An nverse problem soluton for post-processng of PIV data Wt Strycznewcz 1,* 1 Appled Aerodynamcs Laboratory, Insttute of Avaton, Warsaw, Poland *correspondng author: wt.strycznewcz@lot.edu.pl Abstract In the presented paper an applcaton of the nverse method for post-processng of PIV data have been dscussed. The am of the study s to develop a new methodology for fllng of mssng values n the expermental data. The problem concerns fllng of large gaps n the vector velocty feld caused by the lmted optcal access to the test secton of hgh speed wnd tunnel. The nverse procedure nvolves fttng of the PIV data wth a CFD smulaton results. In order to estmate unknown parameters of the flow the procedure utlses a mult-parametrcal dentfcaton algorthm. The paper presents results of the numercal test performed n order to valdate the proposed methodology. The smulated flow over an arfol n a test secton of a hgh speed wnd tunnel has been locally masked n order to reproduce the lmtatons of the actual measurements. The recovered vector velocty fled s n very good agreement wth the masked model data. The results of the numercal studes proved the feasblty of the proposed methodology for post-processng of vector velocty felds corrupted wth large areas of mssng data. The developed method wll be appled for PIV measurements of the flow over an arfol n a hgh speed wnd tunnel. The locaton of the lghtsheet optcs do not allow to perform measurements of the flow over the leadng edge of an arfol, therefore the nverse method has been proposed. Keywords: Partcle Image Velocmetry, Inverse Methods, CFD 1 Introducton Rapd development of the Partcle Image Velocmetry method n last decade allow to ncrease the spatal and temporal resoluton of the measurements. Especally greater accessblty of hgh frequency mpulse lasers and advancements n camera technology allowed to nvestgate transent aerodynamc processes n more detals. Ths holds true also for nvestgaton of the transonc and supersonc flows. Nevertheless, the measurements n hgh speed wnd tunnels can be subjected to some lmtatons. For example, the optcal access to the test chamber of supersonc wnd tunnel s often lmted. Ths cause the lmtatons n the measurements area. Moreover, n many cases the lght sheet optcs cannot be postoned n desred poston due to compact buld of the test chamber. Therefore not all regons of the nvestgated flow can be properly llumnated. For example, n case when the lght sheet optcs s postoned n dstant locaton downstream the flow, the leadng edge of the nvestgated arfol s not llumnated (See Fgure 1). Fg. 1 A scheme of a lghtsheet optcs postonng n a hgh speed wnd tunnel. The lmted access to the test secton do not allow to properly llumnate the whole feld of camera s vew. The lmtatons n lght sheet ntroducton the test secton and the lmted optcal access may cause large gaps n the vector velocty feld measured wth PIV. The effect of a lack of llumnaton n front of a leadng edge of an arfol s shown n Fgure 2. Paper ID:44 1
Fg. 2 Exemplary velocty feld measured n the test secton of a hgh speed wnd tunnel wth lmted optcal access. The freestream Mach number M=.7. Although, the procedures for fllng mssng values had been developed they are not fully applcable for postprocessng results of nvestgaton of compressble flow. In the case of nvestgaton of a flow feld wth large velocty gradents assocated wth a shock wave (see Fgure 2.) applcaton of a method based on medan flterng can reduce the rapd spatal changes n the flow feld. In the present paper a method for fllng mssng PIV data has been proposed. The nverse procedure nvolves fttng of the PIV data wth a CFD smulaton results. The present paper covers the ssue of the nverse problem formulaton, soluton of the exemplary drect problem and mult-parametrcal dentfcaton. Typcal results of the performed numercal studes are shown. The smulated flow over an arfol n a test secton of a wnd tunnel has been locally masked n order to reproduce the lmtatons of the actual measurements. 2 Method In the proposed methodology mssng velocty data where estmated by fttng a soluton of a numercal smulaton V to expermental PIV data Y. The soluton of ths nverse problem s based on the mnmzaton of the ordnary least squares norm n a form: [ Y ] V P ( ) 2 S( P) (1) = I = 1 where: S - s the sum of squares error, P s the vector of unknown parameters, V (P) soluton of a drect problem (CFD velocty feld), Y s a expermental velocty feld (2D PIV data). In the presented work an dentfcaton of unknown parameter P s performed by Levenberg-Marquardt teratve procedure n a form: n+ 1 n n T n n n 1 n T n P = P + [( J ) J + µ Ω ] ( J ) [ Y T ( P )] (2) where: n - s the teraton number, μ n - s a postve scalar named dampng parameter, Ω n - s the dagonal matrx and J - s the matrx of the senstvty coeffcents n the form: T T T ( P) J ( P) = P (3) In the numercal test an dentfcaton of masked veloctes was performed. The expermental PIV data V was smulated wth use of MES software. In order to reproduce the lmtatons of the actual measurements the smulated flow over an arfol n a test secton of a hgh speed wnd tunnel has been locally masked. The Paper ID:44 2
Comsol Multphyscs Flud Flow module was used n order to smulate 2D flow over an arfol n the closed wnd tunnel test secton (See Fgure 3). Fg. 3 Soluton of a drect problem The CFD velocty data was nterpolated n order to smulate the rectangular grd of PIV data. The gaps n the data was created by maskng of the PIV data (Fgure 4.)..4.3.2.1 -.1 -.2 -.3 -.4 -.5-1.5-1 -.5.5 1 1.5 2 2.5 Fg. 4 The CFD smulaton results (blue) was nterpolated to rectangular grd - the smulated PIV data (red) The smulated PIV data was used as an nput data n the performed numercal test. The nlet velocty V nl was estmated. The teratve estmaton procedure was stopped when the followng condton was fulflled: [ Y ] V ( P) 2 S( P) < ε (1) = I = 1 The CFD data from last teraton of the dentfcaton process V(P) was used for fllng the gaps n the nput expermental data Y. Paper ID:44 3
3 Results Typcal result of the estmaton procedure s presented n Fgure 5. The mssng data was flled wth data resulted from estmaton process..4.2 -.2 -.4 -.6-1.5-1 -.5.5 1 1.5 2 Fg. 5 Exemplary result of the fttng procedure. The estmated velocty fled (green) s superposed wth the masked data (red). 1 8 V estmated [m/s] 6 4 2 2 4 6 8 1 V model [m/s] Fg. 6 Plot of the freestream velocty estmated: V estmated versus V model (the smulated results) 4 Summary In the presented work a concept of a method of fllng of large gaps n PIV data was descrbed. The am of the work was to develop a method for replacng of the mssng data n vector velocty feld, especally for post processng of data from hgh speed wnd tunnel tests. The proposed nverse methodology nvolves fttng PIV data wth soluton of drect CFD smulaton of the nvestgated flow. The present paper covered the nverse problem formulaton, exemplary soluton of the drect problem and parametrcal dentfcaton usng Levenberg-Marquardt algorthm. In the presented paper, the results of performed numercal test was Paper ID:44 4
shown. Results show good agreement between the estmated and the masked data (see Fg. 6). It s mportant to underlne, that n presented prelmnary test the PIV data was smulated wth use of smplfed numercal 2D model of a flow. In the next stage of the procedure development, expermental PIV data from wnd tunnel tests (See Fgure 2) and 3D numercal model of the test chamber of the wnd tunnel wll be used. References 1. Anderson J D jr. (211) Fundamentals of aerodynamcs, Ffth edto n SI unts. McGraw-Hll, New York 2. Garca D (21) A fast all-n-one method for automated post-processng of PIV data, Experments n Fluds 5, vol. 5, pp. 1247-1259. do: 1.17/s348-1-985-y 3. Kasamatsu S, Itoh T, Watanabe S, Kuch-sh S, Yasue K (211) A Study on Vsualzaton for EFD/CFD Integraton, IEEE Pacfc Vsualzaton 211, pp. 5-6. 4. Ozsk M. N., Orlande H. R. B. (2) Inverse Heat Transfer, Taylor&Francs, New York 5. Raffel M., Wllert C., Wereley S., Kompenhans J. (27), Partcle Imagne Velocmetry - A Practcal Gude, Sprnger-Verlag Berln Hedelberg 6. Scarano F. (28) Overvew of PIV n Supersonc Flows. In: Schroeder A., Wllert C.E. (ed) Partcle Image Velocmetry, Sprnger-Verlag Berln Hedelberg 7. Schwartz R. J., Flemng, G. A. (27) Vrtual Dagnostcs Interface: Real Tme Comparson of Expermental Data and CFD Predctons for a NASA Ares I-Lke Vehcle, In Proceedngs of the ICIASF 7, pp. 1-12 8. Watanabe S., Kuch-sh S., Aoyama T. (21) A prototype system towards EFD/CFD ntegraton: dgtal/analog-hybrd wnd tunnel, In Proceedngs of the 27TH Internatonal Congress Of The Aeronautcal Scences ICAS21 Paper ID:44 5