integral analyses of the convective heat transfer around ......ichmt international symposium on...

Proceedings of CHT-08ICHMT International Symposium on Advances in Computational Heat Transfer

May 11-16, 2008, Marrakech, Morocco

CHT-08-130

INTEGRAL ANALYSES OF THE CONVECTIVE HEAT TRANSFER AROUND ICEPROTECTED AIRFOILS WITH NON-ISOTHERMAL SURFACES

Guilherme Araujo Lima da Silva∗,§, Otavio de Mattos Silvares∗,∗∗ andEuryale Jorge Godoy de Jesus Zerbini∗

∗Escola Politecnica da Universidade de Sao Paulo, Sao Paulo, Brazil∗∗Instituto Maua de Tecnologia, Sao Caetano, Brazil

§Correspondence author. Email: [email protected]

ABSTRACT The heated airfoil operating under icing conditions has some important characteristicsthat differentiates the problem from the case of adiabatic airfoil subjected to ice growth. In pres-ence of thermal ice protection, the hypothesis of flow over isothermal surfaces, which is assumedby most classic icing codes, may not represent the operation satisfactorily. The present paper imple-mented successfully two modeling strategies that considers streamwise surface temperature variationsin thermal boundary-layer evaluation by integral procedure: 1) solution of the approximated enthalpythickness integral equation assuming flow over a non-isothermal surface; 2) application of superposi-tion principle to thermal boundary-layer solutions to represent the effects of flow over a surface withnon-uniform temperature distribution. The numerical results were compared with isothermal modelresults as well as the experimental results of flat plate in a wind tunnel and two NACA aifoils, the0012 and the 652-0016, operating in icing tunnel under clear air and icing conditions. The streamwisesurface temperature gradient, water evaporation rate variation and the presence of laminar-turbulenttransition, when occurring within the protected area, are effects that are represented adequately by themathematical models.

INTRODUCTION

Some authors [Gelder and Lewis, 1951, Sogin, 1954, Gent et al., 2003] have observed that the con-vective heat transfer coefficient is one of of most important and difficult parameter in aircraft winganti-ice thermal performance estimation. This relevance was also observed in ice shape prediction[Gent et al., 2000, Stefanini et al., 2007].

Most works, found in present research, applied boundary layer integral analysis to icing cylinders[Makkonen, 1985] and airfoils [Wright, 1995, Guffond and Brunet, 1988, Gent, 1990, Wright et al.,1997, Gent et al., 2000], where the main objective is to evaluate ice growth and predict its shape.Those papers use mathematical models that assumes laminar and turbulent flows over isothermal,fully rough icing surface with moderate pressure gradient and no evaporation effects. The laminar-turbulent transition is considered to occur abruptly, i.e., the flows goes from fully laminar to fullyturbulent at the onset position.

Few works applied the momentum and thermal boundary layer integral models to anti-ice simulationsatisfactorily. Morency et al. [1999b] developed the numerical code CANICE A that evaluates the

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heat transfer coefficient considering laminar flow over isothermal surface[Smith and Spalding, 1958],turbulent flow over smooth and non-isothermal surface[Ambrok, 1957] and abrupt laminar-turbulenttransition. Same authors developed other version of the code, CANICE B, and use the experimentaldata of heat transfer coefficient . Only this second code represented satisfactorily the surface tem-perature numerical results. The transient ice protection code developed in Henry [1989] applied theMakkonen [1985] boundary-layer model that is used in ONERA2D icing code[Guffond and Brunet,1988]. Al-Khalil et al. [2001] employed experimental data in his ANTICE code simulations. NeitherLEWICE [Wright, 1995] nor ANTICE heat transfer coefficients were used in validation process. Gentet al. [2003] reported difficulties when applying TRAJICE2 boundary layer model[Gent, 1990, Gentet al., 2000, Makkonen, 1985] to rotorcraft ice protection systems. The authors got overestimatedresults and recommended more research about heat transfer coefficient evaluation[Gent et al., 2003].

Other works tried differential modelling strategies to improve predictions accuracy since integralanalyses appeared to show significant limitations. So did Henry [1992], who used a two-dimensionalfinite difference code to evaluate heat transfer in ice protection transient operation. In the same way,Morency et al. [1999a] published the CANICE FD version that evaluates hair distribution with Ce-beci’s finite difference code[Cebeci and Bradshaw, 1984]. Those works provided numerical resultswith smaller deviation to experimental data than their integral counterparties. This fact may indicatethat finite difference strategies could represent more adequately momentum, heat and mass transferphenomena around heated airfoils operating under icing conditions. A three-dimensional finite ele-ment flow solver has been developed by Croce et al. [2002] to estimate conjugated conduction andconvection heat transfer on thermally protected aircraft wings.

OBJECTIVE

The objective of the present paper is to evaluate the convective heat transfer around ice protectedairfoils by adopting different types for momentum and thermal boundary-layer analysis: 1) non-isothermal integral procedures as developed by Ambrok [1957] for laminar and turbulent boundary-layers, which has been applied to aircraft anti-icing by Silva [2002], Silva et al. [2003]; 2) super-position of isothermal integral boundary layer evaluations in laminar [Lighthill, 1950] and turbulent[Reynolds et al., 1958a] regimes. The numerical results are compared with boundary-layer isothermalmodel results as well as experimental data to assess applicability of the modeling strategies proposedherein.

AIRFOIL ANTI-ICE MATHEMATICAL MODEL

This paper uses the anti-ice thermal model developed by Silva et al. [2007a,b], whom briefly de-scribed the mathematical model, presented some numerical code results and compared with exper-imental data as well as other codes results. The anti-ice system operation simulation applies theFirst Law of Thermodynamics to liquid water flow and airfoil surface and also the Conservation ofMass and Momentum to liquid water flow. The wetness factor estimation, by water film breakdownand rivulets formation, was based in other work [Silva et al., 2006] plus the assumption of constantrivulets spacing.

Solvers The anti-ice simulation problem requires the solution in a sequence of steps : (step 1) veloc-ity and pressure fields around the airfoil; (step 2) droplet trajectories; (step 3) momentum and thermalboundary layers to obtain the coupled heat and mass transfer over the airfoil solid surface and liquidwater flow; (step 4) First Law of Thermodynamics to the liquid water and airfoil solid surface plusthe Conservation of Mass and Momentum to the liquid water flow (film and rivulets) over the airfoil.Both flow field around airfoil and local collection efficiency data were provided by external numericalcodes (steps 1 and 2). The momentum and thermal boundary-layer are evaluated (step 3) in order to

estimate the heat and mass transfer around airfoil over non-isothermal and transpired surfaces with asmooth laminar-turbulent transition occurrence. With data from previous steps, the anti-ice mathemat-ical model (step 4) is able to predict operational parameters like solid surface temperatures, runbackmass flow rate and convection heat transfer coefficient distributions along the airfoil solid surface.The present paper presents modeling strategies for the thermal boundary-layer (step 3). All the othermodels (steps 1,2 and 4 and also momentum boundary-layer) are kept as presented in previous worksand are not included herein [Silva, 2002, Silva et al., 2003, 2006, 2007a].

Domains Figure 1 shows the coordinates system and the five domains used in the present anti-icemathematical model, which are: I) free stream flow; II) gaseous flow; III) momentum or thermalboundary layers; IV) water film flow; V) solid surface. By using this strategy for domain division, themathematical model can be organized and simplified. Only the convective heat transfer estimationis discussed herein, since the objective of the present paper is to assess alternatives for modelingstrategies of the domain (III).

Airfoil solid surface (V)

Water film flow (IV)

y

X

s

-s

Y

Free stream flow (I)

s

y Gaseous flow (II)

Momentum or Thermal boundary layers (III)

Figure 1. Thermal Anti-ice Model Domains

Numerical implementation Figure 2 shows the numerical implementation of anti-ice mathematicalmodel. The thermodynamic solver collects data from user and the from the external flow solver andstarts the calculation. A routine for momentum and thermal boundary layers evaluation provideshair and C f around the airfoil. Then the mass and momentum conservation equations are applied toliquid water film flow as well as the First Law of Thermodynamics is applied separately to airfoilsolid surface and liquid water film. The solution of the system of equations (Twall and mrun) in eachfinite volume is considered satisfactory when the convergence of heat and mass convection fluxes isverified. Thus, the equations are solved in all finite volumes at lower surface from stagnation point tothe airfoil trailing edge. Complementarily, the same solution process is carried out also along uppersurface. The execution of thermodynamic solver is repeated until the continuity of heat flux across thestagnation point is achieved. At the end of solution process, the code is able to estimate the anti-icesystem operational parameters such as solid surface temperatures, runback water mass flow rate and

convection heat transfer coefficient distributions as well as the end of of water film positions at airfoilsolid surface.

j=nstag

to nlower

j=nstag

to nupper

.

Solid surface temperature and runback water mass

flow distributions

until

converges in

Repeat convective heat flux

each finite volume

Repeat for all finite volumes : firstly in lower and after upper surfaces

stagnation heat flux converges

Repeat until

User Flight and icing conditions Heaters configuration and

power density

External solver Flow field and

impingement data

Input files

Mass and Momentum conservation of water film. First Law of

Thermodynamics applied to water and solid surface.

at stagnation

Boundary Layer subroutine

Thermodynamic solver

q ji = q j

i−1

q j= q j− 1

Figure 2. Brief description of numerical implementation

HEAT TRANSFER AROUND HEATED AND WETTED AIRFOILS

The heat transfer around thermally protected airfoil operating under icing conditions is mainly af-fected by heat and mass convection mechanisms but also by conduction, surface wetness factor, en-thalpy associated with runback water flow and droplets impingement.

Water Film Breakdown and Rivulets Formation With a thermal anti-ice activated, the waterdroplets impinge and form a thin water film at leading edge. Then the runback water flows to down-stream regions driven by pressure and shear forces applied by external flow around the airfoil. Thefilm thickness may vary streamwise due to effects of evaporation, external flow pressure gradient,shear stress or heating. If a critical thickness is reached, the water film breaks-up and forms rivulets.The change from film to rivulets flow pattern is marked by a decrease in wetted area because drypatches start to grow between rivulets and the airfoil surface becomes directly exposed to gaseousflow around airfoil. In summary, the rivulet flow affects the effectiveness of anti-ice system because

it decreases the area of heat transfer between water and airfoil surface, and also decrease the area ofheat and mass transfer between water and external flow.

From stagnation point to impingement region limits, the runback water is assumed to flow as a con-tinuous film. Downstream the limits, a wetness factor is calculated by using a rivulets formationmodel [Silva et al., 2006] that adopts the Minimum Total Energy criteria [Mikielewicz and Moszyn-ski, 1975, 1976]. It proposes four equations to find the critical film thickness, the rivulets wetnessfactor Fr, rivulet radius and center-to-center rivulets spacing: 1) conservation of mass in the transi-tion between film and rivulets flow patterns in streamwise direction; 2) conservation of total energyfrom film to rivulet in streamwise direction; 3) rivulet total energy minimization; 4) geometricalrelationships. In present model, the overall wetness factor F is composed by two contributions:

F = Fr ·Fs where 0 6 F 6 1 and F =Awet

Atotalwhere Atotal = Adry +Awet (1)

where the wetness factor Fr is defined as the ratio between the rivulet base width and the distancebetween two rivulets centers λ, Fs is the ratio of streamwise wetted distance by the finite volume totaldistance; Atotal is the total finite volume area exposed to gaseous flow around airfoil. Thus, F is usedto multiply Atotal associated with water and air convective heat and mass transfer terms in First Lawof Thermodynamics applied to both solid surface and runback water flow.

Mass Transfer Blowing Effect By using the convective mass transfer model of Spalding [1963],the water evaporation mass flux is calculated:

m′′evap = gm ·Bm and gm = St ·G ·Le2/3 · ln(1+Bm)Bm

(2)

where Bm is calculated by the following expressions:

Bm =xH2O,S− xH2O,G

xH2O,S−1(3)

The water-vapor partial pressures are evaluated at position S, just above the liquid water film surface,and position G, in the gaseous flow around airfoil. The heat transfer driving force of convectiveevaporative cooling is defined by Spalding [1963]:

Bh =m′′evap

St∗ ·G(4)

The effect of blowing on both laminar and turbulent convective heat transfer is accounted:

St∗

St=

ln(1+Bh)Bh

(5)

This is a coupled heat and mass transfer process where St∗ depends on Bh, Eq. (5), that dependson both m′′evap and St∗, Eq. (4). The iterative calculation process only finishes when First Law ofThermodynamics is satisfied in each finite volume.

Overall Heat Transfer Coefficient The overall heat transfer coefficient U is defined to take into ac-count the effects of convective heat transfer rate across solid-liquid and liquid-gas surfaces interfaces,runback water enthalpy net flux, water droplets impingement enthalpy and evaporation enthalpy:

U =qlost

1 ·∆s ·∆T(6)

qlost = R−1t ·1 ·∆s ·∆T − mevap · (hlv +hwater)+ mimp ·hd + min ·hin− mout ·hout (7)

THERMAL BOUNDARY-LAYER

Non-isothermal Model Silva et al. [2007a] adopted an boundary-layer integral analysis that con-siders flow over non-isothermal surfaces. At stagnation point, the local convective heat transfer isestimated by isothermal integral analysis developed by Smith and Spalding [1958]:

Nu0 =

0.246 ·Re∞ ·d(ue/

V∞

)d (s/c)

∣∣∣∣∣s=s0

1/2

(8)

Ambrok [1957] developed an original expression in order to evaluate laminar local convective heattransfer due to a flow over non-isothermal surfaces with moderate pressure gradient:

Nulam = 0.3 ·Res ·∆T ·(Z sstr

s0

ue ·∆T 2

νairds)−1/2

(9)

The local convective heat transfer in turbulent regime is evaluated by Ambrok [1957]:

Stturb = 0.0125 ·Re−0.25∆2,turb

·Pr1/2 (10)

Alike Narasimha [1990], the present paper assumes that the virtual origin of turbulent boundary co-incides with the transition onset, where the turbulent spots start to appear. It is assumed that inter-mittency γ and enthalpy thickness ∆2,turb start to be different than zero at transition onset str. In sum,the present papers considers that virtual origin of boundary-layer occurs at same position of turbulentbreakdown, where the turbulent spots starts to appear. The turbulent enthalpy thickness is estimatedby Ambrok [1957] approximated solution:

Re∆2,turb ·∆T =[

0.0156 ·Pr−1/2 ·µ−1air ·

Z s

str

G ·∆T 1.25ds]0.8

(11)

Isothermal Model Classic icing codes [Wright et al., 1997] use the integral analysis to evaluatelaminar heat transfer around isothermal surfaces of icing airfoils. That is an acceptable assumptionfor non-heated airfoils subjected to ice formation, since the exposed ice or airfoil surface equilibriumtemperatures are approximately constant. In Smith and Spalding [1958] model, the heat transfercoefficient hair,lam is estimated by evaluating the laminar conduction thickness ∆4,lam:

u2.87eν·∆2

4,lam = 11.68 ·sZ

s0

u1.87e ds and ∆4,lam =

kair

hair,lam(12)

The ∆2,turb of flow over isothermal surfaces can evaluated by a modified version of Ambrok [1957]integral analysis that considers a constant ∆T in Eq. (11):

Re∆2,turb =[

0.0156 ·Pr−1/2 ·µ−1air ·

Z s

str

G ·ds]0.8

(13)

Then, the result of Eq. (13) is replaced in Eq. (10) to estimate Stturb in isothermal model. It isassumed that intermittency γ and enthalpy thickness ∆2,turb start to be different than zero at transitiononset position str.

LAMINAR-TURBULENT TRANSITION

Silva et al. [2007a] adopted the work of Reynolds et al. [1958b] that defines the laminar-turbulenttransition region statistically by a mean position sm and a standard deviation length σ. Both St andC f within transition region are calculated by linear combination of the laminar C f ,lam and Stlam withturbulent C f ,turb and Stturb values.

Within the laminar-turbulent transition region, sm−2 ·σ≤ s≤ sm +2 ·σ, St is defined as:

St(s) = [1− γ(s)] ·Stlam(Res)+ γ(s) ·Stturb(Res) (14)

Analogically, the linear combination procedure is also applied to friction coefficient calculation C f ,i.e., the St(s) is replaced by C f (s) in Eq. (14). The turbulent flow probability γ(Res) is evaluated by:

γ(Res) =Z Res

−∞

(1

Res·√

2 ·π)· exp

(−

Res−Re2sm

2 ·Re2σ

)d(Res) (15)

THERMAL BOUNDARY-LAYER BY SUPERPOSITION APPLICATION

As emphasized by Smith and Spalding [1958], the integral analysis has limitations to predict heattransfer over non-isothermal surfaces since it considers that boundary-layer growth rate depends onlyon local flow conditions. In order to represent the flow history effect induced by streamwise tempera-ture gradient at the wall, the present paper proposes the application of the superposition procedures tothe thermal boundary-layer. The superposition principle was derived analytically by Lighthill [1950]for laminar regime, firstly applied to laminar heat transfer around airfoils with non-isothermal sur-faces by Spalding [1958] and extended to turbulent flow heat transfer by Reynolds et al. [1958b].

The approximation proposed by Ambrok [1957], to estimated the heat transfer in flow over non-isothermal surfaces, may provide non-satisfactory results for engineering purposes, mainly, whenlarge and rapid surface temperature variations occurs [Moretti and Kays, 1965]. More recently, au-thors have agreed that application of superposition principle in integral analysis provides results closeto those of finite difference boundary-layer solvers [Kays and Crawford, 1993, White, 2000, Cebeciand Bradshaw, 1984].

Laminar Regime The temperature difference distribution, ∆T , between the airfoil surface and airrecovery temperature, around the airfoil can be approximated by ramps and steps [Reynolds et al.,1958a]:

∆Tint(s) =N

∑n=1

mn(s−an)+J

∑j=1

b j (16a)

d(∆Tint)ds

(s) =N

∑n=1

mn (16b)

where mn is the ramp inclination at breakpoint n, located at position an; N is total breakpoint number;b j are the steps at breakpoints j, located at position l j; J is the total breakpoints number.

Thus, the laminar Stlam over non-isothermal surfaces is estimated by application of superposition tothe isothermal Stlam,iso values:

Stlam = Stlam,iso

[1+

s∆Tint

N

∑n=1

mnFA

(an

x

)+

1∆Tint

J

∑j=1

b jFS

(l j

x

)](17)

The auxiliary functions are defined:

FA

(an

x

)= (4/3) ·βr(2/3,4/3)−

(1− an

s

)(18a)

FS

(l j

x

)=

[1−(

l j

s

)3/4]−1/3

−1 (18b)

rn = 1−(an

s

)3/4(18c)

βr(2/3,4/3) =Γ(4

3rn)

Γ(2

3

)Γ(4

3rn + 23

) (18d)

where Γ(p) is the Gamma function and βr is the incomplete Beta function, which is the result of theStietjes superposition integral. Equation (17) is a first order approximation of the Lighthill [1950] su-perposition solution for the laminar heat transfer over surfaces with non-uniform surface temperaturedistribution.

The present paper estimates Stlam,iso around the airfoil by the procedure developed by Smith andSpalding [1958], which is adopted in most classic icing codes [Wright et al., 1997], and use it inthe Eq. (17). It is is important to emphasize that the approach adopted herein is similar to theone developed by Spalding [1958] to estimate the thermal boundary-layer with pressure gradient bysuperposition.

Turbulent Regime Reynolds et al. [1958a] developed and validated superposition procedures to beapplied in heat transfer in turbulent flow over non-isothermal surfaces:

Stturb = Stturb,iso

[1+

s∆Tint

N

∑n=1

mnFA

(an

x

)+

1∆Tint

J

∑j=1

b jFS

(l j

x

)](19)

with auxiliary functions:

FA

(an

x

)= (10/9) ·βr(8/9,10/9)−

(1− an

s

)(20a)

FS

(l j

x

)=

[1−(

l j

s

)9/10]−1/9

−1 (20b)

rn = 1−(an

s

)9/10(20c)

βr(8/9,10/9) =Γ(10

9 rn)

Γ(8

9

)Γ(10

9 rn + 89

) (20d)

The present authors follows the recommendations of Moretti and Kays [1965] and use a modifiedAmbrok [1957] procedure, Eqs. (13) and (10), to estimate the thermal boundary layer flow overisothermal surfaces.

SELECTED EXPERIMENTAL CASES

Flat Plate Tests Moretti [1964] measured Stturb of turbulent boundary-layer flow over flat plateswith step-wise and arbitrary wall temperature distributions for an extensive range of pressure gradi-ents. The author concluded that the superposition procedures allied with modified Ambrok [1957]model (isothermal) provided satisfactory results for all cases tested except for those with strong ac-celeration. The present paper uses runs 1 and 24. The run 24 that had one wall temperature steps andflow with V∞≈ 20 m/s, Ttot = 36◦C, P∞ = 102438 Pa and no pressure gradient. Also it uses run 24 thathad four temperature steps at plate wall subjected to flow with initial V∞ ≈ 13 m/s, Ttot = 35.5◦C,P∞ = 104707 Pa and approximately constant pressure gradient. The flat plate cases were chosento verify the superposition implementation and compare modelling strategies. Figure 3 shows thefreestream velocity and wall temperature experimental distributions for runs 1 and 24.

22

24

26

28

30

32

34

36

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 13

14

15

16

17

18

19

20

21

Wal

ll T

empe

ratu

re [C

]

Loca

l Vel

ocity

[m/s

]

x [m]

temperaturevelocity

(a) Run 1

26

27

28

29

30

31

32

33

34

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10

15

20

25

30

35

40

45

50

55

60

65

Wal

ll T

empe

ratu

re [C

]

Loca

l Vel

ocity

[m/s

]

x [m]

temperaturevelocity

(b) Run 24

Figure 3. Velocity and wall temperature test data for flat plate cases [Moretti, 1964]

NACA Airfoil Anti-ice Tests Gelder and Lewis [1951] conducted one of the first investigations ofthe heat transfer from airfoil in clear air and icing in closed circuit NACA Lewis icing tunnel. Thetests used a 1.839 m span by 2.438 m chord NACA 652-0016 airfoil that was adopted previously byNeel and Bergrun [1947] in ice protection flight experiments under similar electrical heating powerdistribution and icing conditions. The authors observed a forward movement of laminar-turbulenttransition induced by water impingement and freestream turbulence level that was higher in tunnelthan flight. Other important experimental evidence noticed was the heating and temperature distribu-tions affects the measured convective heat transfer coefficient significantly. The present paper usesthe NACA test case 8 under icing and clear air conditions. The liquid water content (LWC), me-dian volumetric diameter (MVD) along other icing tunnel and airfoil configuration are presented inTable 1.

Recent Airfoil Anti-ice Tests Al-Khalil et al. [2001] performed anti-icing experiments at closedcircuit Icing Research Tunnel at NASA Glenn Research Center facilities (former NACA Lewis),Cleveland, Ohio, USA, to measure surface temperature and overall heat transfer coefficient distri-butions. The test purpose was to validate ANTICE numerical code results that was developed bythem. This code is a version of NASA’s LEWICE [Wright, 1995] for thermal anti-ice simulation. Theairfoil tested was 1.828 m span by 0.914 m chord NACA 0012 profile with electronically controlledheaters. Each heater element in streamwise direction had one thermocouple, one thermoresistor sen-sor and one heat flux gauge installed. Two tests cases from Al-Khalil et al. [2001] experimental dataset are used herein: case 22A, that is an evaporative condition with runback ending upstream the im-pingement limits; and 67A, that is a wet case with more water running around leading edge. Table 1presents the experimental conditions for both cases.

Table 1.Airfoil Anti-ice Test Conditions

Gelder and Lewis [1951] Al-Khalil et al. [2001]

Parameter 8 icing 8 clear air 22A 67A

V∞, m/s 73.8 73.8 44.7 89.4Ttot , ◦C -6.1 -6.1 -7.6 -21.6α 0◦ 0◦ 0◦ 0◦

LWC, g · m−3 0.5 - 0.78 0.55MV D, µm 11 - 20 20airfoil NACA 652-0016 NACA 652-0016 NACA 0012 NACA 0012chord, m 2.438 2.438 0.914 0.914

Table 2.Boundary Layer Models

Non-isothermal Isothermal and Superposition

Regime author equation author equation

Laminar Ambrok [1957] Eq. (9) Smith andSpalding [1958]

Eq. (12), (19)

Turbulent Ambrok [1957] Eq. (10), (11) modifiedAmbrok [1957]

Eq. (10), (13)and (17)

Transitional Reynolds et al.[1958b]

Eq. (14), (15) Reynolds et al.[1958b]

Eq. (14), (15)

RESULTS

The mathematical models listed in Table 2 were implemented and integrated in anti-ice numericalcode. The results of each model were compared with experimental data sets of Moretti [1964], Gelderand Lewis [1951], Al-Khalil et al. [2001].

Figure 4 presents a comparison between predictions of: Ambrok non-isothermal, modified Ambrokisothermal, superposition principle application and experimental data of Moretti [1964] for flat plate.The analysis of Fig. 4(a) shows that the non-isothermal thermal boundary-layer and superpositionpredictions had acceptable deviations from experimental data. As expected, the isothermal integralanalysis does not represent satisfactorily the St distribution. On the other hand, only the superpositionnumerical predictions were closer to experimental data in run 24, which has a series of abrupt stepsin wall temperature and flow acceleration (pressure gradient variation).

The present code provided surface temperature distributions presented in Fig. 5(a) for case 22A of Al-Khalil et al. [2001] test conditions. The transition mean position, sm, and standard deviation, σ werefixed in non-isothermal, superposition and isothermal models as presented in Table 3. The results ofsuperposition, non-isothermal and isothermal models have the same trend, however, the first is closerto experimental data than the other ones. It is important to emphasize that the ANTICE numericalcode does not evaluate hair. Therefeore Al-Khalil et al. [2001] did not calculated hair and used the onlymeasured heat transfer coefficient distribution. Experimental laminar-turbulent transition position andextension were not measured in icing tunnel tests by Al-Khalil et al. [2001]. Figure 5(b) presents pre-dictions of overall heat transfer coefficient U for all models. The higher differences between models

0

0.001

0.002

0.003

0.004

0.005

0 0.5 1 1.5 2

St

x [m]

non-isothermal AmbrokSt test data

isothermal Ambroksuperposition

(a) Run 1

0

0.001

0.002

0.003

0.004

0.005

0 0.5 1 1.5 2

St

x [m]

non-isothermal AmbrokSt test data

isothermal Ambroksuperposition

(b) Run 24

Figure 4. Predicted Stturb distributions compared to flat plate experimental data of Moretti [1964]

Table 3.Transition Region Parameters for Non-isothermal, Superposition and Isothermal Models

s/c upper s/c lower

Case sm σ sm σ

22A 0.070 0.035 −0.080 0.04067A 0.070 0.007 −0.067 0.007

8 icing 0.070 0.035 −0.036 0.0188 clear air 0.110 0.055 −0.070 0.037

(*) theses cases have geometry, icing and flow parameters presented in Table 1

results are located where the water flow disappears, where the liquid runback water stops flowing.Figure 6(a) shows that convective heat transfer coefficient, hair, estimated by superposition model ismore sensible to changes in temperature surface than the Ambrok’s non-isothermal and, as expected,the isothermal. At the end of water flow position, the local temperature increases significantly, sincethe evaporative cooling process ceases and only convective heat transfer takes place. As the temper-ature peaks, caused by end of water film, were attenuated in superposition predictions, it decreaseddeviations between its results to experimental data. Figure 6(b) presents the runback water flow andFig. 8(a) the overall surface wetness factor F . Those results indicates that water evaporates close tothe impingement limits and flows as a continuous film. The end of runback flow, impingement andheating limits in upper and lower surfaces are presented in Table 4.

Figure 9 shows the temperature and overall heat transfer coefficients results for case 67A of Al-Khalil et al. [2001]. As expected in wet cases like 67A, the predicted hair, runback water flow andF distributions have smaller variations than in the fully evaporative case 22A. The most significantchanges were due to transition occurrence since the surface temperatures distribution has small vari-ations along wetted region in both upper and lower airfoil surfaces. However, Fig. 7(a) shows thatthe history of heat transfer coefficient around the airfoil was estimated slightly different by isother-mal, non-isothermal and superposition models. The distinct flow history, estimated by each model,affected the laminar-turbulent transition process, as show in Figs. 7(a) and 9(b). Both effects, localheat transfer coefficient distribution and transition process, caused the runback water flow to disap-pear more downstream in superposition model than other ones. The limits of wetted area predictedby each model can be verified in Figs. 7(b) and 8(b). For all models used in case 67A, the transition

-20

0

20

40

60

80

100

120

140

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Streamwise distance per airfoil chord, s/c

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iso

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experimentalU non-iso

U superU iso

(b) Overall and convective heat transfer coefficients

Figure 5. Case 22A - Present code predictions compared with experimental data and ANTICEnumerical results of Al-Khalil et al. [2001]

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Figure 6. Case 22A - Present code predictions compared with ANTICE numerical results ofAl-Khalil et al. [2001]

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superiso

(b) Runback water

Figure 7. Case 67A - Present code predictions compared with ANTICE numerical results ofAl-Khalil et al. [2001]

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Figure 8. Present code predictions for overall wetness factor - F

parameters were fixed in both upper and lower surfaces as shown in Table 3.

Table 4.Runback Flow, Impingement and Ice Protected Area Limits

s/c upper side s/c lower side

Case model runback impingement heaters runback impingement heaters

22A non-iso. 0.025 0.031 0.113 −0.025 −0.031 −0.102super 0.029 −0.027iso. 0.027 −0.025

67A non-iso. 0.071 0.037 0.113 −0.069 −0.037 −0.102super 0.077 −0.077iso. 0.071 −0.071

8 icing non-iso. 0.101 0.071 0.576 −0.131 −0.131 −0.190super 0.101 −0.131iso. 0.101 −0.131(*) theses cases have geometry, icing and flow parameters presented in Table 1

The classic anti-icing experiments of Gelder and Lewis [1951] was simulated with the present anti-icenumerical code. This legacy data set has not been used by researchers in icing field since a long time.It presents significant surface temperature variations due to asymmetrical and non-uniform electricalheating distribution. This unique characteristic is not usually found in modern ice protection systemsbut it collaborates to the present analysis about non-isothermal wall effects on thermal boundary-layerbehavior. The predicted surface temperatures for icing and clear air tests are shown respectively inFigs. 10(a) and 12(a). Most deviations, between numerical results and experimental data, may becaused, according to the present authors view, because of three factors: 1) measurement uncertaintiesin terms of thermocouples streamwise position, calibration and installation; 2) the authors did nothave heat flux gauges installed around airfoil and measured only the electrical power provided to theheaters, thus, the thermal losses were not determined experimentally; 3) the abrupt step in heating ats/c≈ 0.3 caused a significant effect on experimental hair that can not be reproduced neither by non-isothermal Ambrok model nor superposition due to integral analysis intrinsic limitations. The localtemperature distribution prediction is considered acceptable for ice protection system engineeringpurposes. Figure 10(a) shows one temperature experimental measurement close to stagnation pointthat could not be predicted for the condition 8-icing of Gelder and Lewis [1951]. By analyzingthe results for condition 8-clear air, which is shown in Fig. 12(a), and comparing with Fig. 10(a),the present authors suspects that the deviation between numerical results and experimental data nearstagnation may be caused by one or a combination of the reasons listed above.

The convective heat transfer coefficients predicted by present model, Figs. 10(b) and 12(b), agreedsatisfactorily with experimental data. In both clear air and icing conditions, the only point thatpresents higher deviations is at the region of the heating step at s/c≈ 0.3, where the power density issuddenly increased. Such heating step located far downstream the impingement limit is not commonin modern aircraft wing anti-ice systems. The superposition model predicted hair distribution withsame trend of experimental data, including a sharp but not too intense variation of hair at beginningof heating step. Both icing and clear air hair distributions around airfoil is shown in Fig. 13. Thesefigures and Table 3 are in agreement with Gelder and Lewis [1951] experimental observations thatturbulent regime was triggered just downstream the stagnation; the onset position in icing was locatedmore upstream than clear air condition; as well as extension of laminar-turbulent transition regionwas not negligible in both cases. The authors also indicated that during flight under natural icing with

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experimentalU non-iso

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Figure 9. Case 67A - Present code predictions compared with experimental data and ANTICEnumerical results of Al-Khalil et al. [2001]

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Figure 10. Icing condition 8 - Present code predictions compared experimental data of Gelder andLewis [1951]

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Figure 11. Icing condition 8 - Runback and surface wetness predictions

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Figure 12. Clear air condition 8 - Present code predictions compared experimental data of Gelderand Lewis [1951]

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Figure 13. Condition 8 - Convective heat transfer coefficient and electrical power densitydistributions

similar heating, airfoil configuration and ambient conditions, the onset occurred more downstreamand transition region is shorter than observed in tunnel.

Table 4 show the the runback, impingement and ice protected area limits for condition 8 under clearair and icing conditions. Only the condition 8-icing was run in tunnel under icing conditions, i.e.,with flow loaded with supercooled water droplets. Therefore there is no runback results in Table 4for condition 8 clear air. For same reason, the Fig. 11 presents only the runback and wetness factorresults for the condition 8-icing. As presented in Fig. 11(a), both impingement area and total amountof runback water in condition 8-icing have lower levels than cases 22A and 67A. As the temperaturedistribution in wet region, where runback water exists, does not have significant variation and theheat transfer coefficient of isothermal, non-isothermal and superposition models are approximatelythe same, the runback results does not differ also. Figure 11 shows that the impingement limits areclose to stagnation point, therefore, a very short region is covered by film flow, where wetness factorequals to unity F = 1. Also Fig. 11 shows that the extension of rivulets flows, where 0 < F < 1, isnegligible in both upper and lower airfoil sides.

CONCLUSIONS

The present work implemented heat transfer convective model and compared their results with differ-ent and recognized data sets as well as operational conditions. It was verified that both non-isothermalor superposition integral analyses can be applied successfully to wing thermal anti-ice numerical sim-ulation. On the other hand, the direct application of an isothermal model, as commonly adopted by iceprotection engineers and researchers, presents limitations for anti-ice system performance prediction.The superposition principle combined with isothermal integral boundary-layer models can providemore conservative results than non-isothermal and isothermal integral models due to its greater sen-sitivity to wall temperature variations.

Prediction improvements were noticed at end of water flow positions, high streamwise temperaturegradient regions, abrupt heating steps, end of thermally protected area and at wet regions, wherethe airfoil surface is fully (continuous film) or partially (rivulets) covered by water flow. In the wetcases, when temperature distribution has smaller variations than evaporative cases, the present workconcluded that laminar-transition transition occurrence is the most significant effect on heat transfercoefficient, surface temperature and water evaporation. On the other hand, the approximate thermalboundary-layer analysis over non-isothermal surfaces is considerably faster in terms of computationaltime, which gives more flexibility for conception and pre-design studies of ice protection systems.

The use of a legacy experimental data set, which has not been used for numerical code validationpurposes in recent literature, reaffirm the validity of those experiments, verify applicability of thepresent numerical tool and may demonstrate the robustness of the mathematical model to representthe physical phenomena.

The heated airfoil operating under icing conditions has some important characteristics that differenti-ates the problem from the case of adiabatic airfoil subjected to ice growth. In presence of thermal iceprotection, the boundary-layer flow over isothermal surfaces, hypothesis assumed by most classic ic-ing codes, may not represent the operation adequately. The streamwise surface temperature gradient,water evaporation rate variation and the occurrence of transition, within the protected area, are effectsthat must be represented adequately by the mathematical models.

Additionally, the present study indicates that non-isothermal integral models, which have been appliedby present authors, may predict wing anti-ice operational parameters, such as end of water film andtemperature distribution, with satisfactory accuracy for cases similar those tested herein. However, thedeviations obtained by integral models herein suggest that research about other calculation methodsshould be pursued as well as other models to approximate satisfactorily the phenomena found.

ACKNOWLEDGMENT

G. A. L. da Silva wishes to acknowledge Fundacao de Amparo a Pesquisa do Estado de Sao Paulo(FAPESP) for the financial support received by the doctoral grant 07/00419-0.

REFERENCES

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