor-
responding to dry, mixed and wet ice accretion.
Present study is an effort to predict ice shapes combin-
ing established approaches for flow field, droplet trajectory,
collection efficiency and ice accretion calculations. In this
context, a computer code is developed in FORTRAN
programming language. Inputs to the problem are the
ambient temperature Ta, freestream velocity V?, liquid
water content (LWC) of air qa, droplet median volume
diameter dp (MVD), total icing time texp, angle of attack aand the airfoil geometry. The liquid water content (LWC)
is the weight of liquid water present in a unit volume of air,
often expressed in g/m3. The droplet median volume
diameter (MVD), which is often given in lm (microns) is a
term used to describe the droplet size. It is the droplet size
at which one-half of the given volume consists of larger
droplets and one-half consists of smaller droplets.
The solution starts with the calculation of the pressure
distribution around the given airfoil shapes using a panel
method. The same calculation serves to determining air and
droplet velocities anywhere in the flow field. Droplets are
‘‘fired’’ at a plane far upstream (10 chords) and their tra-
jectories are calculated by integrating the equations of
motion in differential form in two dimensions. Distance
between two adjacent particles is taken as 10-4 m in the
present calculations. Impact locations and droplet collec-
tion efficiencies on the airfoil surface are thus found.
Prediction of the heat transfer coefficient distribution plays
an important role in icing predictions. Characteristics of the
viscous flow such as the skin friction coefficient distribu-
tion over the airfoil surface are determined by using
empirical relations either based on experimental results or
solutions of the Integral Boundary–Layer Equation. From
the obtained data, it is possible to calculate convective heat
transfer coefficients using Reynolds’ Analogy. Depending
on parameters like the freestream velocity, ambient tem-
perature, liquid water content and collection efficiency,
rime ice, glaze ice or mixed ice accumulates on the airfoil
surface. Rime ice prediction involves a simple mass bal-
ance since droplets freeze immediately on impact. Under
milder conditions glaze ice develops, involving a layer of
water lying on top of a layer of ice. There is evidence that
glaze ice is always preceded by a thin layer of rime ice and
the transition from rime to glaze ice is smooth, i.e.,
freezing fraction reduces smoothly from unity to the
equilibrium value, which is less than unity. In the original
Messinger method, this transition is sudden, resulting in an
underprediction of the ice thickness. Therefore, the
Extended Messinger Model used in this study reflects the
physics of the problem better. Under gravitational and/or
aerodynamic forces, water layer may flow downstream
(called runback water) or may be shed.
The calculations may be done either in one-layer mode,
where the ice shapes are predicted in one step for the entire
duration of texp, or in multi-layer mode where texp is divi-
ded into segments (or layers). In the multi-layer mode,
flowfield, droplet trajectory and ice calculations are
Heat Mass Transfer
123
repeated for each layer. This approach allows the effect of
ice shapes on flowfield and droplet trajectories to be taken
into account, thus reflecting the physics of the problem
more realistically. It also allows cases with varying ambi-
ent and icing conditions to be treated, like climbing flight
which is a novelty of the current study. This feature allows
icing computations to be performed for the entire flight
profile of an airplane. Another important aspect of the
present study is that, although multi-element airfoils have
been treated by other researchers in the past, an extensive
parametric study like the one presented in this study does
not exist to the authors’ best knowledge. Therefore, the
present study fills an important gap in the literature.
Section 2 describes the solution method, briefly
explaining the flowfield, droplet trajectory, droplet collec-
tion efficiency and convective heat transfer coefficient
calculations, and the Extended Messinger Model. Section 3
is devoted to code validation, where the ice shapes
obtained in the current study are compared to experimental
and numerical ice shapes reported in the literature. In
Section 4, several realistic icing scenarios are studied and
the resulting ice shapes for single and two-element con-
figurations are presented. Finally, Section 5 summarizes
the study and points out important conclusions.
2 Problem formulation and solution method
In this section, the method developed for ice accretion
calculations is summarized. A brief flowchart of the cal-
culation procedure and the developed program is presented
in Fig. 1.
2.1 Flow field solution: panel method
In order to determine the pressure distribution around the
airfoil and provide the air velocities required for droplet
trajectory calculations, a panel method [5] is employed in
this study. In this method, the geometry is discretized by
quadrilateral panels each being associated with a singu-
larity element of unknown but constant strength. The
developed code uses N panels to solve for N singularity
strengths using the flow tangency boundary condition at the
surfaces. A velocity potential can then be constructed for
any point in the flow field using the calculated singularity
strengths. The velocity components at the given point are
the x-, y- and z-derivatives of this velocity potential. The
velocity distribution around the airfoil is also used for
boundary-layer calculations in order to determine the
convective heat transfer coefficients. Results of the devel-
oped code are compared with experimental data [13] in
Fig. 2 for single and two-element cases. Although there is a Fig. 1 Flowchart of the present calculation procedure
Heat Mass Transfer
123
slight disagreement of the pressure coefficients especially
on the flap, the results are accurate enough for the purposes
of this study. The mentioned disagreement can be attrib-
uted to the fact that the flow may be separated close to the
trailing edge of the flap, a phenomenon that cannot be
modeled by the panel method used in this study.
2.2 Droplet trajectories and the collection efficiencies
The following assumptions are made for the formulation of
the equations of motion for the water droplets:
• Droplet sizes are small, hence remain spherical,
• The flowfield is not affected by the presence of the
droplets,
• Gravity and aerodynamic drag are the only forces
involved.
These assumptions are valid for dp B 500 lm. These
assumptions are safe, as droplet sizes of 25 lm or greater
are found in only 4% of encounters [4]. The governing
equations for droplet motion are:
m€xp ¼ �D cos c; ð1Þ
m€yp ¼ �D sin cþ mg; ð2Þ
c ¼ tan�1 _yp � Vy
_xp � Vx; ð3Þ
D ¼ 1=2qV2CDAp; ð4Þ
V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_xp � Vx
� �2þ _yp � Vy
� �2q
: ð5Þ
In the above expressions, CD is the droplet drag coefficient,
Vx and Vy are the components of the flow field velocity at
the droplet location and _xp; _yp; €xp; €yp are the components
of the droplet velocity and acceleration. Atmospheric
density and droplet cross-sectional area are denoted by qand Ap.
The drag coefficients of the droplets are calculated using
the following drag law [3]:
CD ¼ 1þ 0:197Re0:63 þ 2:6� 10�4Re1:38; Re� 3500;
CD ¼ ð1:699� 10�5ÞRe1:92; Re [ 3500; ð6Þ
where, Re = qVdp/l is the Reynolds number based on
droplet diameter dp and relative velocity V, while l is the
atmospheric viscosity. This parameter is calculated using
Sutherland’s viscosity law as a function of ambient
temperature [12]. The pattern of droplet impact on the
airfoil determines the amount of water that impacts the
surface and the region subject to ice growth. The local
collection efficiency is defined as the ratio of the area of
impingement to the area through which water passes at
some distance upstream of the airfoil. The local collection
efficiency can be defined as:
b ¼ dyo
ds� Dyo
Ds; ð7Þ
where dyo is the distance between two water droplets at the
release plane and ds is the distance between the impact
locations of the same two droplets on the airfoil, see Fig. 3.
Examples of particle trajectories are presented in Fig. 4 for
single and two-element configurations. Note that, for the
two-element configuration, the icing problem is influenced
by much larger number of droplets compared to the single-
element case.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
0
1
2
3
Present StudyExperimental
Cp
Cp
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-1
0
1
2
3
Present studyExperimental
a
b
x/c
x/c
Fig. 2 Comparison of pressure distributions obtained from panel
method and experiment (NACA 23012 section for airfoil and flap).
a Single element airfoil, a = -0.27�; b airfoil with an external flap,
a = -1.05�
Heat Mass Transfer
123
Effects of key parameters like chord, droplet size, air-
speed and angle of attack on the collection efficiency b are
illustrated in Fig. 5. Results in Fig. 5 suggest that smaller
airframes, lower speed and larger drop sizes increase the
collection efficiency, hence possibility of ice formation.
Smaller size increases collection efficiency as the body
creates a smaller obstacle for the incoming droplets and the
deviation of the droplets away from the body is not suffi-
cient for them to avoid it. In other words, when a droplet is
heading for a small body, it is more likely that it will
impact it compared to a case of a larger body. Greater
droplet size and airspeed increases the collection effi-
ciency, as both of these parameters increase droplet inertia.
The more inertia a droplet has, the more difficult it will be
to deviate it from the body. On the other hand, angle of
attack determines the region of ice accumulation. The
highest collection efficiency occurs at the lowest angle of
attack (in magnitude) depicted, although the corresponding
region subject to icing is the lowest.
In the developed tool, droplet trajectory calculations
consume more than 90% of the CPU time. In a multi-layer
calculation, as the droplet calculations are repeated at each
layer, total CPU time is proportional to the number of
layers. In this study, it is found that the best compromise
between computational time and accuracy is a four-layer
approach regardless of the exposure time, as higher number
of layers does not improve the accuracy in a significant
manner.
2.3 Calculation of convective heat transfer coefficients
The current study employs an Integral Boundary Layer
Method for the calculation of the convective heat transfer
coefficients. This method enables calculation of the details
of the laminar and turbulent boundary layers fairly accu-
rately. Transition prediction is based on the roughness
Reynolds number, Rek = qUkks/l, where ks is the rough-
ness height and Uk is the local airflow velocity at the
roughness height calculated from the following expression
[11]:
Uk
Ue¼ 2
ks
d� 2
ks
d
� �3
þ ks
d
� �4
þ 1
6
d2
ma
dUe
ds
ks
d1� ks
d
� �3
: ð8Þ
In the above expression, Ue is the flow velocity outside the
boundary-layer at the roughness location and s is the
streamwise distance along the airfoil surface starting at the
stagnation point. Roughness height is calculated from ks ¼ð4rwlw=qwFsÞ1=3
[13], where rw, qw and lw are the
surface tension, density and viscosity of water,
respectively. Fraction of the airfoil surface that is wetted
by water droplets is denoted by F, while s denotes local
surface shear stress. The boundary layer thickness is given
by [12]:
d ¼ 315
37hl: ð9Þ
Laminar momentum thickness is computed using
Thwaites’ formulation [12]:
h2l
m¼ 0:45
U6e
Z
s
0
U5e ds: ð10Þ
-1 -0.5 0 1-0.5
0
∆yo
∆s-0.25
x (m)
y (m
)
0.25
0.5
0.5
Fig. 3 Definition of collection efficiency
x(m)
y(m
)
-2 -1 0 1 2-1.2
-0.8
-0.4
0
0.4
0.8
1.2
y(m
)
-2 -1 0 1 2-1.2
-0.8
-0.4
0
0.4
0.8
1.2
x (m)
a
b
Fig. 4 Particle trajectories for a NACA 4412 airfoil, V? = 92.6 m/s,
a = 4o. a one-element; b two-element
Heat Mass Transfer
123
For laminar flow Rek B 600, the equation of Smith and
Spalding is adopted to calculate the convective heat
transfer coefficient [3]:
hc ¼0:296kU1:435
effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mR s
0U1:87
e dsq ; ð11Þ
where k is the conductivity of air. This parameter is cal-
culated by using viscosity computed from Sutherland’s
viscosity law, assuming constant Prandtl number and spe-
cific heat. Note that expression (11) is not dependent on
roughness.
For turbulent flow Rek [ 600, the method of Kays and
Crawford is employed [3]. The turbulent convective heat
transfer coefficient is evaluated from:
hc ¼ StqUeCp; ð12Þ
where Cp is the specific heat of air. The Stanton number
can be calculated from:
St ¼ Cf =2
Prt þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðCf =2Þ=Stk
p ; ð13Þ
where Prt = 0.9 is the turbulent Prandtl number. The
roughness Stanton number is calculated from:
Stk ¼ 1:92Re�0:45k Pr�0:8; ð14Þ
where Pr = lCp/k = 0.72 is the laminar Prandtl number.
The skin friction is calculated from the Makkonen relation:
Cf
2¼ 0:1681
lnð864ht=ks þ 2:568Þ½ �2: ð15Þ
The turbulent momentum thickness is computed from:
ht ¼0:036m0:2
U3:29e
Z
s
str
U3:86e ds
0
@
1
A
0:8
þhtr; ð16Þ
where htr is the laminar momentum thickness at transition
location.
β
-0.16 -0.12 -0.08 -0.04 0 0.040
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8c = 0.5 mc = 1.0 mc = 2.0 m
β
-0.2 -0.16 -0.12 -0.08 -0.04 0 0.040
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8d = 10 md = 20 md = 40 m
µpp
p
µµ
β
-0.16 -0.12 -0.08 -0.04 0 0.040
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8V = 50 m/sV = 100 m/sV = 200 m/s
inf
inf
inf
β
-0.12 -0.08 -0.04 0 0.04 0.08 0.120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
α = 0α = 4α = −6
o
oo
s/c s/c
s/c s/c
b
c d
a
Fig. 5 Effect of various parameters on the droplet collection efficiency (NACA 0012 airfoil). a Effect of chord; b effect of droplet size; c effect
of freestream velocity; d effect of angle of attack
Heat Mass Transfer
123
The boundary-layer calculations start at the leading edge
and proceed downstream using the marching technique for
the upper and lower surfaces of the airfoil. Transition is
fixed at the streamwise location where Rek = 600,
according to Von Doenhoff criterion.
2.4 Extended Messinger model
The ice shape prediction is based on the standard method of
phase change or the Stefan problem. The phase change
problem is governed by four equations: energy equations in
the ice and water layers, mass conservation equation and a
phase change condition at the ice/water interface [8]:
oT
ot¼ ki
qiCpi
o2T
oy2; ð17Þ
ohot¼ kw
qwCpw
o2hoy2
; ð18Þ
qi
oB
otþ qw
oh
ot¼ qabV1 þ _min � _me;s; ð19Þ
qiLFoB
ot¼ ki
oT
oy� kw
ohoy; ð20Þ
where h and T are the temperatures, kw and ki are the thermal
conductivities, Cpw and Cpi are the specific heats and h and
B are the thicknesses of water and ice layers, respectively.
In Eq. (19), qabV?, _min and _me;s are impinging, runback
and evaporating (or sublimating) water mass flow rates for a
control volume (panel), respectively. In Eq. (20), qi and LF
denote the density of ice and the latent heat of solidification
of water, respectively. Ice density is assumed to have two
different values for rime ice (qr) and glaze ice (qg), see
Table 1. The coordinate y is normal to the surface. In order
to determine the ice and water thicknesses together with the
temperature distribution at each layer, boundary and initial
conditions must be specified. These are based on the fol-
lowing assumptions [8]:
1. Ice is in perfect contact with the airfoil surface, which is
taken to be equal to the air temperature, Ta in this study:
T 0; tð Þ ¼ Ts ¼ Ta: ð21Þ
2. The temperature is continuous at the ice/water
boundary and is equal to the freezing temperature:
T B; tð Þ ¼ h B; tð Þ ¼ Tf : ð22Þ
3. At the air/water (glaze ice) or air/ice (rime ice)
interface, heat flux is determined by convection (Qc),
radiation (Qr), latent heat release (Ql), cooling by
incoming droplets (Qd), heat brought in by runback
water (Qin), evaporation (Qe) or sublimation (Qs),
aerodynamic heating (Qa) and kinetic energy of
incoming droplets (Qk):
Glaze ice :�kwohoy¼ QcþQeþQdþQrð Þ
� QaþQkþQinð Þ at y¼ Bþ h:
ð23Þ
Rime ice :�kioT
oy¼ QcþQsþQdþQrð Þ
� QaþQkþQinþQlð Þ at y¼ B:
ð24Þ
4. Airfoil surface is initially clean:
B ¼ h ¼ 0; t ¼ 0: ð25Þ
In the current approach, each panel constituting the
airfoil is also a control volume. The above equations are
written for each panel and ice is assumed to accumulate
perpendicularly to a panel. This is an extension of the one-
dimensional model described by Myers [8] to two-dimen-
sional, which is accomplished by taking mass and energy
terms due to runback water flow in the conservation
equations into account, see Eq. (19). The differences
between the present model and the one described by Myers
are subtle. The main difference is related to the multi-layer
approach, where the exposure time is subdivided into
segments or layers. In this case, the boundary condition
given in Eq. (25) is modified so that each layer except the
first one starts with a non-zero ice thickness. Another
Table 1 Parameter values used in the calculations
Symbol Definition Value
Cp Specific heat of air 1,006 J/kg K
Cpi Specific heat of ice 2,050 J/kg K
Cpw Specific heat of water 4,218 J/kg K
eo Saturation vapor pressure constant 27.03
g Gravitational acceleration 9.81 m/s2
ki Thermal conductivity of ice 2.18 W/m K
kw Thermal conductivity of water 0.571 W/m K
Le Lewis number 1/Pr
LF Latent heat of solidification 3.344 9 105 J/kg
LE Latent heat of evaporation 2.50 9 106 J/kg
LS Latent heat of sublimation 2.8344 9 106 J/kg
Pr Laminar Prandtl number of air 0.72
Prt Turbulent Prandtl number of air 0.9
e Radiative surface emissivity of ice 0.5–0.8
lw Viscosity of water 1.795 9 10-3 Pa s
qr Density of rime ice 880 kg/m3
qg Density of glaze ice 917 kg/m3
qw Density of water 999 kg/m3
rr Stefan–Boltzmann constant 5.6704 9 10-8
rw Surface tension of water 0.072 N/m
Heat Mass Transfer
123
difference which is less important is that the present model
takes the heat loss due to radiation into account.
2.4.1 Rime ice growth and temperature profile
Rime ice thickness can be obtained directly from the mass
conservation Eq. (19) as water droplets freeze immediately
on impact [8]:
B tð Þ ¼ qabV1 þ _min � _ms
qr
t: ð26Þ
It has been shown that, for ice thicknesses less than
2.4 cm (which is mostly the case), the temperature
distribution is governed by [8]:
o2T
oy2¼ 0: ð27Þ
Integrating the above equation twice and applying
conditions given in Eqs. (21) and (24) yields the
temperature distribution in the rime ice layer as:
TðyÞ ¼ Ts
þ Qa þ Qk þ Qin þ Qlð Þ � Qc þ Qd þ Qs þ Qrð Þki
y:
ð28Þ
2.4.2 Glaze ice growth
It has been shown that, if ice and water layer thicknesses
are less than 2.4 cm and 3 mm (which is the case for most
applications), respectively, the temperature distributions in
the ice and water layers are governed by [8]:
o2T
oy2¼ 0;
o2hoy2¼ 0: ð29Þ
After integrating above equation twice and employing
conditions (21) and (22), the temperature distribution in the
ice becomes:
TðyÞ ¼ Tf � Ts
Byþ Ts: ð30Þ
The temperature distribution in the water layer is
obtained by integrating Eq. (29) twice and employing
conditions (22) and (23):
hðyÞ ¼ Tf þQa þ Qk þ Qinð Þ � Qc þ Qd þ Qe þ Qrð Þ
kwðy
� BÞ:ð31Þ
Mass conservation Eq. (19) is integrated once to obtain
the expression for water height, h:
h ¼ qabV1 þ _min � _me
qw
ðt � tgÞ �qg
qw
ðB� BgÞ; ð32Þ
where Bg and tg are the ice thickness and the corresponding
time at which glaze ice first appears, respectively. When
Eq. (32) is substituted into the phase change condition in
Eq. (20), a first order ordinary differential equation for the
ice thickness is obtained:
qgLFoB
ot¼
ki Tf � Ts
� �
B
þ kwQc þ Qe þ Qd þ Qrð Þ � Qa þ Qk þ Qinð Þ
kw:
ð33Þ
During transition from rime ice to glaze ice, ice growth
rate must be continuous:
oB
ot
�
rime
¼ oB
ot
�
glaze
at B ¼ Bg or t ¼ tg ð34Þ
Using Eqs. (26) and (33) yields:
tg ¼qr
qabV1 þ _min � _msub
Bg: ð36Þ
In order to calculate the glaze ice thickness as a function
of time, Eq. (33) is integrated numerically, using a Runge–
Kutta–Fehlberg method.
2.4.3 Energy terms
The energy terms appearing in the above equations need to
be expressed in terms of the field variables. Although
convective heat transfer (Qc) and latent heat (Ql) are the
most prominent terms, all relevant energy terms are con-
sidered here, and used in the computer program developed.
Bg ¼ki Tf � Ts
� �
qabV1 þ _min � _msubð ÞLF þ Qa þ Qk þ Qinð Þ � Qc þ Qd þ Qe þ Qrð Þ ; ð35Þ
Heat Mass Transfer
123
In the subsequent formulation, Tsur is the temperature at the
ice surface (for rime ice) or the water surface (glaze ice).
• Convective heat transfer at the water surface (Qc):
Qc ¼ hcðTsur � TaÞ: ð37Þ
• Cooling by incoming droplets (Qd):
Qd ¼ qabV1CpwðTsur � TaÞ: ð38Þ
• Evaporative heat loss (Qe):
Qe ¼ veeoðTsur � TaÞ; ð39Þ
where ve is the evaporation coefficient and eo = 27.03.
Evaporation coefficient is expressed as [8]:
ve ¼0:622hcLE
CpPtLe2=3; ð40Þ
where Pt is the total pressure of the airflow.
• Sublimation heat loss (Qs):
Qs ¼ vseoðTsur � TaÞ; ð41Þ
Sublimation coefficient vs is expressed as [8]:
vs ¼0:622hcLS
CpPtLe2=3: ð42Þ
• Heat loss due to radiation (Qr):
Qr ¼ 4errT3a ðTsur � TaÞ; ð43Þ
where e is the surface emissivity and rr is the Stefan-
Boltzmann constant.
• Aerodynamic heating term Qað Þ :
Qa ¼rhcV2
12Cp
; ð44Þ
where r is the adiabatic recovery factor (r ¼ Pr1=2 for
laminar flow, r ¼ Pr1=3 for turbulent flow).
• Kinetic energy of incoming droplets (Qk):
Qk ¼ qabV1V212; ð45Þ
• Energy brought in by runback water (Qin):
Qin ¼ _minCpwðTf � TsurÞ; ð46Þ
where _min is the mass flow rate of the incoming runback
water.
• Latent heat of solidification (Ql):
Ql ¼ qrLFoB
ot: ð47Þ
With these definitions, it is possible to express
Eqs. (28), (31), (33) and (35) in terms of the airfoil
surface temperature (Ts) and ambient temperature (Ta)
only.
2.4.4 Rime ice temperature distribution
Equation (28) becomes:
TðyÞ ¼ Q0r þ Q1rTs
ki � Q1rByþ Ts; ð48Þ
where
Q0r ¼ qrLFoB
otþ qabV1
V212þ rhc
V21
2Cpa
þ qabV1CpwTa
þ hcTa þ 4errT4a þ vse0Ta þ _minCpwTf ;
ð49Þ
Q1r ¼ qabV1Cpw þ hc þ 4errT31 þ vse0 þ _minCpw: ð50Þ
2.4.5 Glaze ice temperature distribution and ice growth
rate
Equation (31) can be written as:
hðyÞ ¼ Q0 þ Q1Tf
kw � Q1hhþ Tf ; ð51Þ
where
Q0 ¼ qabV1V212þ rhc
V21
2Cpa
þ qabV1CpwTa þ hcTa
þ 4errT4a þ vee0Ta þ _minCpwTf ; ð52Þ
Q1 ¼ qabV1Cpw þ hc þ 4errT31 þ vee0 þ _minCpw: ð53Þ
Equation (33) can be written as:
qgLFoB
ot¼ ki
Tf � Ts
B� kw
Q0 þ Q1Tf
kw � Q1h: ð54Þ
Equation (35) can be written as:
Bg ¼ki Tf � Ts
� �
qgLFqabV1þ _min� _msub
qr
� �
þ Q0 þ Q1Tf
� �
: ð55Þ
2.4.6 Freezing fractions and runback water
Freezing fraction for a given control volume (or a panel in
this case) is the ratio of the amount of water that solidifies
to the amount of water that impinges on the control volume
plus the water that enters the panel as runback water.
Rime ice: FF ¼ qrB
qabV1 þ _minð Þt : ð56Þ
Glaze ice : FF ¼qrBg þ qg B� Bg
� �
qabV1 þ _minð Þt : ð57Þ
Runback water mass flow rate:
_mout ¼ 1� FFð Þ qabV1 þ _minð Þ � _me: ð58Þ
This becomes _min for the neighboring downstream
panel. It is assumed that, all unfrozen water passes to the
Heat Mass Transfer
123
next downstream panel for the upper surface. For the lower
surface, it is assumed that all the unfrozen water is shed [2].
2.4.7 Evaporating or sublimating mass
Evaporating or sublimating mass is given as [11]:
_me;s ¼0:7
Cpa
hcpv;sur � pv;1
P1
� �
: ð59Þ
pv,sur and pv,? are the vapor pressures at the ice or water
surface and the ambient air, respectively. These are
computed from [11]:
pv ¼ 3386 0:0039þ 6:8096� 10�6 �T2 þ 3:5579� 10�7 �T3� �
;
ð60Þ�T ¼ 72þ 1:8 T � 273:15ð Þ: ð61Þ
3 Code validation
In order to validate the developed tool, the obtained results
are compared with experimental ice shapes [10] over a
NACA 0012 airfoil. Four test cases with significantly
varying ambient temperatures are selected (Ta = -27.8, -
19.8, -13.9 and -6.7�C) for comparison. Numerical data
obtained by different research groups corresponding to
these conditions are available in the literature [14], which
is a reason why they were selected. The experimental and
numerical data have been reproduced from Wright et al.
[14]. Geometric and flow conditions corresponding to these
cases are presented in Table 2.
In Fig. 6, ice shapes corresponding to Ta = -27.8�C are
presented. Figure 6a compares the ice shapes obtained in
the current study with those reported by Olsen et al. [10].
The conditions are rime ice conditions as made evident by
the obtained ice shapes. As can be observed, the developed
code predicts the experimental shape and the iced region
very well. For the current study, the results are presented
for one-layer and four-layer calculations. One-layer cal-
culations predict a slightly higher ice volume, but both
predictions are similar and agree well with the experi-
mental shape.
In Fig. 6b, obtained ice shapes are compared with those
obtained numerically by DRA, NASA and ONERA,
respectively. All the codes (including the current one)
predict similar shapes, all agreeing fairly well with the
experimental one. As rime ice shapes are obtained from a
simple algebraic equation, the good agreement observed
here is expected.
In the case presented in Fig. 7, the temperature is higher,
but all other parameters remain the same as in Fig. 6.
Again, the results of the current study are presented using
one-layer and four-layer calculations. There is a marked
difference between the results of two approaches as illus-
trated in Fig. 7a. One-layer calculations predict a typical
glaze ice shape, while four-layer calculations predict a
typical rime ice shape. However, it is the ice shape pre-
dicted by the four-layer approach that agrees well with the
experimental shape.
In Fig. 7b, obtained ice shapes are compared with those
obtained numerically by DRA, NASA and ONERA,
respectively. The results of the current code and DRA
Table 2 Geometric and flow conditions for code validation
calculations
Variables Value
a, angle of attack 4�c, airfoil chord 0.53 m
V?, freestream velocity 58.1 m/s
p?, ambient pressure 95610 Pa
qa, liquid water content 1.3 g/m3
texp, exposure time 480 s
dp droplet diameter 20 lm
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)DRANASAONERA
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)Present Study (1 layer)
b
a
x (m)
x (m)
y (m
)y
(m)
Fig. 6 Comparison of ice shape predictions for NASA 27 case
(Ta = -27.8�C)
Heat Mass Transfer
123
agree well. The results of these two codes also agree better
with the experimental data compared to others.
Although the temperature is low in this case, the high
LWC obviously results in glaze ice at least in some
regions, hence runback water. One-layer calculation dem-
onstrates a horn shape, which is typical of glaze ice. This is
due to runback water that piles up and freezes at the
observed location. The convective heat transfer loss is high
there due to high local velocity, resulting in the shape
observed. However, one-layer calculation obviously over-
estimates the runback water and predicts the observed horn.
On the other hand, in four-layer calculations, the amount of
runback water is less for each layer, which obviously
freezes before piling up somewhere downstream, explain-
ing the absence of the horn and the smoother ice shape that
actually occurs.
Figure 8 illustrates the ice shapes obtained for a milder
condition corresponding to Ta = -13.9�C. Again, four-
layer calculations predict a fairly similar shape to the
experimental shape, while one-layer calculations mispre-
dict not only the shape but also the extent of the iced region
as can be seen in Fig. 8a.
Among the numerical data reported by other research
groups, the results reported by DRA are the closest to the
ones obtained by the current study and the experiments, as
can be observed in Fig. 8b.
Finally, Fig. 9 presents ice shapes corresponding to
Ta = -6.7�C. The high temperature combined with the
high LWC produce typical glaze ice conditions. The
experimental and numerical ice shapes also support this.
Again the ice shape obtained in the current study using a
four-layer approach agree well with the experimental
shape, reproducing the prominent horn just downstream of
the leading edge. All the numerical predictions capture this
feature well, as made evident in Fig. 9b.
4 Icing calculations for various flight scenarios
In this section, results of the calculations for various real-
istic flight conditions are presented. The scenarios
considered correspond to climb, cruise, loiter and several
descent regimes. The descent regimes include two-element
airfoil configurations with a main airfoil component and a
y(m
)
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)DRANASAMap 7
y(m
)
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)Present Study (1 layer)
b
a
x (m)
x (m)
Fig. 7 Comparison of ice shape predictions for NASA 28 case
(Ta = -19.8�C)
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)DRANASAONERA
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)Present Study (1 layer)
b
a
x (m)
x (m)
y (m
)y
(m)
Fig. 8 Comparison of ice shape predictions for NASA 29 case
(Ta = -13.9�C)
Heat Mass Transfer
123
flap. Different flap settings at different angles of attack are
analyzed. The flight conditions and the corresponding icing
parameters are presented in Table 3. The airfoil considered
is NACA 4412.
The LWC values are read from the so-called altitude
overlays, depicted in Fig. 10, which is reproduced from
Jeck [4]. The figure shows the observed limits for LWC in
stratiform clouds as a function of altitude above ground
level (AGL). The x-axis represents the horizontal extent in
nautical miles, defined as HE = V? * texp. The figure
represents data for MVD = 15 lm since 77% of all
droplets encountered in real atmospheric conditions are
within 15 ± 5 lm [4].
4.1 Climb scenario
In this scenario, flight phases of takeoff and climb to cruise
altitude is considered. It is assumed that the airplane is
climbing at a constant rate with an airspeed of 200 knots
(102.9 m/s). It is also assumed that the aircraft travels about
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)DRANASAONERA
-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175-0.05
-0.025
0
0.025
0.05
0.075
0.1CleanExperimentalPresent Study (4 layers)Present Study (1 layer)
b
a
x (m)
x (m)
y (m
)y
(m)
Fig. 9 Comparison of ice shape predictions for NASA 30 case
(Ta = -6.7�C)
Table 3 Atmospheric and flight conditions for Figs. 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18
h (ft) a (deg) V? (knots) Ta (�C) LWC (g/m3) HE (nm) texp (s)
Climb Layer 1: 5000 6 200 Layer 1: -2.5 Layer 1: 0.6 Layer 1: 20 Layer 1: 360
Layer 2: 10000 Layer 2: -7.5 Layer 2: 0.8 Layer 2: 20 Layer 3: 360
Layer 3: 15000 Layer 3: -12.5 Layer 3: 0.6 Layer 3: 20 Layer 3: 360
Layer 4: 20000 Layer 4: -17.5 Layer 4: 0.4 Layer 4: 20 Layer 4: 360
Cruise 10000 2 240 -7.5 0.384 80 1200
Loiter 10000 7 180 -7.5 0.832 10 1800
Descent Layer 1: 20000 6 200 Layer 1: -17.5 Layer 1: 0.4 Layer 1: 20 Layer 1: 360
Layer 2: 15000 Layer 2: -12.5 Layer 2: 0.6 Layer 2: 20 Layer 3: 360
Layer 3: 10000 Layer 3: -7.5 Layer 3: 0.8 Layer 3: 20 Layer 3: 360
Layer 4: 5000 Layer 4: -2.5 Layer 4: 0.6 Layer 4: 20 Layer 4: 360
Descent with flap 5000 0 or 8 160 -2.5 0.6 20 450
***********
****
**********
100 101 1020
0.2
0.4
0.6
0.8
1
1.2
1.4
h=10000 ft15000 ft
5000 ft
20000 ft
2500 ft
LWC
(g/m
^3)
HE(nm)
Fig. 10 Altitude overlays for meteorological data (reproduced from
[4]
Heat Mass Transfer
123
20 nautical miles in each 5,000 ft thick cloud layer centered
at 5,000, 10,000, 15,000 and 20,000 ft AGL. The airplane is
in continuous stratiform icing conditions from 2,500 ft
AGL to 22,500 ft AGL and the cloud base is at 0�C.
Assuming that temperature decreases at a rate of 1�C/
1,000 ft, the cloud ceiling is at -20�C. It is also assumed
that the droplets throughout the entire scenario have a MVD
of 15 lm. This scenario is the same as the one defined by
Jeck [4]. The details of the flight and cloud conditions for
this scenario are given in the first row of Table 3.
The computations are performed using the four-layer
approach defined above, each of the layers corresponding
to each of the 5,000 ft thick cloud layers. Such a sce-
nario could be treated only with a multi-layer approach
since the flight and icing conditions are varying with
time.
The ice shape obtained for this scenario is illustrated in
Fig. 11. The first part of the figure depicts the general
layout, while the second part shows a zoomed view of the
iced region. The ice shape is rather smooth, suggesting
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.48
-0.32
-0.16
0
0.16
0.32
0.48
-0.2 -0.1 0 0.1 0.2 0.3-0.12
0
0.12
0.24a b
x (m)x (m)
y (m
)
y (m
)
Fig. 11 Ice shape for a climb scenario
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.48
-0.32
-0.16
0
0.16
0.32
0.48
-0.2 -0.1 0 0.1 0.2 0.3-0.12
0
0.12
0.24a b
x (m)x (m)
y (m
)
y (m
)
Fig. 12 Ice shape for a cruise scenario
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.48
-0.32
-0.16
0
0.16
0.32
0.48
-0.2 -0.1 0 0.1 0.2 0.3-0.12
0
0.12
0.24a b
x (m)x (m)
y (m
)
y (m
)
Fig. 13 Ice shape for a loiter scenario
Heat Mass Transfer
123
that the accumulated ice is mostly rime. The ice mass
accumulation during this scenario is 2.55 kg/m.
4.2 Cruise scenario
In this scenario, the airplane is assumed to be cruising at
10,000 ft altitude with a velocity of 240 knots (123.5 m/s).
The ambient temperature is -7.5�C and the distance cov-
ered is 80 nm. From Fig. 10, the corresponding LWC is
0.384 kg/m3 assuming MVD = 15 lm throughout the
flight. The details of the flight and cloud conditions are
given in the second row of Table 3.
Figure 12 shows the details of the resulting ice shape
again using the four-layer approach. The ice shape is again
rather smooth, although it is significantly different from
the previous situation. The resulting shape suggests that
the ice formation mostly consists of rime ice, possibly due
to low LWC. The ice accumulation during this flight is
1.5310 kg/m.
4.3 Loiter scenario
In this scenario, the airplane is assumed to be orbiting at
an altitude of 10,000 ft with a velocity of 180 knots
(92.6 m/s). The ambient temperature is -7.5�C and the
horizontal extent of the cloud is assumed to be 10 nm,
yielding a LWC of 0.832. Again a MVD of 15 lm is
assumed for the entire flight. The details of the flight and
cloud conditions are given in the third column of Table 3.
In Fig. 13, the resulting ice shapes are illustrated. The ice
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.48
-0.32
-0.16
0
0.16
0.32
0.48
-0.2 -0.1 0 0.1 0.2 0.3-0.24
-0.12
0
0.12
0.24a b
x (m)x (m)
y (m
)
y (m
)
Fig. 14 Ice shape for a descent scenario
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.24
-0.18
-0.12
-0.06
0
0.06
a b
cx (m)x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 15 Ice shapes for a two-element configuration in a descent scenario (a = 0�, df = 30�)
Heat Mass Transfer
123
shape is significantly rough compared to the two previous
cases, suggesting that at least some portions of the
accumulated ice are glaze. This is expected as the low
ambient temperature and high LWC yield typical glaze
ice conditions. The total ice accumulation during this
scenario is 2.22 kg/m.
4.4 Descent scenario without flap
In this scenario, the airplane is assumed to descent from a
cruise altitude of 22,500–2,500 ft. It is assumed that the
airplane is descending at a constant rate with an airspeed of
200 knots (102.9 m/s). It is also assumed that the aircraft
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.24
-0.18
-0.12
-0.06
0
0.06
a b
cx (m)x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 16 Ice shapes for a two-element configuration in a descent scenario (a = 0�, df = 45�)
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.3
-0.24
-0.18
-0.12
-0.06
0
a b
cx (m)
x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 17 Ice shapes for a two-element configuration in a descent scenario (a = 0�, df = 60�)
Heat Mass Transfer
123
travels about 20 nautical miles in each 5,000 ft thick cloud
layer centered at 20,000, 15,000, 10,000, 5,000 ft AGL.
The airplane is in continuous stratiform icing conditions
from 22,500 ft AGL to 2,500 ft AGL and the cloud ceiling
is at -20�C. Assuming that temperature decreases at a rate
of 1�C/1,000 ft, the cloud base is at -0�C. It is also
assumed that the droplets throughout the entire scenario
have a MVD of 15 lm. This scenario is exactly the mirror
image of the climb scenario. The details of the flight and
cloud conditions are given in the fourth row of Table 3.
The computations are performed using the four-layer
approach defined above, each of the layers corresponding
to each of the 5,000 ft thick cloud layers. The ice shape
obtained is depicted in Fig. 14. The ice is typical rime,
since it follows the contours of the airframe well and is
rather smooth. The ice has a mass of 2.1 kg/m, which is
significantly less than the one predicted during the climb
scenario.
4.5 Descent scenarios with flap
Six scenarios with different angles of attack and flap set-
tings are treated in this section. Both the main airfoil and the
flap have a NACA 4412 section. In all of these scenarios the
airplane is assumed to descend from 7,500 ft AGL to
2,500 ft AGL. The atmospheric conditions during this
maneuver are represented by the conditions at 5,000 ft AGL
(which makes this scenario equivalent to cruising at 5000 ft
AGL). The speed of the airplane is 160 knots (82.3 m/s) and
the horizontal distance covered is 20 nm. The ambient
temperature and the LWC are -2.5�C and 0.6, respectively.
The ice shapes presented in Figs. 15, 16 and 17 are all for 0�angle of attack for flap angles of 30�, 45� and 60�, respec-
tively. A common feature of the ice shapes is that the main
airfoil element accumulates ice slightly downstream of the
leading edge at the bottom surface only, while the flap
accumulates ice over almost the entire bottom surface. It is
also noteworthy that, ice accumulates around the leading
edge for a low flap setting (df = 30�) but not for higher
settings (df = 45� and 60�). Table 4 shows the ice mass that
accumulates for these scenarios. It can be noticed that even
though the flap is much smaller in size compared to the
main airfoil, it accumulates as much ice as the airfoil or in
some cases higher amount of ice. Both the main airfoil and
the flap accumulate more ice as the flap angle is increased
for a = 0�.
Table 4 Ice masses for descent scenarios with flaps
Ice mass for main
airfoil (kg/m)
Ice mass for
flap (kg/m)
a = 0�df = 30� 0.46 0.73
df = 45� 0.99 0.75
df = 60� 1.04 0.81
a = 8�df = 30� 0.96 0.76
df = 45� 0.88 0.80
df = 60� 0.80 0.82
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.24
-0.18
-0.12
-0.06
0
0.06
a b
cx (m)
x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 18 Ice shapes for a two-element configuration in a descent scenario (a = 8�, df = 30�)
Heat Mass Transfer
123
Meanwhile, for a = 8�, the trends slightly change. The
ice shapes for these scenarios are depicted in Figs. 18, 19
and 20. The main airfoil element accumulates less ice as
the flap angle is increased. The flap still accumulates more
ice as its angle is increased, just like the case for a = 0�.
Here, ice accumulates around the leading edge not only for
the df = 30� case but also the df = 45� case.
5 Conclusions
Ice shape and ice mass predictions are performed over
single and two-element airfoil configurations using the
Extended Messinger Model. The validation study indicates
that the developed tool is capable of predicting ice shapes,
iced regions and ice masses very well, for a wide range
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.24
-0.18
-0.12
-0.06
0
0.06
a b
cx (m)x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 19 Ice shapes for a two-element configuration in a descent scenario (a = 8�, df = 45�)
-0.4 0 0.4 0.8 1.2 1.6 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.2 -0.1 0 0.1 0.2 0.3 0.4-0.12
-0.06
0
0.06
0.12
0.18
0.24
1.4 1.5 1.6 1.7 1.8-0.3
-0.24
-0.18
-0.12
-0.06
0
a b
cx (m)x (m)
x (m)
y (m
)
y (m
)
y (m
)
Fig. 20 Ice shapes for a two-element configuration in a descent scenario (a = 8�, df = 60�)
Heat Mass Transfer
123
of0ambient temperatures, atmospheric conditions and
geometries.
It is shown that, especially for mild icing conditions (i.e.,
high ambient temperatures and high LWC) multi-layer
approach yields significantly more accurate ice shapes.
Such an approach is also shown to simulate icing conditions
that vary with time like climbing or descending flight. Icing
calculations in varying conditions is being reported for the
first time, according to the authors’ best knowledge.
In two-element cases, it is observed that the smaller ele-
ment, i.e., the flap experiences icing over its entire bottom
surface and accumulates a comparable ice mass as the main
airfoil element itself. This is primarily due to the size-
dependence of the collection efficiency. The collection
efficiency increases significantly with decreasing size. This
may be a reason explaining why aircraft control character-
istics often become unpredictable when there is ice adhering
to the control surfaces, as due their smaller size, the tail
surfaces will start accumulating ice before the main wing.
The developed tool could be used for designing proper
de/anti-icing equipment on aircraft as ice masses and iced
regions are fairly well predicted.
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