Experimental investigation and correlation development of jet impingement heat transfer with two rows of aligned jet holes on an internal surface of a wing leading edge

Ji YU,Long PENG,Xueqin BU,Xioin SHEN,*,Guiping LIN,Lizhn BAI

aSchool of Aeronautic Science and Engineering,Beihang University,Beijing 100083,China

bInstitute of Engineering Thermophysics,Chinese Academy of Sciences,Beijing 100190,China

KEYWORDS Correlation;Experiment;Jet impingement heat transfer;Two rows of aligned jet holes;Wing anti-icing

Abstract Extensive experimental studies on the heat transfer characteristics of two rows of aligned jet holes impinging on a concave surface in a wing leading edge were conducted,where 50000≤Rej≤ 90000,1.74≤ H/d≤ 27.5,66°≤ α≤ 90°,and 13.2≤ r/d≤ 42.03.The finding was that the heat transfer performance at the jet-impingement stagnation point with two rows of aligned jet holes was the same as that with a single row of jet holes or the middle row of three-row configurations when the circumferential angle of the two jet holes was larger than 30°.The attenuation coefficient distribution of the jet impingement heat transfer in the chordwise direction was so complicated that two zones were divided for a better analysis.It indicated that:the attenuation coefficient curve in the jet impingement zone exhibited an approximate upside-down bell shape with double peaks and a single valley;the attenuation coefficient curve in the non-jet impingement zone was like a half-bell shape,which was similar to that with three rows of aligned jet holes;the factors,including Rej,H/d and r/d,affected the attenuation coefficient value at the valley significantly.When r/d was increased from 30.75 to 42.03,the attenuation rates of attenuation coefficient increased only by 1.8%.Consequently,experimental data-based correlation equations of the Nusselt number for the heat transfer at the jet-impingement stagnation point and the distribution of the attenuation coefficient in the chordwise direction were acquired,which play an important role in designing the wing leading edge anti-icing system with two rows of aligned jet holes.

1.Introduction

Jet impingement has been widely used in many engineering applications to enhance the local heat transfer rate from the target surface.Notable applications of jet impingement are the metal annealing,glass tempering,food and paper drying,gas turbine blade and electronics cooling,and aircraft antiicing.Martin,1Jambunathan et al.2and Zuckerman et al.3reviewed the jet impingement heat transfer.Many researchers have studied the effect of the nozzle layout,the shape and size of the nozzle,the nozzle to impingement surface distance,the jet Reynolds number,the impingement angle,and the curvature of impinging surface on the jet impingement heat transfer for a better understanding of the physical mechanism.4–17

To ensure the flight safety in the icing conditions,anti-icing or de-icing system is essential for both civil and military aircraft,and wing leading edge hot-air anti-icing system with piccolo tube has been widely employed to date.Fig.1 shows the schematic of a typical hot-air anti-icing system in the wing leading edge,where,H is the piccolo tube-to-surface distance.The hot air from the engine jets through the holes on the piccolo tube to heat the surface of the wing leading edge when aircraft is flying over the icing cloud to avoid ice accretion.Fig.2 shows the schematic of the two rows of jet holes impinging on the internal surface of the hot-air anti-icing system,where,α is the jet impingement angle,θ is the circumferential angle of jet holes on the piccolo tube,x is the chordwise arc length from the geometry stagnation point,y is the coordinate in the chord direction.The jet impingement is much more complicated than that on a cylindrical concave surface due to its continuous variable curvature characters in the leading edge and the holes’layout on the piccolo tube(usually two or three rows of holes are applied)as shown in Fig.2.However,only very limited studies including the numerical study and the experimental study18–34have been reported thus far.

Fig.1 Schematic of wing leading edge hot-air anti-icing cavity with two rows of jet holes.

Wright18made a review on the heat transfer correlations of jet impingement that could be applicable to piccolo tube antiicing system,and the Goldstein correlation7was suggested to evaluate the Nusselt number for a piccolo tube anti-icing system.Fregeau,Saeed and Paraschivoiu19–21studied the thermal performance of a 3D hot-air jet flow impinging on a normal semicircular concave surface and surfaces of a typical aircraft wing/slat with a single row or two rows of staggered jet holes.The simulation results showed that the tube-to-surface distance,the circumferential angle of the jet holes and the jet Mach number affected the Nusselt number significantly.Liu and Feng14studied the flow and heat transfer of impingement cooling on internal turbine leading edge with a single row of circle jets by numerical method.They investigated the influence of the jet nozzle position and jet Mach number.Guan et al.22conducted the computational study to investigate the jet impingement heat transfer by a row of tab-excited jets on a wedge-shaped concave surface from a hot-air anti-icing configuration of an engine inlet strut.It was found that the optimum non-dimensional jet-offset distance was about 1.5 and the heat transfer was enhanced with the increase of tab number as well as tab penetration ratio.

Fig.2 Schematic of jet impingement on a concave surface from a piccolo tube.

In 1970,Jusionis23investigated experimentally the jet flow and heat transfer on an enclosed surface for aircraft antiicing applications using a piccolo tube with a single row of jet holes,from which the correlation of the average heat transfer coefficient over the target surface was acquired.Brown et al.24performed an experimental study to present a better understanding of the jet impingement heat transfer characteristics in an aircraft nacelle anti-icing system employing a piccolo tube with three rows of jet holes.A correction of the average heat transfer coefficient over the jet impingement region was developed.Papadakis et al.25–27conducted a parametric study to analyze the jet impingement heat transfer performance of a bleed air ice-protection system with three rows of aligned jet holes using both experimental and numerical methods.It was found that the tube-to-surface distance and the jet hole locations affected the impinging surface temperature significantly.In 2014,Imbriale et al.15studied the jet impingement heat transfer performance with a single row of jets impinging on a wing leading edge surface by experimental method,and a modified average Nusselt number over the impinging surface was proposed based on Meola’s correlation.28More recently,Guan et al.29–31performed experimental and numerical investigations to study the conjugated heat transfer on leading edge of a wedge-shaped concave wall internally impinged by hot jets from corrugated orifice plate29and from chevron nozzle.30,31It was found that the corrugated impinging plate had a significant impact on improving the conjugated heat transfer performance in the vicinity of concave wall leading edge.It was also found that the chevron jet could improve the heating effectiveness due to the jet core velocity increase and more intensive jet fluctuation caused by the presence of chevrons,especially under a small jet Reynolds number or a smaller jet-to-leading edge distance.Yang et al.32conducted experimental and computational study to investigate the heat transfer characteristics of a single one row jet impingement on the concave surface of the leading edge of a NACA0015 airfoil.Eight different turbulence models were investigated and verified.It was found that the heat transfer was enhanced with H/d=10,S/d=30,and θ =15°,where d is the hole diameter and S is the distance between the adjacent jets.Streamlines and velocity distributions were obtained to better understand the heat transfer results.In our previous work,the jet impingement heat transfer characteristics on a variable-curvature concave surface in a wing leading edge for anti-icing applications33,34were investigated.An optimal relative tube-to-surface distance of 4.5 was determined under Rej=51,021 for the jet impingement.33The optimal relative tube-to-surface distance was within 4–5.75,which decreased with increasing jet Reynolds number.34Meanwhile,the curves of the local Nusselt number exhibited different shapes for jet impingement with two or three rows of aligned jet holes on the piccolo tube due to different intensity of interference between adjacent air jets.33For the jet impingement with three rows of aligned jet holes,it was found that the attenuation coefficient curve of jet impingement heat transfer in the chordwise direction exhibited an approximate bell shape with a peak located at the stagnation point(x=0).34The experimental correlations of the Nusselt number at the stagnation point(x=0)and the attenuation coefficient distribution in the chordwise direction were developed for jet impingement with three rows of aligned jet holes.34

As briefly reviewed above,investigations of the jet impingement heat transfer and the related correlations have been proposed mainly for single one jet or one row of jet holes.With regard to the aircraft anti-icing applications,jets from multiple rows of hole become more effective,while the studies on impingement for two or three rows of jet holes on a piccolo tube are still less established.This paper will focus on the heat transfer of jet impingement on a variable-curvature concave surface of a wing leading edge with two rows of aligned jet holes.The effects of the jet Reynolds number,the relative tube-to-surface distance,the jet impingement angle,and the relative chordwise arc length in the jet impingement zone on the heat transfer will be analyzed.In addition,experimental data-based correlation equations of the Nusselt number for the heat transfer at the jet-impingement stagnation point and the attenuation coefficient distribution in the chordwise direction will be presented,which play an important role in designing a wing leading edge hot-air anti-icing system with two rows of aligned jet holes.

2.Experimental apparatus

The experimental facility,as shown in Fig.3,33was composed of an air supply system,a test unit including an anti-icing cavity and piccolo tubes with two rows of aligned jet holes,a heating system and a data measurement and acquisition system.The air supply system consisted of the air compressor,dry filter,buffer tank,temperature control module and some connection pipes providing high pressure air with fixed temperature for the test section.The 2D cross sectional shape of the antiicing cavity and the piccolo tube,as shown in Fig.1,was from a real regional jet.The spanwise length of the anti-icing cavity was 500 mm.The materials of the cavity and the piccolo tube were aluminum-magnesium alloy(5A06)and stainless steel,respectively.The heating system was composed of a power distribution cabinet and a thin film electric heater as shown in Fig.4,17which was attached directly to the outer surface of the anti-icing cavity to provide a uniform heat flux of about 5600 W/m2.The data measurement and acquisition system was composed of a gas flow meter,a pressure transmitter and a temperature transmitter for the supplied air measurement,type T thermocouples for the cavity surface temperature measurement,a power meter for the cavity surface heat flux measurement,and an Agilent 34970 module and a computer for the data acquisition.

As shown in Fig.5,33the anti-icing cavity was designed as a detachable assembly,so that it can be equipped with different piccolo tubes at different locations conveniently.Different piccolo tubes can be installed in the cavity via the through hole in the left-side panel,and accurate positioning can be achieved by the positioning pin in the piccolo tube and the positioning hole in the right-side panel.Meanwhile,the desired tube-to-surface distance H can be adjusted by replacing the left-side panel with different through hole locations.

In order to minimize the heat loss to the ambience,the rubber sponge insulation material with the thickness of about 10 cm was employed to cover the thin film heater,and the heat loss was estimated to be within 2.40%,which can be safely neglected.The type T thermocouple junctions were placed in the blind holes with a depth of 2.5 mm in the outer surface and fixed by adhesives with good heat conduction and electrical insulation.Because all the thermocouple wires were arranged outside the anti-icing cavity,the flow field in the cavity was not affected.As the thermocouple junctions were placed only 0.5 mm from the inner surface of the leading edge,it was validated to accurately represent the temperatures at the inner surface.Fig.633shows the detailed thermocouple arrangement along the chordwise direction of the leading edge.Two-column layout of thermocouples in the spanwise direction was arranged on the same plane with the middle two adjacent columns of jet holes on piccolo tube,individually.The relevant parameter ranges in the experiments are listed in Table 1,where Cnis the spanwise distance between adjacent jet holes,d is the jet hole diameter,H is the tube-to-surface distance,r is the chordwise arc length in the jet impingement zone,and α is the jet impingement angle.The supply pressure was set to be larger than 180000 Pa to ensure that the velocity of the hole exit reached the speed of sound.

Fig.3 Experimental facility.33

Fig.4 Constantan film heater.17

3.Data processing and error analysis

Fig.5 Disassembly sketch of hot-air anti-icing cavity.33

The inverse heat transfer method was adopted in the experiment,i.e.,the heat was transferred to the impingement air in the wing anti-icing cavity from outside,which was just opposite to the actual situation where heat was transferred from the hot air in the wing anti-icing cavity to outside.The outer surface of the anti-icing cavity was heated using a constant heat flux,and high pressure air with a temperature of about 300 K was introduced into the cavity as a cooling medium,so as to study the fluid flow and jet impingement heat transfer in the wing anti-icing cavity.The heat transfer was determined to be steady when the variation of all the temperatures recorded was negligible.

Fig.6 Temperature measurement points distribution in chordwise direction.33

Table 1 Parameter range in experiments.

According to the experimental principle,the local convective heat transfer coefficient hxis determined as

where q is the heat flux on the outer surface of the leading edge,Twxis the local temperature on the inner surface of the leading edge,and Tawis the adiabatic wall temperature.

The local Nusselt number can be defined by Eq.(2):

where λ is the thermal conductivity of air.

The jet Reynolds number is written as

where Gmis the mass flow rate of air,N is the number of jet holes, ρ is the density of air,and μ and νare kinetic and dynamic viscosities,respectively.

More importantly,the jet total temperature Tjreplaces Tawin Eq.(1).The reasons are that it’s hard to measure Tawdirectly without affecting an influence on the jet impingement flow field in the anti-icing cavity and Tjis close to Tawin many cases(the differences between them are generally less than 1.5 K34).

With regard to the distribution characteristics of the local heat transfer performance in the chordwise direction of the leading edge,the attenuation coefficient ξxis defined in Eq.(4):

The measurement uncertainties of directly measured parameters,such as the voltage,resistance,temperature and flow rate,are shown in Table 2.All uncertainties of h,Nu and ξ are less than 10.0%.

4.Experimental system validation

To guarantee the validity and reliability of the experimental results achieved in this work,a comparison with other available experimental data was made.As the test model in this work represents a real wing anti-icing system applied in a regional jet,while most published experimental and numerical results were corresponding to jet impingement on a flat or cylindrical surface,it is very difficult to find an appropriate example to compare with our results.The only similar example for comparison was found from the work by Brown et al.24,who experimentally investigated a full-scale,2D model of an aircraft nacelle anti-icing system.The profile of the impingement surface as well as the operating conditions from Ref.25was quite similar to those in this work,and Fig.7 shows the comparison between our results and those from Brown et al.In the experiment made by Brown et al.,d=1.5 mm,H/d=5,while in our experiment,d=2.0 mm,H/d=6.63.As shown in Fig.7,our experimental results for averaged Nusselt number exhibit a very similar trend with those from Brown et al.,but are slightly lower caused by a higher relative tube-to-surface distance in our experiment.The reason of theunderestimation of the Nusselt number can be explained as follows.According to the previous studies,34the optimal relative tube-to-surface distance was within 4–5.75.As the relative tube-to-surface distance is larger than the optimal value,the momentum and velocity in the jet core zone drop with increasing H/d values due to the interaction between the jet and the surrounding air,resulting in the decrease of the heat transfer on the surface and thus the underestimation of the Nusselt number in this work.This comparison validates the experiment approach and the accuracy of the experimental results in this work.

Table 2 Uncertainties of measuring equipment.

Fig.7 Validation of experimental results.

5.Experimental results and analysis

In this work,the heat transfer performance at the jetimpingement stagnation point and the attenuation distribution characteristics in the chordwise direction were taken into consideration in the jet impingement heat transfer on a variable curvature concave surface in a wing leading edge with two rows of aligned jet holes on a piccolo tube.For the first aspect,the heat transfer performance at the jet-impingement stagnation point with two rows of aligned jet holes was compared with that with a single row of jet holes and the middle row of three-row configurations.For the latter one,the influences of the jet Reynolds number Rej,the relative tube-to-surface distance H/d and the relative chordwise arc length in the jet impingement zone r/d on the distribution characteristics of the attenuation coefficient ξxwere investigated.The jet impingement zone was defined as a region between the two stagnation points on the concave surface.The correlations of Nusselt number at the jet-impingement stagnation point and the attenuation coefficient were developed on the basis of experimental data.

5.1.Heat transfer performance at stagnation point

Fig.8 Influence of circumferential angle of jet holes on Nusselt number in chordwise direction.

Fig.9 Nusselt number in chordwise direction with a single row of jet holes and three rows of aligned jet holes.

Although the jet interference before impingement could exert an effect on the heat transfer of a jet array impinging on the surface,the experimental results from Fig.833indicated that the heat transfer at the stagnation point with two rows of aligned jet holes was not affected by the jet interference when the circumferential angle of jet holes on the piccolo tube was larger than 30°,that is,the jet impingement zone r/d was larger than 8.1.Thus,the Nusselt number at the jet-impingement stagnation point with two rows of aligned jet holes was equal to that with a single row of jet holes to a certain extent.Actually,the heat transfer at the middle jet-impingement stagnation point with three rows of aligned jet holes was almost the same as that with a single row of jet holes,as shown in Fig.9 where the data were extracted from our previous work.33Note that the Nusselt number at the jet-impingement stagnation point that is comparable between different jet hole configurations should be the data at the same position on the surface.Therefore,it was concluded that the heat transfer at the jetimpingement stagnation point with a single row of jet holes and that with two or three rows of aligned jet holes were similar when the points were in the same position.The influences of the parameters on the heat transfer at the jet-impingement stagnation point are not repeated here,which can be found in our previous work.33,34

5.2.Attenuation distribution characteristics in chordwise direction

Fig.10 Comparison between two and three rows of aligned jet holes for Nusselt number.

Compared with the attenuation distribution characteristics in the chordwise direction for the jet impingement with three rows of aligned jet holes as shown in Fig.10,those with two rows of aligned jet holes were different and more complicated.For the jet impingement with two rows of aligned jet holes,the curve of the Nusselt number in the impingement zone exhibited double peaks and a single valley where the two peaks located on the upper and lower surfaces of the wing leading edge individually.This phenomenon can be illustrated as follows.Due to the large distance between stagnation points for the jet impingement with two rows of aligned jet holes,the interaction between the two jets was weak.The heat transfer in the central area was not improved like that brought by the jet impingement with three rows of aligned jet holes at the middle stagnation point where x=0 mm.In addition,the jet with two rows of aligned jet holes impinged the target surface away from the wing geometric stagnation where x=0 mm.As a result,the oblique jet caused a main flow toward the aft side of leading edge,and a constrained flow toward the forward side,as illustrated in Fig.11.Thus,the heat transfer of the forward side was significantly affected and became weaker,especially at the front stagnation point where x=0 mm,developing a valley value of the Nusselt number.

Fig.11 A constrained flow in forward side of leading edge due to oblique jet.

In this paper,two zones including the jet impingement zone and the outside zone were distinguished in analyzing the jet impingement heat transfer attenuation distribution characteristics with two rows of aligned jet holes,as shown in Fig.10.The negative-value zone on the abscissa denotes the lower surface of anti-icing cavity,while the positive-value zone the upper surface.The two jet-impingement stagnation points were set as the reference and the start points of the attenuation distribution.It was found in Fig.10 that the heat transfer distribution outside the jet impingement zone with two rows of aligned jet holes was similar to that with three rows,which was in a half-bell shape.In this paper,we mainly discuss the influences of Rej,H/d and r/d on the distribution characteristics of the attenuation coefficient ξxin the jet impingement zone and the correlation equations of ξxin these two zones in the chordwise direction.

5.2.1.Effect of jet Reynolds number on attenuation coefficient

Fig.12 indicates the distribution of attenuation coefficient in the impingement zone in the chordwise direction of the wing leading edge with different jet Reynolds numbers.The curves exhibited double peaks and a single valley.The peaks located on the jet-impingement stagnation points on the target surface,i.e.,the end of jet impingement zone,indicating the position where the largest Nusselt number Nustagwas.The valley was on the position x/d=0,the most front of the leading edge,indicating the position where the lowest Nusselt number Nu0was.The ξxcurves against x/d in the impingement zone exhibited an upside-down bellshape attenuating from the jet-impingement stagnation point to the middle location x/d=0,as shown in Fig.12.It was concluded that the jet Reynolds number affected the attenuation coefficient in the impingement zone greatly,and the attenuation rate of the attenuation coefficient from both jet-impingement stagnation points to the middle increased with the jet Reynolds number,showing a larger difference between the peak and valley value.When the jet Reynolds number increased from 34241 to 85340,the attenuation coefficient at the valley location decreased by nearly 5.5%.This phenomenon can be illustrated below.The larger the jet Reynolds number was,the larger the jet turbulence level at the stagnation point was.Moreover,the velocity of the wall jet was much lower than that of the core,so that the influence of the jet Reynolds number on the interaction between the two wall jets was not obvious.Because of the above reasons,there was an increased difference in the peak and valley value with an increasing jet Reynolds number,resulting in a faster attenuation rate of the attenuation coefficient from the jet impingement stagnation point to the middle.This conformed to the previous results that the Nusselt number decreased at a much faster rate in the chordwise direction with a higher jet Reynolds number.33,34

Fig.12 Influence of jet Reynolds number on attenuation coefficient in impingement zone.

As shown in Fig.12,when the jet Reynolds number was small,for example Rejwas equal to 34241 and 45850,the attenuation coefficient of the jet impingement heat transfer varied slowly at the beginning,from about the jet impingement stagnation points to the position x/d=±15,and then decreased sharply and became flat near the stagnation point of the leading edge where x/d=0.While when the jet Reynolds number was large,for example Rejwas equal to 85340,the attenuation coefficient declined rapidly at the very beginning and was flat around the center position.The reason was that the turbulence level at the jet-impingement stagnation point was much larger than that around the wing leading edge stagnation point when the jet Reynolds number was large,resulting in much more difference of heat transfer between these two positions.That is,the smaller the jet Reynolds number was,the more slowly the turbulence level in the jet impingement zone varied,resulting in smaller slope in the attenuation curves ξxvs x/d.

5.2.2.Effect of relative tube-to-surface distance on attenuation coefficient

Fig.13 shows the distribution of the attenuation coefficient in the impingement zone in the chordwise direction of the wing leading edge with different relative tube-to-surface distances H/d.It was observed that the relative tube-to-surface distance had a similar effect with that of the jet Reynolds number,i.e.the smaller the relative tube-to-surface distance was,the faster the attenuation rate of the attenuation coefficient was from both jet-impingement stagnation points to the middle,showing a larger difference between the peak value and the valley value.When the relative tube-to-surface distance declined from 9.88 to 5.75,the attenuation coefficient at valley location decreased by 1.5%.

Fig.13 Influence of relative tube-to-surface distance on attenuation coefficient in impingement zone.

It was also found that,when the relative tube-to-surface distance was large,such as H/d=9.88,the curve of attenuation coefficient varied slowly near the jet-impingement stagnation points,and then decreased greatly from both jetimpingement stagnation points to the middle.In contrast,the curve of attenuation coefficient declined rapidly from the very first when the relative tube-to-surface distance was small,such as H/d=5.75.The main reason was similar to the influence of jet Reynolds number on attenuation coefficient.The turbulence level at the jet-impingement stagnation point was much larger than that around the center position when the relative tube-to-surface distance was small.Thus,the attenuation coefficient reduced rapidly from the jet-impingement stagnation point to the center position.In addition,the development of the jet flow was perhaps another reason for the influence of the relative tube-to-surface distance on the attenuation coefficient.According to the previous work,34the optimal H/d is in the range of 4 to 5.75 indicating that the jet flow is fully developed before impinging onto the target surface when H/d is larger than 5.75.Under this condition,the target surface is in the downstream zone of the potential core.As H/d increased,the wall jet on the surface became weaker,resulting in the weaker variation of attenuation coefficient.

5.2.3.Effect of relative chordwise arc length in jet impingement zone on attenuation coefficient

Fig.14 shows the attenuation coefficient in the impingement zone in the chordwise direction of the wing leading edge with various relative arc lengths r/d.It indicated that the effect of relative chordwise arc length on the attenuation coefficient was much different from those of the jet Reynolds number and the relative tube-to-surface distance.Firstly,the relative arc length obviously affected the range of the impingement zone in the chordwise direction.The larger the relative arc length was,the wider the impingement zone was.Secondly,the relative arc length also significantly affected the attenuation coefficient in the impingement zone.The larger the relativearc length was,the bigger the difference in the attenuation coefficient value between the peak and the valley was.However,the attenuation rates of attenuation coefficient from both jet-impingement stagnation points to the middle were nearly the same as different relative arc length.For instance,when r/d was increased from 30.75 to 42.03,the attenuation rates of attenuation coefficient increased only by 1.8%.

In addition,when the relative arc length was small,such as r/d=10.12,the attenuation rate of the heat transfer performance was large even around the middle geometry stagnation point(x/d=0).However,with the increase of the relative arc length,the variation of the attenuation coefficient around the position x/d=0 was not obvious.For instance,when r/d=42.03,the variation of attenuation coefficient was only 0.1 around the position x/d=0.It was because that the larger the relative arc length was,the smaller the interaction between the two wall jets was,so that the heat transfer around the middle geometry stagnation point was not affected by jets and largely remained unchanged.

Fig.14 Influence of relative arc length on attenuation coefficient in impingement zone.

6.Correlation equations

6.1.Correlation equation of stagnation Nusselt number

Since the stagnation Nusselt number was verified to be described as that of stagnation Nusselt number for the jet impingement with three rows of aligned holes as stated earlier,the correlation equation of the stagnation Nusselt number in relation of jet Reynolds number,relative tube-to-surface distance and jet impingement angle was written as follows34:

The above correlation was valid for a wide range of parameters,i.e.,50000≤Rej≤90000,2.48≤H/d≤13.8,66°≤ α≤ 90°,and the average difference between the calculated and experimental results(Nucaland Nuexp)was within 6%,as shown in Fig.15.

6.2.Correlation equation of attenuation coefficient

On the basis of the correlation equation of the stagnation Nusslet number,it is of high interest to propose a correlation for the attenuation coefficient ξx,in order to quickly and accurately obtain the distribution characteristics of the jet impingement heat transfer in the chordwise direction on the surface of the anti-icing cavity.For the structure of two rows of aligned jet holes where these two jet impingement stagnation points distributed on both sides of the front leading edge,the jet impingement heat transfer performance and the attenuation law are more complicated than those of a single row or three rows of aligned jet holes.The impingement zone and the outside region beyond the impingement zone were divided and investigated.Regarding the Nusselt number at the jet stagnation point as the datum point,the attenuation coefficient ξxof these two jet stagnation points were both 100.

Fig.15 Comparison between calculated and experimental results for stagnation Nusselt number.

For three dimensionless parameters including the jet Reynolds number,the relative tube-to-surface distance and the relative arc length,the correlation of attenuation coefficient was developed.34As explained above,the ξxvs x/d curve in the region beyond the impingement zone in the chordwise direction on the surface of the wing leading edge for the structure of two rows of aligned jet holes was similar to that for the structure of three rows of aligned jet holes,which was like a half-bell shape.The ξxvs x/d curve in the impingement region exhibited an approximate upside-down bell shape.In order to ignore the effect of the slot structure as shown in Fig.1 on the heat transfer on the upper surface,which made this issue much more complex,and on a basis of conservative design,the experimental data of attenuation coefficient on the upper surface were replaced by the one on the lower surface and used for fitting.After analyzing the characteristics of the curve of the attenuation coefficient,the Boltzmann function was selected to fit the experimental data of the attenuation coefficient in the impingement region and the Gauss function to fit that in the outside zone.

Boltzmann function was applied in the impingement region:

where M1and M2are the initial and final values,respectively,means the minimum and maximum value of ξx;x0is the center;dx is the time constant.

According to Eq.(6),non-linear fitting technique was applied to fit the experimental data,each of which yielded a certain A1,A2,x0and dx value.In terms of Rej,H/d and r/d,exponential functions were applied to fit M1,M2,x0and dx,respectively,as shown in Eqs.(7)–(10).

It was noted that ξxmight be a little larger than 100 at the stagnation point calculated by Eq.(6).If it occurred,the attenuation coefficient was considered as 100.Eqs.(6)–(10)were used to predict ξxin the impingement region,which were applicable in the following parameter range:5×104≤Rej≤9×104,5.75≤H/d≤13.75 and 26.88≤r/d≤34.8.

Gauss function was applied for the outside region beyond the impingement zone:

where ξ0is the value of ξxat the farthest position from the stagnation point;xstagis the coordinate of the stagnation point and defined as 0;the attenuation height Hais expressed aswhich is equal to ξstag- ξ0and ξstagis 100 in this paper;w istimes the Gauss fitted curve width at the half of the attenuation height Ha/2.

The attenuation coefficient M was defined as M=2/w2,and the correlation of ξxoutside the impingement zone was deduced as Eq.(12).

In Eq.(12),non-linear fitting technique was applied to fit the experimental data,each of which yielded a certain M and a value.In terms of Rej,H/d and r/d,exponential functions were applied to fit M and a,respectively,as shown in Eqs.(13)and(14).

Fig.16 Comparison of results from experiments and correlation Eq.(6)for ξx.

By substituting Eqs.(13)and(14)into Eq.(12),the correlation of ξxoutside the impingement zone was derived,which was applicable in the following parameter range:5×104-≤Rej≤9×104,5.75≤H/d≤27.5 and 13.2≤r/d≤42.03.

To validate the correlation of attenuation coefficient,a comparison between the experimental data and the calculation results was conducted based on the correlation equation,as is presented in Fig.16.The largest deviation position appeared at the middle,i.e.,the valley location.The deviations were all lower than 2%,indicating that the correlation was effective within the proposed parameter ranges.For this specific structure with two rows of aligned jet holes,the correlation can be applied to predict the distribution characteristics of the jet impingement heat transfer in the impingement region on the surface of the anti-icing cavity,which helps to guide the design and the performance assessment for a wing leading edge hotair anti-icing system.

7.Conclusions

Extensive experimental studies on the heat transfer characteristics of two rows of aligned jet holes impinging on a variable curvature concave surface in a wing leading edge were conducted in this work.Conclusions have been summarized as follows:

(1)The heat transfer performance at the stagnation point with two rows of aligned jet holes for aircraft antiicing application was close to that with a single jet hole or the middle jet of three-row configurations.

(2)The distribution curves of the attenuation coefficient in the chordwise direction were divided into two zones for analysis.In the jet impingement zone,it exhibited an approximate upside-down bell shape with double peaks and a single valley,and in the non-jet impingement zone,it was similar to that with three rows of aligned jet holes.In the impingement zone,the attenuation rate of attenuation coefficient increased with the rise of the jet Reynolds number and the decline of relative tube-to-surface distance.The larger the relative chordwise arc length was,the larger the difference in attenuation coefficient value between the peak and the valley was.However,the attenuation rates of attenuation coefficient increased only by 1.8%when r/d was increased from 30.75 to 42.03.

(3)Based on the experimental data,the correlation equations for both the stagnation Nusselt number and the distribution of the attenuation coefficient in the chordwise direction were acquired, which were applicable to the following parameter ranges:5×104-≤ Rej≤9×104, 5.75≤ H/d≤13.75, 66°≤α≤90°and 26.88≤r/d≤34.8.

As the piccolo tube with two or three rows of jet holes is generally adopted by many aircraft engineers in the antiicing system,this work provides a way to design and evaluate the performance of the aircraft wing leading-edge anti-icing system with two rows of jet holes on piccolo tube.

Acknowledgement

This work was supported by the National Natural Science Foundation of China(No.51206008).