Fast Algorithm for Prediction of Airfoil Anti-icing Heat Load *

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1 Energy and Power Engneerng, 13,, do:.4236/epe.13.4b09 Publshed Onlne July 13 ( Fast Algorthm for Predcton of Arfol Ant-cng Heat * Xueqn Bu, Ru Yang, Ja Yu, Xaobn Shen, Gupng Ln School of Aeronautc Scence and Engneerng, Behang Unversty, Beng, Chna Emal: buxueqn@buaa.edu.cn Receved January, 13 ABSTRACT any flght and cng condtons should be consdered n order to desgn an effcent ce protecton system to prevent ce accreton on the arcraft surface. The ant-cng heat load s the basc knowledge for the desgn of a thermal ant-cng system. In order to help the desgn of the thermal ant-cng system and save the desgn tme, a fast and effcency method for predcton the ant-cng heat load s nvestgated. The computaton flud dynamcs (CFD) solver and the essnger model are appled to obtan the snapshots. Examples for the calculaton of the ant-cng heat load usng the proper orthogonal decomposton (POD) method are presented and compared wth the CFD smulaton results. It s shown that the heat loads predcted by POD method are n agreement wth the CFD computaton results. oreover, t s obvously to see that the POD method s tme-savng and can meet the requrement of real-tme predcton. Keywords: Ant-cng; Heat ; Proper Orthogonal Decomposton 1. Introducton * Supported by the Natonal Natural Scence Foundaton of Chna (Grant No ). Arcraft cng s a thermodynamc phenomenon that leads to the formaton of ce on arcraft n flght. Severe nflght cng s a serous hazard that destroys the clean aerodynamc shape of the arfol, resultng n ncreased drag, decreased lft, reduced arcraft stablty, performance and controllablty. Thermal ant-cng system, used as the most popular means of ce preventon n cng condtons, operates to keep the surface temperature and the water collected by the arcraft above the freezng pont. Durng the process of the desgn of the ant-cng system, many flght and cng condtons should be consdered for the calculaton of the water droplet mpngement and the ant-cng heat load to have knowledge of how about the most severe condton and how much heat flux should be suppled for ant-cng. Generally, the CFD method s used for the ar flow smulaton around the obect and then the thermal analyss are appled to calculate the heat load [1] consderng the whole flght and cng condtons under FAR Appendx C. Ths would lead to very large computaton work and expend much computaton tme. In order to help the desgn and save the desgn tme, a fast and effcency method for predcton the ant-cng heat load s nvestgated n ths paper. The proper orthogonal decomposton (POD) [2] technque s a powerful reduced-order model (RO), whch can effectvely reduce the degree of freedom for physcal problems wth hgh precson, thus reducng the calculaton tme and savng the data storage sgnfcantly. POD has been used extensvely n the feld of flow and heat transfer computaton [3,4] to predct these problems accurately n very short tme. In ths paper the CFD method s used to obtan the snapshots (samples) for the POD analyss. The POD method s studed and used for the fast predcton of the heat loads. The effects of the varables for the POD predcton of the heat load are nvestgated and the results are compared wth the CFD smulaton results. 2. POD ethodology POD method can rapdly reconstruct the flow nformaton applyng a data set called snapshots or samples of the physcal propertes. The essence of the POD method s the extracton of a set of egenfunctons whch can best descrbes the domnant features of the flow or heat transfer. Once these egenfunctons are calculated, the approxmate states of the flow feld and heat transfer parameters can be obtaned by superposton of them. POD method requres a set of snapshots whch can be obtaned ether from numercal calculatons or tests. In ths paper, the snapshots are obtaned from the CFD analyss. The snapshot U, a vector whch descrbes a certan physcal feld, can be any type of the flow var- Copyrght 13 ScRes.

2 494 X. Q. BU ET AL. able, such as velocty, pressure, ce shape [,6], or ant-cng heat load, as n ths paper. If each U has N sample values, a set of snapshots forms a N matrx U, whch can be expressed by: 1 U U,, U. (1) As mentoned above, POD method presents the man characterstcs of the flow by usng a set of egenfunctons. Ths means that a set of proper orthogonal decomposton modes, should be found, and then U can be expressed by:, (2) where s a set of sample coeffcents for the proper orthogonal decomposton modes, can be calculated by: =1 = (, U ), (3) th where denotes the proper orthogonal decomposton mode, and denotes the snapshot. th All the form the proper orthogonal decomposton modes matrx : 1,,, (4) where s the number of snapshots. The matrx, of the same dmensonalty as U, satsfes the constrant. The elements of the modes are defned as: U V, () where V s the egenvector matrx of C : CV V, (6) s a dagonal matrx storng the egenvalues of the postve defnte covarance matrx C. The entres of C are defned as: N k k k 1 C U U. (7) Thus, the problem of solvng the proper orthogonal decomposton modes changes nto that of solvng the egenvalues and egenvectors of the matrx C. Obvously, C s a matrx, so the matrx to be solved becomes much smaller, reducng from a sze N (the number of grd ponts) to a sze (number of snapshots). Ths s one of the advantages of usng the POD technque. After the proper orthogonal decomposton modes and sample coeffcents are obtaned from the matrx C whch can be calculated from the snapshot matrx U, the approxmate states of the flow feld can be obtaned by lnear superposton of the proper orthogonal decomposton modes, shown as follows: pod pod. (8) =1 To predct a feld whch s not part of the snapshots, an nterpolaton method must be employed to obtan the POD POD coeffcents. The lnear nterpolaton method s employed to compute the coeffcents n ths paper. oreover, the magntude of each egenvalue determnes the energy contaned n the correspondng mode. The rato of the energy contaned n a certan mode can be measured by: Energy. (9) 1 Therefore, f necessary, the modes that contan less energy can be neglgble, resultng n the reducton of proper orthogonal decomposton modes. The approxmaton by POD calculaton of the lnear combnaton of egenvectors can be wrtten as: l pod pod, () =1 where l s the proper orthogonal decomposton modes used for the constructon of the approxmaton by POD, so l. If some proper orthogonal decomposton modes are gnored, the flow feld reconstructed by POD wll be dfferent from that reconstructed by all modes. The naccuracy can be controlled by the total energy of all the used proper orthogonal decomposton modes. The most tme cost durng the calculaton by POD s that of solvng the egenvalues and egenvectors of the matrx C, and the total calculaton process only takes a few seconds. In ths paper, the effectveness of the POD method n predcton the ant-cng heat load on an arfol s nvestgated and the effects of the ambent temperature, flght velocty and lqud water content (LWC) on the PODreconstructed ant-cng heat load are studed. 3. Ant-cng Heat Predcton for Snapshots The predcton of the ant-cng heat load generally has four steps: 1) ar flow feld calculaton, 2) water droplet mpngement calculaton, 3) external convectve heat transfer coeffcent predcton, 4) ant-cng heat load analyss. In ths paper, for the snapshots, the Euleran method s used for the calculaton of the water droplet mpngement. The boundary layer ntegrated method s used for the calculaton of the external heat transfer coeffcent. The ant-cng heat loads smulaton s based on the essnger model. The Computatonal Flud Dynamc (CFD) tool Fluent and ts User Defned Functon (UDF) are appled to mplement the above computatons for the snapshots. Detals about the method and analyss can be Copyrght 13 ScRes.

3 X. Q. BU ET AL. 49 found n Reference [1,7,8] wrtten by authors. 4. POD Valdaton and Results Analyss In ths paper, the POD method, at frst, s appled to the predcton of ant-cng heat loads for 2D vscous flow under a one-varable problem, here beng ambent temperature. The calculaton process of the POD method s ntroduced n detal. Then, the ant-cng heat loads predctons by POD under two-varable and three-varable problems have been nvestgated. An arfol whch has 344 control volumes s taken nto consderaton for the study. The ant-cng heat load samples of each control volume on the arfol under certan flght and cng condtons are obtaned by the CFD method. Besdes three parameters whch are the ambent temperature, the velocty and the LWC, the other parameters n all cases are the same, whch are: μm as the mean volume dameter, 2 as the angle of attack, 90KPa as the ambent pressure, as the surface temperature of the arfol One Parameter POD Applcaton for Heat It s well known that the ambent temperature has a great nfluence on ant-cng heat loads. Therefore, frstly, the effect of the ambent temperature on the POD-predcted ant-cng heat loads on an arfol s nvestgated and the effectveness of the POD method s studed. In ths secton, the LWC and the flght velocty are 0. 6 g/m 3 and 90 m/s ndvdually. The ambent temperature s from - to -. The ant-cng heat loads, whch are obtaned by CFD method, at the ambent temperature of -, -, -, -, -, - are taken as samples. So a matrx forms and = 6. The ant-cng heat loads under varous ambent temperatures, predcted by the CFD method, are shown n Fgure temp= - temp= - temp= - temp= - temp= - temp= - Fgure 1. Ant-cng heat loads at dfferent ambent temperatures (CFD method). Accordng to the sample matrx U, the correspondng matrx C can be calculated usng Equaton (7), and C s a 6 6 matrx because there are 6 snapshots. The results of C matrx are showed n Table 1. Then the ordered egenvectors and egenvalues of matrx C are calculated from Equaton (6). The egenvalues are showed n Table 2. It can be found that the frst egenvalue s much larger than others from Table 2, whch means that the energy contaned n the correspondng mode s very large. It can be concluded that the frst mode s vtal to the predcton and t determnes the broad contour of the ant-cng heat loads dstrbuton. Next, the ultmately used proper orthogonal decomposton modes can be obtaned by orthogonalzng the matrx, whch can be calculated from Equaton (). Here all the proper orthogonal decomposton modes are used for calculaton. The sample coeffcent for the correspondng proper orthogonal decomposton mode can be obtaned usng Equaton (3). The results are shown n Table 3. It s obvous that each sample coeffcent for the frst proper orthogonal decomposton mode s much larger than other modes, ndcatng that the frst mode makes a sgnfcant contrbuton on the POD calculaton. Table 1. C matrx values Table 2. Egenvalues. case Λ Table 3. Sample coeffcents. а1 а2 а3 а4 а a Copyrght 13 ScRes.

4 496 X. Q. BU ET AL. After the above calculatons, the POD model has been bascally establshed. The ant-cng heat loads at the ambent temperature of -23, -12 are predcted usng the POD approach. The predcted results are compared wth the results from CFD method. The POD coeffcents POD are nterpolated by the lnear method, and the predcted ant-cng heat loads are calculated usng Equaton (). Fgure 2 shows the heat loads comparsons between the POD solutons and the CFD method, whch ndcates that the results nearly the same between the POD and CFD method under the same condtons. Therefore, the POD method can be appled very well to predct the heat load when one parameter changes Two Parameters POD Applcaton for Heat In ths secton, the POD method s used to predct the ant-cng heat loads under two-varable problem, here beng the ambent temperature and flght velocty. The LWC s 0.6 g/m 3. The samples are obtaned at the ambent temperature of -, -, -, -, -, - and the flght velocty of 90, 0, 1, 1, 1, 1 m/s. Thus the number of the snapshots s 36. All the 36 proper orthogonal decomposton modes are appled for the predcton of the heat load. Fgure 3 compares the heat load results obtaned usng POD solutons to that acqured through the CFD method. Case 1 ndcates that the ambent temperature s -23 and the flght velocty s 1 m/s. Case 2 ndcates that the ambent temperature s -12 and the velocty s 1 m/s. It s obvous that the heat loads predcted by the POD method ft well wth the CFD smulated results except a lttle devaton at the locaton where a large drop occurs. That mght because that the runback water evaporates completely at that place and the heat load decreases remarkably, whch causng non-dervablty, leadng to the temp= -23 (CFD) temp= -23 (POD) temp= -12 (CFD) temp= -12 (POD) Fgure 2. Ant-cng heat loads comparson of sngle parameter case1(cfd) case1(pod) case2(cfd) case2(pod) Fgure 3. Ant-cng heat loads comparson of two parameter devaton of heat load between The soluton tme taken by the POD method s less than two seconds also Three Parameters POD Applcaton for Heat At last, the POD method s used to predct the ant-cng heat loads under three-varable problem, here beng the ambent temperature, flght velocty and LWC. To generate the varous samples, the ambent temperature, flght velocty and LWC are selected as follows. Therefore, n total, 216 snapshots are computed. Temperature = -, -, -, -, - and -( ) Velocty = 90, 0, 1, 1, 1 and 1(m/s) LWC = 0.3, 0.4, 0., 0.6, 0.7 and 0.8(g/m 3 ) From calculatng the egenvectors and egenvalues of matrx C, 216 proper orthogonal decomposton modes s obtaned. Because the number of proper orthogonal decomposton modes s very large, the frst seventeen modes whch contan more than 99.9% energy of all the modes are chosen for the predcton of the ant-cng heat loads. It s consdered to be able to accurately descrbe the ant-cng heat loads. The comparsons of the results between the three parameters POD soluton and the CFD method are shown n Fgure 4. Case 1 ndcates that the ambent temperature s -23, the flght velocty s 1 m/s and the LWC s 0. g/m 3. Case 2 ndcates that the ambent temperature s -12, the velocty s 1 m/s and the LWC s 0.6 g/m 3. ore devatons can be seen near the locatons where are non-dervablty. But the devatons are n the acceptable regon. Lkewse, the soluton tme taken by the POD method s almost equal to the one-varable problem above, means tme-savng. Copyrght 13 ScRes.

5 X. Q. BU ET AL case1(cfd) case1(pod) case2(cfd) case2(pod) Fgure 4. Ant-cng heat loads comparson of three parameter. Conclusons The POD approach for reduced-order modelng s appled to the predcton of the ant-cng heat loads for a two-dmensonal arfol. Consderng the nfluence of the ambent temperature, flght velocty and LWC, the ant-cng heat loads predctons by the POD method under one-varable, two-varable and three-varable problems have been nvestgated. The snapshots for the POD predcton are obtaned by the CFD method. To valdate the POD method for the heat load predcton, the heat load results from the POD solutons are compared wth those from the CFD method. It can be concluded that the POD approach can predct the ant-cng heat loads remarkably well besdes some slght dsagreements at the locatons where water evaporates totally. Comparng the tme t takes va POD method and CFD solver, t can be obvously found that the POD method s much tme- savng. It should be studed further on how to select the samples to reduce the number of the snapshots. REFERENCES [1] X. Q. Bu, G. P. Ln, Y. X. Peng and J. Yu, ethod for Calculaton of Ant-cng Heat s, Acta Aeronautca et Astronautca Snca, Vol. 27, No. 2, 06, pp [2] Z. Ostrowsk and R. A. Bałeck, Estmaton of Constant Thermal Conductvty by Use of Proper Orthogonal Decomposton, Computatonal echancs, Vol. 37, No., 0, pp do:.07/s y [3] P. Dng and W. Q. Tao, Reduced Order odelng wth the Proper Orthogonal Decomposton, Journal of Engneerng Thermophyscs, Vol., No. 6, 09, pp [4] X. H. Wu, W. Q. Tao, et al., Reduced Order odel for Fast Computaton of Incompressble Flud Flow Problem, Proceedngs of the CSEE, Vol., No. 26,, pp [] K. Nakakta and W. G. Habash, Toward Real-tme Aero-cng Smulaton of Complete Arcraft va FEN- SAP-ICE, Journal of Arcraft, Vol. 47, No. 1,, pp do:.14/1.477 [6] S. K. Jung, S.. Shn, R. S. yong and T. H. Cho, An Effcent CFD-based ethod for Arcraft Icng Smulaton Usng a Reduced Order odel, Journal of echancal Scence and Technology, Vol., No. 3, 11, pp do:.07/s [7] X. Q. Bu, J. Yu and G. P. Ln, Electrothermal Arfol Ant-cng System Smulaton, The Hong Kong Insttuton of Engneers Transactons, Vol. 17, No. 2, 09, pp [8] X. Q. Bu, Numercal Smulaton of the Hot-ar Ant-cng System, Ph.D thess, Beng Unversty of Aeronautcs and astronautcs, Beng, Chna,. Copyrght 13 ScRes.