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Aircraft Icing
“2-D Ice accretion prediction”
Prof. Dr. Serkan ÖZGEN
Dept. Aerospace Engineering, METU
Fall 2015
Outline
• Flow field solution
• Droplet trajectories and collection efficiencies
– Governing equations
– Droplet impact
– Parameters effecting collection efficiency
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Boundary-layer calculations
Ice growth model; Extended Messinger Model
Modified geometry
Runback water
Single and multi-step approach
Validation results
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Flow Field Solution
The flow field solution is needed to determine:
• The velocity and pressure distribution on the
surface of the geometry for subsequent
boundary-layer calculations and determination of
aerodynamic performance.
• Off-body velocities to be used for droplet
trajectory calculations.
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Flow Field Solution
Alternatives:
• Panel method coupled with a boundary-layer
solution: less accurate but fast and cheap.
• Navier-Stokes: more accurate but slow and
expensive.
No siginificant improvement has been observed by
using a Navier-Stokes solver instead of a panel code
 use a Hess-Smith panel code.
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Flow Field Solution
• In the panel method, the geometry is divided into quadrilateral
panels each associated with a source singularity element
together with a vortex singularity that is constant for all panels.
• The strengths of the singularities are constants and are
unknowns of the problem. The developed computer program
uses N quadrilateral panels to solve for N+1 singularity
strengths using the flow tangency boundary condition at the
collocation points of the panels and an additional equation is
introduced for the Kutta condition. The collocation points are
the centroids of each panel.
• Once the singularity strengths are calculated, one can construct
a velocity potential and hence calculate the air flow velocity
components at any location in the flow field including the
boundaries of the geometry. The velocity components at a
given point are the x-, y-derivatives of the velocity potential
constructed at that point.
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Flow field solution
NACA 23012, α=-0.27o
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Flow field solution
NACA 23012, α=-1.05o
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Droplet trajectories and collection
efficiencies
For droplet trajectories, Lagrangian approach is used
with the following assumptions:
• Droplets are assumed to be spherical.
• The flow field is not affected by the presence of the
droplets.
• Gravity and aerodynamic drag are the only forces
acting on the droplets.
These assumptions are valid for droplet sizes where
𝑑𝑝 ≤ 500 𝜇𝑚.
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Droplet trajectories and collection
efficiencies
The governing equations for droplet trajectories are:
𝑚𝑥𝑝 = −𝐷 cos 𝛾,
𝑚𝑦𝑝 = −𝐷 sin 𝛾 + 𝑚𝑔,
𝛾=
𝑦𝑝 −𝑉𝑦
−1
tan
𝑥𝑝 −𝑉𝑥
,
1
2
2
𝐷 = 𝜌𝑉𝑟𝑒𝑙
𝐶𝐷 𝐴𝑝 ,
𝑉𝑟𝑒𝑙 =
(𝑥𝑝 − 𝑉𝑥 )2 +(𝑦𝑝 − 𝑉𝑦 )2
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Droplet trajectories and collection
efficiencies
Definitions of parameters:
𝐶𝐷 : droplet drag coefficient,
𝑉𝑥 , 𝑉𝑦 : components of the flow field velocity at the droplet
location,
𝑥𝑝 , 𝑦𝑝 : components of the droplet velocity,
𝑥𝑝 , 𝑦𝑝 : components of the droplet accelaration,
𝜌: atmospheric density,
𝐴𝑝 =
𝜋 2
𝑑 :
4 𝑝
droplet cross-sectional area.
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Droplet trajectories and collection
efficiencies
The drag coefficients of the droplets are calculated from:
𝑅𝑒
𝐶𝐷
24
= 1 + 0.197𝑅𝑒 0.63 + 2.6 ∗ 10−4 𝑅𝑒1.38 ,
𝑅𝑒
𝐶𝐷
24
= 1.699 ∗ 10−5 𝑅𝑒1.52 ,
𝑅𝑒 < 3500,
𝑅𝑒 > 3500.
𝑅𝑒 = 𝜌𝑉𝑑𝑝 𝜇
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Droplet trajectories and collection
efficiencies
Droplet impact:
𝑥𝑖+1 , 𝑦𝑖+1
𝑥𝑑𝑝 , 𝑦𝑑𝑝
𝑥𝑑 , 𝑦𝑑
𝑥𝑖 , 𝑦𝑖
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Droplet trajectories and collection
efficiencies
𝑙1 = 𝑥𝑑 − 𝑥𝑑𝑝
𝑛1 = 𝑦𝑑 − 𝑦𝑑𝑝
vector along particle trajectory
𝑙2 = 𝑥𝑖+1 − 𝑥𝑖
𝑛2 = 𝑦𝑖+1 − 𝑦𝑖
vector along panel
𝑙1 𝑡1 + 𝑙2 𝑡2 = 𝑥𝑖 − 𝑥𝑑𝑝
system of two equations
𝑛1 𝑡1 + 𝑛2 𝑡2 = 𝑦𝑖 − 𝑦𝑑𝑝
with two unknowns
𝑙1
𝑛1
𝑥𝑖 − 𝑥𝑑𝑝
𝑙2 𝑡1
= 𝑦 −𝑦
𝑡
𝑛2 2
𝑖
𝑑𝑝
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Droplet trajectories and collection
efficiencies
𝑡1 =
𝑡2 =
𝑛2 𝑥𝑖 −𝑥𝑑𝑝 −𝑙2 (𝑦𝑖 −𝑦𝑑𝑝 )
(𝑙1 𝑛2 −𝑙2 𝑛1 )
𝑙1 𝑦𝑖 −𝑦𝑑𝑝 −𝑛1 (𝑥𝑖 −𝑥𝑑𝑝 )
(𝑙1 𝑛2 −𝑙2 𝑛1 )
; along the trajectory,
; along the panel.
Additional constraints to prevent fake impact locations:
0 ≤ 𝑡1 ≤ 1,
0 ≤ 𝑡2 ≤ 1.
Exact impact location:
𝑥𝑖𝑚𝑝 = 𝑥𝑖 + 𝑙2 𝑡2 ,
𝑦𝑖𝑚𝑝 = 𝑦𝑖 + 𝑛2 𝑡2 ,
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Droplet trajectories and collection
efficiencies
Collection efficiency:
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Particle trajectories
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Droplet collection efficiency
(effect of airframe size)
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Droplet collection efficiency
(effect of droplet size)
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Droplet collection efficiency
(effect of freestream velocity)
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Droplet collection efficiency
(effect of angle of attack)
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Droplet trajectories and collection
efficiencies
Factors increasing collection efficiency:
• Smaller airframe size; geometry creates a smaller
obstacle for the incoming droplets and the deviation
of the particles away from the body is not sufficient
for them to avoid it.
• Greater droplet size; increases droplet inertia and
droplets follow ballistic trajectories  collection
efficiencies increase, impingement zone gets wider.
• Higher airspeed; increases droplet inertia.
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Boundary-layer calculations
• Integral boundary layer method is used for the
calculation of the convective heat transfer
coefficients.
• With this method, the details of the laminar and
turbulent boundary layers are calculated fairly
accurately.
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Boundary-layer calculations
• Transition prediction is based on the roughness Reynolds number:
𝑅𝑒𝑘 =
𝜌𝑈𝑘 𝑘𝑠
𝜇
𝑘𝑠 : roughness height,
𝑈𝑘 : local flow velocity at the roughness height:
𝑈𝑘
𝑈𝑒
=2
𝑘𝑠
𝛿
−2
𝑘𝑠 3
𝛿
+
𝑘𝑠 4
𝛿
+
1 𝛿 2 𝑑𝑈𝑒 𝑘𝑠
6 𝜈 𝑑𝑠 𝛿
1−
𝑘𝑠 3
𝛿
𝑈𝑒 : flow velocity outside the boundary-layer at the roughness
location.
𝑠: streamwise distance along the airfoil surface starting at the
stagnation point.
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Boundary-layer calculations
• Roughness height (NASA model):
𝑘𝑠
= 0.00117𝑘𝑉∞ 𝑘𝐿𝑊𝐶 𝑘 𝑇∞ 𝑘𝑑𝑝
𝛿
𝑘𝑉∞ = 0.4286 + 0.0044139𝑉∞ ,
𝑘𝐿𝑊𝐶 = 0.5714 + 0.2457 ∗ 1000𝐿𝑊𝐶 + 1.2571 ∗ (1000𝐿𝑊𝐶)2
𝑘 𝑇∞
𝑇∞
= 46.8384
− 11.2037
1000
𝑘𝑑𝑝 = 1 𝑖𝑓 𝑑𝑝 ≤ 20 𝜇𝑚,
𝑘𝑑𝑝 = 0.033 𝑖𝑓 20𝜇 ≤ 𝑑𝑝 ≤ 50𝜇𝑚,
𝑘𝑑𝑝
5 1
= − 𝑑𝑝 ∗ 106 𝑖𝑓 𝑑𝑝 > 50𝜇𝑚.
3 30
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Boundary-layer calculations
• Laminar boundary-layer thickness:
315
𝛿=
𝜃𝑙
37
• Laminar momentum thickness (Thwaites formulation):
𝜃𝑙2
0.45 𝑠 5
= 6
𝑈𝑒 𝑠 𝑑𝑠
𝜈
𝑈𝑒 (𝑠) 0
• For laminar flow, 𝑅𝑒𝑘 ≤ 600, method of Smith & Spalding for the
calculation of the convective heat transfer coefficient:
ℎ𝑐 =
0.296𝑘𝑈𝑒1.435
𝑠
𝜈 0 𝑈𝑒1.87 𝑠 𝑑𝑠
𝑘: thermal conductivity of air.
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Boundary-layer calculations
• For turbulent flow, 𝑅𝑒𝑘 ≥ 600, method of Kays & Crawford
for the calculation of the convective heat transfer coefficient:
ℎ𝑐 = Stρ𝑈𝑒 𝐶𝑝
• Stanton number:
𝑆𝑡 =
𝐶𝑓 /2
𝑃𝑟𝑡 + (𝐶𝑓 2) /𝑆𝑡𝑘
𝑃𝑟𝑡 = 0.9: turbulent Prandtl number.
• Roughness Stanton number:
𝑆𝑡𝑘 = 1.92𝑅𝑒𝑘−0.45 𝑃𝑟 −0.8
𝑃𝑟 = 0.72: laminar Prandtl number.
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Boundary-layer calculations
• Skin friction coefficient is calculated from Makkonen relation:
𝐶𝑓
0.1681
=
2
ln 864 𝜃𝑡 𝑘𝑠 + 2.568
2
• Turbulent momentum thickness:
0.036𝜈 0.2
𝜃𝑡 =
𝑈𝑒3.29
0.8
𝑠
𝑈𝑒3.86 𝑠 𝑑𝑠
+ 𝜃𝑡𝑟
0
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Boundary-layer calculations
• 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.
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Boundary-layer calculations
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Boundary-layer calculations
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Extended Messinger Model
See lecture notes.
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Modified geometry
• After the ice growth rates given by equation (10) for rime ice
and equation (52) for glaze ice are solved, an ice thickness
results for a given control volume (panel).
• In order to obtain the modified geometry with ice formation,
it is assumed that the ice grows perpendicularly to the
surface.
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Modified geometry
Ψ𝑖−1
𝑥𝑖+1 , 𝑦𝑖+1
Ψ𝑖
𝑥𝑖 , 𝑦𝑖
Ψ𝑖 : panel inclination angle
𝑥𝑖 = 𝑥𝑖 + 𝐵𝑖 sin
Ψ𝑖−1 +Ψ𝑖
2
𝑦𝑖 = 𝑦𝑖 + 𝐵𝑖 cos
Ψ𝑖−1 +Ψ𝑖
2
𝑥𝑖−1 , 𝑦𝑖−1
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Runback water
• Under glaze ice conditions, freezing fraction is less than one
(FF < 1), meaning that only a fraction of the impinging water
freezes.
• Remaining water may either flow downstream as runback
water or may be shed due to high shear and gravity.
• According to Fortin et al:
– Upper surface: All the unfrozen water passes to the neighboring
downstream panel as runback water,
– Lower surface: All unfrozen water is shed.
• Including the runback water introduces some modifications to
the Messinger Model formulation.
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Single and multi-step approach
• Computing the ice shapes in a single cycle or step for the
entire duration of the exposure may yield unrealistic and
incorrect ice shapes.
• This is particularly true for:
– Glaze ice conditions where runback water effect is prominent,
– Icing exposures where the icing conditions are varying, i.e. climbing
flight, descending flight.
– Long exposures.
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Single and multi-step approach
• In the current solution method, an unsteady problem is solved by a
quasi-steady approach.
• Therefore a continuous flow of water, impingement, ice and water
accumulation has to be analyzed in as small as possible intervals to
represent these faithfully but this increases computational time.
• With a multi-step approach, the effect of ice formation on the flow
field, droplet trajectories, ice formation and runback water are
taken into account to some degree.
• At each computational step, flow field solution, droplet trajectories,
collection efficiencies, energy terms and the ice formation routines
are repeated.
• Experience shows that Δ𝑡 = 60 − 120 seconds is a good
compromise between computational time and accuracy.
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Validation results
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Validation results (Ta=-27.8o C)
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Validation results (Ta=-19.8o C)
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Validation results (Ta=-13.9o C)
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Validation results (Ta=-6.7o C)
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Multi-step approach (Ta=-27.8o C)
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Multi-step approach (Ta=-19.8o C)
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Multi-step approach (Ta=-13.9o C)
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Multi-step approach (Ta=-6.7o C)
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Airfoil with flap
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Particle trajectories
(airfoil with flap)
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Ice shapes in two element airfoil
(δf=10o)
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Ice shapes in two element airfoil
(main airfoil, δf=10o)
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Ice shapes in two element airfoil
(flap, δf=10o)
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Ice shapes in two element airfoil
(δf=30o)
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Ice shapes in two element airfoil
(main airfoil, δf=30o)
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Performance variation
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