ice accretion simulation on multi-element airfoils using ...ae716/jhmt2.pdf · s. o¨ zgen Æ m....

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


qgLFoB

ot¼ ki

Tf � Ts

B� kw

Q0 þ Q1Tf

kw � Q1h: ð54Þ


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


b

a

x (m)

x (m)


(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


-0.05 -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


b

a

x (m)

x (m)

y (m

)y

(m)


(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


-0.05 -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


b

a

x (m)

x (m)

y (m

)y

(m)


(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



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




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

)


-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

)


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

)


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

)


-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

)


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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