Comparative Analysis of Velocity Decomposition Methods for Internal Combustion Engines

Transcription

1 Open Journal of Flud Dynamcs,,, Publshed Onlne September ( Comparatve Analyss of Velocty Decomposton Methods for Internal Combuston Engnes Semh Ölçmen, Marcus Ashford, Phlp Schnestsky, Mebougna Drabo Department of Aerospace Engneerng and Mechancs, The Unversty of Alabama, Tuscaloosa, USA Department of Mechancal Engneerng, The Unversty of Alabama, Tuscaloosa, USA Emal: Receved May 5, ; revsed June 9, ; accepted July 5, ABSTRACT Dfferent sgnal processng technque performances are compared to each other wth regard to separatng the mean and fluctuatng velocty components of a smulated one-dmensonal unsteady velocty sgnal comparable to sgnals observed n nternal combuston engnes. A smulaton sgnal wth known mean and fluctuatng components was generated usng expermental data and generc turbulence spectral nformaton. The smulaton sgnal was generated based on observatons on the measured velocty data obtaned usng LDV n a motored Brggs-and-Stratton engne at about 6 RPM. Expermental data was used as a gude to shape the smulated sgnal mean velocty varaton; fluctuatng velocty varatons wth specfed spectrum and standard devaton was used to mmc the turbulence. Cyclc varatons were added both to the mean and the fluctuatng velocty sgnals to smulate prescrbed cyclc varatons. The smulated sgnal was ntroduced as nput to the followng algorthms: ensemble averagng; hgh-pass flterng; Proper-Orthogonal Decomposton (POD); Wavelet Decomposton (WD) and Wavelet Decomposton/Prncpal Component Analyss (WD/PCA). The results were analyzed to determne the best method n correctly separatng the mean and the fluctuatng velocty nformaton, ndcatng that the WD/PCA performs better compared to other technques. Keywords: Proper-Orthogonal Decomposton; Wavelet Decomposton; Prncpal Component Analyss; LDV; Sgnal Processng. Introducton Understandng the nature of turbulence n nternal combuston engnes s mportant n studyng the underlyng mechansms between turbulence and related phenomena, such as nose generaton or lean/clean combuston []. Combuston qualty depends on the turbulent mxng of fuel and ar; turbulence defnes the flame speed, burnng temperature and the emsson of pollutants from combuston processes. Much research has been undertaken to study the n-cylnder flow felds usng many dfferent expermental technques [], ncludng hotwre anemometry [3], partcle trackng velocmetry (PTV) [4-7], partcle mage velocmetry (PIV) [7-9] and LDV [-8]. Among these technques, LDV has been wdely used snce LDV can be easly adapted to study flow felds wthn hard to reach geometres such as the valve ext flow or, the flow nsde complex bowl-pston confguratons [9]. Lterature survey ndcates that prevous research n studyng the n-cylnder flow felds s manly focused on analyzng the data collected n varous types of engnes usng varous methods. Erdl et al. [3] used a Rcardo E6/T varable compresson engne under motored condtons at 5 and RPM and obtaned data usng hot-wre anemometry for varous engne confguratons. Data were analyzed usng a turbulence flter, ensemble averagng and POD technques. Roudntzky et al. [] appled POD on -D tme-resolved PIV data obtaned n a sngle transparent cylnder at RPM to determne the mean component, coherent structures and random Gaussan fluctuatons. Fogleman et al. [] appled the POD technque to datasets obtaned n nternal combuston engnes to emphasze the tumble breakdown nstablty. Sullvan et al. [] used ensemble averagng, hghpass flterng and the Wavelet Decomposton n a comparatve fashon to analyze the characterstcs of turbulence n an L-Head research engne usng LDV data obtaned at 9 RPM. Ther analyss showed that there were no major dfferences between the mean velocty values plotted as velocty vs. crank-angle profles, but the power spectral densty plots showed consderable dfference at frequences hgher than Hz n the range of - Hz. The turbulence energy calculated by the ensemble-averagng technque was about twce the one calcu- Copyrght ScRes.

2 S. ÖLÇMEN ET AL. 7 lated usng hgh-pass flterng and WD technques. Ancmer et al. [3] proposed a WD based nose flterng technque to remove the cyclc varatons that can occur n the mean velocty to separate the turbulence and the mean velocty varatons. Park et al. [8] report LDV measurements made under motored condtons at RPM n a sngle cylnder of a V6 3. L optcal engne, and analyss made usng contnuous and dscrete WD n addton to ensemble averagng and hgh-pass flterng technques. Söderberg et al. [4] dscussed the correlaton between the heat release and turbulence measurements obtaned near the spark plug by a two-component LDV system on a four valve spark gnton engne usng WD technque. Measurements were made n a sngle-cylnder verson of a Volvo engne under skp-fred condtons at about 5 RPM usng fve dfferent camshafts. Results ndcate that combuston rate ncreases wth ncreased hgh turbulence rate and results do not change wth the use of dfferent wavelet functons. Sen et al. [5] measured the pressure varaton n a four-stroke, sngle-cylnder Aprla/Rotax spark gnton engne under sx dfferent loadng condtons to determne the maxmum pressure varaton n each cycle. The maxmum pressure values obtaned were then analyzed usng Morlet wavelets to determne the perodcty n the sgnals. The sgnals were shown to become turbulent at lower cycle counts wth ncreased torque loadng. Zhang et al. [6] dscuss the dffcultes nvolved n descrbng turbulence by conventonal methods and use an RI-splne-6 wavelet to defne the unsteady turbulence ntensty usng data obtaned n a constantvolume cylndrcal vessel equpped wth optcal access to measure the unburned flow velocty usng the LDV technque and the flame speed. They further study the effect of the turbulence ntensty on the turbulent burnng velocty. A bref summary of the prevous research gven wth selected references and the references ncluded n these publcatons ndcate that ensemble averagng; hgh-pass flterng; Proper-Orthogonal Decomposton; and Wavelet Decomposton technques are commonly used n prevous research n comparson to each other wthout the exact knowledge of the performance of the technque n determnng the turbulence. In the present work an unsteady sgnal generated as sum of the mean and fluctuatng parts wth known statstcs s used to compare dfferent analyss technques and evaluate the performance of these methods. The sgnal was generated based on the sngle component velocty data obtaned n a Brggs- Stratton engne usng an LDV probe. In the followng sectons frst the research outlne s gven, expermental data obtaned n a Brggs-and-Stratton engne s brefly descrbed. Secton dscusses the smulated sgnal. Secton 3 s devoted to descrbng the analyss methods. Comparson of the performance of analyss methods s gven n Secton 4, and conclusons derved from the present work are gven n Secton 5... Research Outlne In the current research a smulated sgnal representng the velocty sgnal of an nternal combuston engne was used to determne the performance of exstng methods n calculatng the mean and the fluctuatng varatons of ths sgnal n tme. Research was focused on determnng the effects of sgnal duraton and sgnal ampltude on the performance of the methods. Smulaton sgnal wth prescrbed mean and fluctuatng varatons n tme was generated based on sngle velocty component data obtaned n a motored engne. A modulated sne wave and multple Gaussan pulses were used to generate the mean velocty sgnal. Smulaton mean velocty sgnal ampltude was adjusted such that the arthmetc average of the sgnal at degree crank angle ncrements would closely follow the expermental data varaton. Low frequency modulatons were employed on the mean sgnal to represent cyclc varatons that mght be present n the mean velocty durng engne operaton. Fluctuatng velocty sgnal wth a known spectral varaton, essentally constant below Hz and varyng as f 5/3 above Hz, was generated to mmc the velocty spectra observed n other expermental work [3,7-9]. Ampltude modulatons mmckng cyclc varatons were also employed n generatng the fluctuatng velocty sgnal to approxmately match the standard devaton varatons observed n measured velocty data at dfferent crank angle ranges. The smulaton sgnal was then used to compare the performance of dfferent exstng methods n separatng the mean and the fluctuatng velocty components from each other, both n the tme doman and n the averaged sense. Tme-dependent mean and fluctuatng velocty nformaton was sought snce t was consdered that relatons between combuston processes and turbulence relevant to engne effcency would be tme-dependent. In order to determne the performance of the methods under dfferent condtons, smulaton sgnal duraton and the fluctuatng velocty ampltude were parametrcally vared. Two scenaros were smulated: ) Sgnal duraton was changed between.6 s and. s, whle the smulated fluctuatng velocty sgnal standard devaton,, was essentally kept constant; ) the sgnal duraton was kept as s, whle the fluctuatng velocty sgnal standard devaton was vared between 4 and 4. Tme dependent mean and fluctuatng velocty sgnals obtaned usng each method were compared to the smulated mean and fluctuatng velocty sgnals by determn- Copyrght ScRes.

3 7 S. ÖLÇMEN ET AL. ng the correlaton coeffcent between the mean and the fluctuatng velocty sgnals separately. Correlaton coeffcent values and the fluctuatng velocty standard devaton calculated by each method were used to determne the best method n separatng the mean and the fluctuatng velocty components from each other... Expermental Data The unsteady flow smulaton sgnal used n the current paper was based on expermental observatons made by the authors usng laser-doppler velocmetry data obtaned wthn a motored Brggs and Stratton sngle-cylnder ar-cooled engne desgned for low emsson operaton. Engne specfcatons are gven n Table. Crankshaft angular poston was determned va BEI H5 ncremental encoder wth.5 crank angle (CA) resoluton. In ths paper, CA corresponds to top-dead-center (TDC) at the end of the exhaust stroke. Cylnder pressure data were measured wth a Kstler 64A pezoelectrc pressure transducer coupled to a Kstler Type 5 charge amplfer. Engne data were recorded by a Natonal Instruments USB-6 data acquston system. For each test, the engne was motored nomnally at 6 RPM (RPM vared between 58-6) usng ts starter and a freshly charged V battery contnuously replenshed by a 4 amp charger. The LDV probe and the system used n the present work to collect the data are descrbed n the paper by Esrgemez and Ölçmen [3]. In the present work, sngle component velocty measurements were made usng the two-smultaneous velocty component, mnature LDV probe that was placed n the spark-plug hole of the Brggs and Stratton engne (Fgure ). Data was collected by a TSI-FSA-4 frequency-based processor and reduced by an n-house code to obtan the tme varaton of a velocty at a locaton near the spark-plug locaton. Probe volume wth 7 µm 7 µm.3 mm sze was located 3.6 mm away from the pston when t s at ts top-dead-center. Tme algnment of the LDV and the engne data was accomplshed va an n-house code. Fgure shows the expermental data non-dmensonalzed wth the maxmum value of the ensemble-averaged mean velocty. Postve velocty values ndcate flow nto the engne, such as the velocty values observed durng Table. Brggs and Stratton test engne relevant data. Model Number 5647 Combuston Chamber Type L Head Bore [mm] 87.3 Stroke [mm] Dsplacement [cc] exhaust valve footprnt pressure transducer a ccess port 8.8 ntake valve footprnt spark plug 5. laser beam access port LDV probe Fgure. The combuston bowl, crossng of the schematc laser beams ndcate the measurement probe volume. Dstances shown have mm unts. the ntake stroke ( - 8 degree range). Expermental data general characterstcs were next used to specfy the general characterstcs of the smulated sgnal, rather than analyzng the data, smlar to the approach taken by Kevlehan et al. [3], where the authors used multple methods to dentfy the vortcal structures they generated by a test functon. Data was used to calculate the mean velocty values n degree crank angle ncrements usng ensemble averagng (see Secton 3.). Dfference between the tme-dependent velocty and the mean velocty at a gven crank angle was further used to calculate the standard devaton of the data, data.5, (Secton 3.). The data was further used to adjust the standard devaton of the fluctuatng component of the smulated sgnal. Thus measured data was used both to adjust the smulated sgnal mean and fluctuatng velocty sgnals. The expermental data was used to ad n defnng the smulaton sgnal; the smulaton sgnal was not ntended to replcate the expermental data. The data acquston rate durng the experments was not hgh enough to capture the frequency varaton of the sgnal. Thus the data were plotted as velocty vs. the crank-angle range to obtan the velocty dstrbuton. The data rate also vared wth the crank angle resultng n more samples at dfferent crank-angle ranges. Thus ths precursory expermental data was used only to defne the general characterstcs of the smulaton sgnal.. Sgnal Generaton The goal of the present work was to compare analyss methods and determne the best method that could be used to separate the tme varyng mean velocty and the tme varyng fluctuatng velocty sgnals from each other. Thus the LDV data was used n a lmted role to adjust Copyrght ScRes.

4 S. ÖLÇMEN ET AL. 73 Non-dmensonal velocty Fgure. LDV data and smulated sgnal mean velocty varaton wth the crank angle. and match the varatons of the mean and fluctuatng components of the smulated sgnal to the expermental data. Fgure shows the smulated mean velocty varaton and the collected data. The turbulent part of the sgnal was generated usng a prescrbed power spectrum as descrbed n Secton.. Ths approach was taken snce the LDV measurement technque results n data that s unequally spaced n tme, requrng pre-processng technques pror to the applcaton of the methods dscussed n the current paper [3]. Although LDV technque has been employed n unsteady flows for the last three decades, ts nherently temporally rregular data acquston contnues to be nvestgated n obtanng spectrum from such data [33,34]. A data stream of one second duraton was smulated for use n the dscusson of the analyss methods descrbed n Secton 3 for easness of presentaton of the results. However, the Matlab code wrtten could be used to generate any desred tme length of data. Longer duraton smulated sgnal data has been generated and used n further analyss as dscussed n Secton 4 of the current paper... Mean Velocty Sgnal A snusodal wave wth unty ampltude was used as the base sgnal. The ampltude of the sgnal was vared to mmc the cycle-to cycle varatons that may be observed durng an engne run (Fgure 3(a)). The sgnal frequency was chosen as Hz, correspondng to ten crank revolutons per second (fve full-engne cycles per second), and 6 RPM, a typcal dle speed. The perod of the sgnal was kept constant assumng the RPM of the engne does not change durng the analyss. Next, Gaussan pulses were added to vary the sgnal locally n tme smlar to the ensemble-averaged mean velocty varaton of the expermental data. Gaussan pulses are used snce t allows sgnal generaton wth desred magntude and bandwdth at a prescrbed pont n tme. Multple Gaussan pulses (yg) were used as needed to effectvely defne the smulated sgnal. The smulated mean velocty varaton s expressed as:.*sn.4*π* *sn*π* U t t t t yg yg yg3 yg4 yg5 () t *t () denotes the varaton of crank angle n tme. Gaussan pulses were generated usng: yg C t td *π * bw * f c * exp( )*cos*π* f *.3 c t td 4*ln C = magntude, td = tme delay, t = tme, f c = cut-off frequency, bw = bandwdth of the sgnal. Fve Gaussan pulses together wth dfferent parameters were used. The parameters are gven n the Table. Table. Gaussan-modulated snusodal pulse parameters. Parameters/functons yg yg yg3 yg4 yg5 Magntude Tme delay Cut-off frequency Bandwdth (3) Copyrght ScRes.

5 74 S. ÖLÇMEN ET AL. Fgure 3(b) shows the sum of sgnals yg through yg5. The sum of the Gaussan pulse sgnals and the base sgnal descrbng the smulated mean velocty varaton n tme s shown n Fgure 3(c)... Turbulence Sgnal The fluctuatng velocty sgnal was generated usng a generc turbulence normalzed power spectrum that s also observed n IC engne flow felds, hw f f 5/3 3* f f [3,7,8,35] and usng the Yule-Walker relatons [36], where, f s the frequency, f s the hghest frequency n the spectrum (f = 36 Hz), and hw s the normalzed power of the sgnal at a partcular f/f [37]. The Yule-Walker relatons calculate the coeffcents of an auto-regressve model (a dgtal flter) to alter a Gaussan whte-nose sgnal (Fgure 4(a)) wth a standard devaton of unty to obtan a sgnal wth normalzed prescrbed power-spectrum at desred frequences, such as the frequences prescrbed n the functon hw gven by Equaton (4). Once ths flter s appled n tme doman to a Gaussan whte-nose sgnal, a tme-seres sgnal wth the prescrbed power spectrum s obtaned (Fgure 4(b)). In the present work, forty-fve Yule-Walker flter coeffcents were used. Tme-doman flter used was a flter that would result n a zero-phase dstorton n the fltered sgnal. A random number generator stream was chosen to generate the Gaussan whte-nose so that at every program run the random number stream would ntalze to the same value for the repeatablty of the results. The (4) tme step used n sgnal generaton was, t 7 s. Next, modulatons were employed on the fltered Gaussan sgnal to smulate cyclc varatons that mght be present n the fluctuatng velocty. It s beleved that the cyclc varatons n mean quanttes would result n cyclc varatons of the turbulence n such unsteady flows. Fltered Gaussan whte nose ampltude was changed usng a modfer sgnal (Fgure 4(c)):.*sn π* * MS t yg yg yg (5) The sgnal s composed of sne modulated Gaussan pulses, where yg6 and yg7 are descrbed by Equaton (3), and the parameters gven n Table 3. Cyclc varatons were consdered snce the turbulence transport s drectly related to the spatal gradents of correlatons and velocty products calculated usng the mean and fluctuatng velocty values and fluctuatng pressure as descrbed by the transport equaton for the Reynolds stresses [38]. Effect of unsteadness on the development of turbulence has been subject to multple papers, such as, n acceleratng ppe flows [39], pulsatng channel flows [38], or flow over arfols under deep stall condtons [4], where unsteadness causes unsteady varatons n turbulence quanttes. Cyclc varatons n turbulence were also contemplated by Enotads et al., [9]; Fansler, [8]; however no explct relatons were suggested. Ampltude of the fltered and modfed Gaussan nose was next adjusted to match the fluctuatng velocty varatons observed n the expermental data at dfferent crank angles. For ths purpose ensemble averaged standard devaton varaton of the expermental fluctuatng velocty data wth the crank angle was calculated and the fltered and modfed Gaussan nose ampltude was ad- U m base (a) U Gauss tme (s) tme (s) (b) U m (c) tme (s) Fgure 3. Smulated mean-velocty sgnal: (a) Base sgnal; (b) Gaussan pulses; (c) Composte mean-velocty sgnal. Copyrght ScRes.

6 S. ÖLÇMEN ET AL. 75 fltered whte nose Gaussan whte nose (a) tme (s) (b) tme (s) (c) MS tme (s) (d) u' power tme (s) power~f -5/3 (e) frequency (Hz) Fgure 4. (a) Gaussan whte nose; (b) Fltered Gaussan whte nose; (c) Modfer sgnal; (d) Smulated fluctuatng velocty sgnal; (e) Sngle sded spectrum of the fluctuatng velocty sgnal. Table 3. Gaussan-modulated snusodal pulse parameters used n fluctuatng velocty modfer sgnal. Parameters/functons yg6 yg7 yg8 Magntude.8.5 Tme delay Cut-off frequency Bandwdth 3 justed so that the standard devaton of the smulated sgnal would follow the varatons of the expermental data at dfferent crank angle ranges. Overall root-meansquare of the smulated sgnal and the expermental data were also matched. Turbulence sgnal s defned as: fltered Gaussan nose*.6.8* u t MS (6) and has the standard devaton as same as of the orgnal data, data.5 (Fgure 4(d)). Whle Equaton (5) descrbes the ampltude modfer sgnal, the use of t together wth Equaton (6) allows mposng perodc varatons on the fluctuaton sgnal. Sngle-sded power spectrum of the fluctuatng velocty sgnal (Fgure 4(e)) ndcates that the power decreases as ~f 5/3, as prescrbed by Equaton (4), wth the effect of ampltude modfcatons by Equaton (5) observed n - Hz range..3. Composte Sgnal The tme-dependent smulated velocty varaton s expressed as: Copyrght ScRes.

7 76 S. ÖLÇMEN ET AL. U t U t u t (7) where, t, s the crank angle varyng n tme between and 7 for a full cycle of a four stroke engne; denotes the mean velocty value at a crank angle; u t denotes the fluctuatng velocty. Fgure 5(a) shows the composte sgnal. Fgure 5(b) shows the ensemble averaged mean velocty sgnal and the tmevaryng sgnal presented as a functon of the crank angle. Sngle-sded power spectrum of the composte sgnal s shown n Fgure 5(c). The power contaned n the frequences above Hz decreases as ~f 5/3..4. Statstcal Quanttes The average mean-velocty varaton wth crank-angle usng only the orgnal mean velocty sgnal s defned as: U U t N (8) where denotes the prescrbed crank angles at whch the mean velocty values are calculated,,,,,79 ; ; N, s the number of samples n the range t for the whole duraton of the sgnal. Ths averagng assumes that the cycle-to-cycle varatons n the mean velocty can be gnored and the mean velocty wthn a prescrbed crank-angle range can be represented wth a sngle mean value. Fgure 6(a) shows the composte velocty varaton wth the crank angle. A standard devaton related to the mean velocty varaton s defned usng dfferent representatons of the mean velocty: U t U or,mean N (9) Ths value s dfferent than zero due to cyclc varaton n the orgnal mean velocty sgnal (Fgure 6(b)). Fgure 6(b) ndcates that presentng the mean velocty sgnal wth averaged values wthn crank angle ntervals results n the calculaton of a standard devaton dfferent than zero, whch could be wrongfully nterpreted as turbulence sgnal. The fluctuatng velocty sgnal u t was next used to obtan the standard devaton at prescrbed crank angle ntervals, the absolute value of the fluctuatng velocty, and an overall average standard devaton, gven wth the followng equatons, respectvely. or u t N () or u t u t () (a) U + u' tme (s) (b) U + u' - power (c) frequency (Hz) Fgure 5. Smulated composte sgnal: (a) Sgnal varaton n tme; (b) Sgnal velocty dstrbuton wth crank angle (scatter plot) and mean velocty calculated usng the orgnal mean (lne); (c) Sngle-sded spectrum of the composte sgnal. Copyrght ScRes.

8 S. ÖLÇMEN ET AL. 77 (a) U + u' or, mean or (b) (c) (d) u' tme (s) Fgure 6. Smulated sgnal statstcs: (a) Sgnal velocty dstrbuton wth crank angle; (b) Mean velocty standard devaton (Equaton (9)) calculated usng the orgnal mean and the ensemble averaged mean velocty; (c) Standard devaton varaton wth the crank angle; (d) Absolute value of the fluctuatng velocty n tme. where or,ave u t N () or descrbes an ensemble averaged varaton of the fluctuatng velocty sgnal (Fgure 6(c)), the or,ave defnes a sngle standard devaton value for the fluctuatng velocty sgnal ( or,ave.5 for the current sgnal). The u or shows the varatons of the sgnal from the mean n the absolute sense (Fgure 6(d)). 3. Analyss Methods-Descrpton/Dscusson Dfferent methods used n the analyss of the smulaton sgnal were, ensemble averagng, hgh-pass flterng, Proper Orthogonal Decomposton (POD) also known as Prncpal Component Analyss (PCA) [4], Wavelet Decomposton (WD), WD and PCA together (WD/PCA). In ths secton these methods are descrbed and some analyss results about ther use are gven. Methods dscussed n ths secton are the most commonly used methods n the analyss of cycle-to-cycle varaton of engne data. Usually multple technques are employed wthn the same research. Whle the ensemble-averagng method s the most commonly used method [5,7,4-45], hgh-pass flterng method has been frequently employed [,7,8,44-46]. Lterature survey ndcates that POD, WD and the WD/PCA methods have been less frequently employed [3,8,-5]. Prevous research on cyclc varablty ndcates that the tme varyng velocty n hghly unsteady IC engne flows has usually been expressed as sum of mean and fluctuatng components, Ut U()? t u t. In general, the mean velocty component s consdered as the component causng the cyclc varablty n the sgnal; although cyclc varablty n the fluctuatng velocty has also been contemplated [8,9]. The methods used by dfferent researchers dffer n how the mean and the fluctuatng veloctes are determned. In ensemble-averagng method t s assumed that the mean velocty does not vary from one cycle to the next and arthmetc averages of the veloctes at specfed crank angle ntervals can be used to descrbe the mean velocty varaton, U t U u en t en. In the hgh-pass (hp) flterng method [7], (named as cycle resolved analyss by Lou and Santavcca, [44] and Fraser and Bracco [45]; named as cyclc averagng by Sulvan et al. [], tme varyng velocty sgnal s fltered at a pre-determned cut-off frequency to calculate the low-pass (lp) fltered sgnal and ths sgnal s referred to as the mean velocty sgnal, U t U t u hp t. A varant of ths technque uses a polynomal curve fttng to obtan the mean lp velocty component [47]. Copyrght ScRes.

9 78 S. ÖLÇMEN ET AL. Another varant of hgh-pass flterng method, named as fltered ensemble averagng by Fansler [8] (named as phase averagng, Erdl et al. [3]), uses three terms to descrbe the velocty sgnal. In ths method the dfference between the low-pass fltered velocty and the ensemble-averaged mean velocty s defned as low frequency (LF) fluctuatng velocty and the u (t) s then consdered as composed of a hgh frequency (HP) component, U t U ulf t u HP t en. In ths varant the mean velocty s defned as, U t U en ulf t en, and the fluctuatng velocty as, u t uhp t. In essence hgh-pass flterng and the fltered ensemble averagng methods are equvalent to each other, and n each method dfferent cut-off frequences could be used at each cycle; although usually one frequency s used for the whole sgnal. In the WD method the velocty sgnal s decomposed nto detals, where the sum of the detals gves the orgnal velocty sgnal. The fluctuaton velocty nformaton contaned n the detal decreases wth the ncreasng detal level. In ths technque a hgh order detal s consdered as the mean velocty sgnal. In the POD method, velocty sgnal s re-expressed usng new axes found as the egenvectors of a covarance matrx formed usng the velocty sgnal. The egenvectors and the assocated egenvalues descrbe new coordnate axes on whch the data le the closest; the larger the egenvalue the closer the data s to the egenvector. In a reverse transformaton to the orgnal axes, neglectng the varatons of sgnal along the axes wth low egenvalues allows defnng the mean velocty component. In the WD/PCA method frst the velocty sgnal s decomposed nto detals and then the POD s appled on these detals. Smlar to the WD method a hgh order detal s then consdered as the mean velocty sgnal. Addtonal unque methods, such as turbulence flterng by Erdl et al. [3], dscuss possblty of separatng the mean and the fluctuatng veloctes wthn the frequency doman usng a flter functon. In the current research, both the mean and the fluctuatng veloctes were consdered to be tme dependent and both were consdered to have cyclc varatons. In each method fluctuatng velocty sgnals were next used to study turbulence statstcs. 3.. Ensemble Averagng In the ensemble averagng method, tme dependent velocty s defned as the sum of the ensemble-averaged mean velocty and the fluctuatng velocty: U t U u en t en (3) t, s the crank angle; denotes average where, value;, are the prescrbed crank angles at whch the mean velocty values are calculated. Ensemble averagng assumes that the mean flow varaton from one cycle to another s neglgble and an average velocty value can be used to represent the mean value at a crank angle. The average velocty at a crank angle s calculated as the arthmetc mean of the velocty values wthn a prescrbed crank angle nterval: U en t U (4) N,,,,79; ; N, s the number of samples n the range t for the whole U (Equaton (4)) s duraton of the sgnal. The dfferent than the U (Equaton (8)), snce the ensemble averaged mean velocty s obtaned usng the composte sgnal rather than the mean velocty varaton alone. Fgure 7(a) shows both ensemble averaged meanvelocty varatons calculated usng the composte sgnal and usng only the mean component of the composte sgnal (shown as smooth lne). A standard devaton s calculated usng the dfferences between the ensemble-averaged mean velocty and the mean velocty of the orgnal sgnal at the prescrbed crank angles usng the whole duraton of the sgnal as: en,mean U t U en ( N ) Fgure 7(b) shows the en (5) en,mean varaton calculated usng Equaton (5). The varaton shown s dfferent than the varaton shown n Fgure 6(b) calculated usng Equaton (9) due to the dfferences n the mean velocty defntons used. Fgure 7(b), as Fgure 6(b), ndcates to the fact that ensemble averagng results n large velocty fluctuatons due to predctng the mean velocty varaton ncorrectly by gnorng cycle-to-cycle varatons. The dfference between the orgnal sgnal mean and the ensemble mean (Equaton (4)) results n a fluctuatng sgnal. The fluctuatng velocty s next used to obtan standard devaton of the fluctuatng velocty at prescrbed crank angles: en u en N t (6) A standard devaton, en, ave usng the whole sgnal s calculated by: en, ave u t N (7) Fgure 7(c) shows the dfference between the standard Copyrght ScRes.

10 S. ÖLÇMEN ET AL. 79 U en, U U U en (a) (b) en,mean (c) ( en / or ) Fgure 7. (a) Ensemble averaged mean velocty varatons, U and U en ; (b) en Dfference between en and the or nondmensonalzed by or.,mean gven by Equaton (5); (c) devatons for the fluctuatng veloctes normalzed by the standard devaton of the orgnal sgnal. In the applcaton of the technque, smulated data wth prescrbed duratons were used to calculate the mean velocty and the standard devaton varaton wth crank angle. Calculated varatons for dfferent sgnal duratons dffer from each other snce the smulated sgnal contans cyclc varatons that affect the calculated values. Calculaton of the mean and the standard devaton varaton by averagng the data values over the span of the sgnal rules out the use of the technque to obtan tme-resolved varatons. 3.. Hgh-Pass Flterng In the hgh-pass flterng method the smulated sgnal s splt nto hgh and low frequency components at a cut-off frequency, f c, chosen by observaton (f c = 3 Hz). The tme dependent sgnal s low-pass fltered to obtan the mean velocty varaton n tme, U t (Fgure low pass 8(a)). The dfference between the orgnal sgnal and the low-pass fltered sgnal s consdered to be the fluctuatng velocty sgnal. hpf (8) U t U t u t low pass In the present work a sngle frequency value s used for the full length of the sgnal to separate the low and the hgh frequency contents. A Fast-Fourer-Transform s appled to the orgnal sgnal to determne the power spectrum of the sgnal and to subsequently determne a cut-off frequency. The orgnal tme seres sgnal was next fltered usng a sxth order Chebyshev flter n the tme doman to obtan the low frequency component. A relaton exsts between the mean velocty dfferences and the fluctuatng velocty dfferences calculated usng the orgnal velocty sgnal and the velocty sgnals calculated by the method: U t U t u t u t (9) hpf low pass Velocty dfferences are presented n Fgure 8(b). Hgh frequency sgnal was then used to calculate the fluctuatng velocty standard devaton at dfferent crank angles, and an overall average standard devaton: hpf hpf N u t () ( ) u t N () hpf, ave hpf Cut-off frequency f c = 3 Hz, results n hpf, ave =.957. Dfferent cut-off frequency values result n dfferent hpf, ave values; for f c =, 5,, and 5 Hz, the standard devatons are hpf, ave =.344,.46, Copyrght ScRes.

11 8 S. ÖLÇMEN ET AL. U - U low pass, u' hpf - u' U low pass, U tme (s) t U t tme (s) U low pass (a) (b) hpf / or - (c) Fgure 8. (a) Mean velocty varaton obtaned usng hgh-pass flterng method and the smulated sgnal mean velocty varaton (sold lne n red); (b) Varatons of U t U t, and u low pass hpf t u t wth tme; (c) Dfference between hpf and the or nondmensonalzed by or..3, and.9, respectvely. Wth the use of a 5 second long sgnal the values obtaned for the f c =, 5,, 5 and 3 Hz, are hpf, ave =.64,.9,.,.954,.788 respectvely, ndcatng that the sgnal length does have an effect on the values calculated, but the effect s not sgnfcant at hgh frequency end. Fgure 8(a) shows the mean velocty calculated by the hgh-pass flterng technque and the mean velocty varaton of the orgnal sgnal n tme. Fgure 8(c) shows the dfference between the standard devatons for the fluctuatng veloctes normalzed by the standard devaton of the orgnal sgnal Proper-Orthogonal Decomposton (POD) The proper orthogonal decomposton (POD) was frst ntroduced by Lumley [48] to dentfy coherent structures n turbulent flows. POD, also known as the Prncpal Component Analyss technque [4] allows one to represent data ponts each defned wth N coordnates n a new N dmensonal coordnate system such that the data ponts now le closer to the new axes, allowng varatons along preferred drectons to be determned [49]. For example data maybe the realzaton of three fluctuatng velocty components obtaned n tme, where the fluctuatng veloctes are the coordnates defnng the data pont obtaned n tme. In the representaton of the data n the new coordnate system, the axes are sorted such that the data le closer to the frst axs than the other axes and the data le closer to the second axs than to the thrd axs and so on. Representaton wth the new varables s sometmes referred to as sortng the data by ther energy content. One of the advantages of the POD technque s that after the representaton of the data n the new coordnate system, one can then approxmate the data by reducng the number of coordnates used, say by neglectng the less energetc coordnates n representng the data. For more detals and examples of POD technque on flud mechancs measurements, the reader s referred to Tropea et al. [5]. In the numercal calculatons of the POD technque multple sgnals collected smultaneously are frst stored n a data matrx, X, j, and the data matrx s then used to calculate a covarance matrx., * T, j, j Cov j X X X X () where, X denotes the average of the data values n the th row, thus X s a dfferent value for each row; superscrpt T denotes the transpose of the matrx; covar- Copyrght ScRes.

12 S. ÖLÇMEN ET AL. 8 ance matrx, Cov, j s an M M square matrx. The next step s calculaton of the egenvalues and the egenvectors of the covarance matrx. Egenvalues, k, where k = to M, and the egenvector matrx V, where k, = to M and k = to M, of the covarance matrx, are calculated usng Matlab. An egenvector correspondng to an egenvalue, k s lsted n the k th column of the egenvector matrx Vk,. The egenvectors of the covarance matrx correspond to the set of orthogonal axes (new coordnates) to represent the data. Once the egenvectors are sorted n order correspondng to the egenvalues of the covarance matrx, the projecton of the orgnal data along the egenvector drectons represents the data n these new orthogonal coordnates. Durng the process the sum of the square of the dstances between the new axes (along the egenvector drectons correspondng to egenvalues n descendng order) and the data are mnmzed. Projecton of the orgnal data along the drecton of the unt egenvectors can be obtaned usng: T, *, j Proj V k X X (3) Ths process can be reverted to obtaned the orgnal sgnal X, j, and can be recalculated usng: T X V * V * X X X (3), j k, k,, j The computatons n Equaton (4) also work f some of the egenvectors correspondng to less energetc egenvalues are omtted (set to zero) durng the reversal process, leavng behnd a sgnal close to the orgnal sgnal that has been cleaned of outler data ponts. In the present work, frst the complete velocty sgnal, wth each row of the matrx representng a full cycle of a four stroke engne spannng 7 of crank angle, wth the frst 7 sgnal (correspondng to the frst cycle) n the frst row, next 7 sgnal n the next row and so on. In descrbng was converted nto a data matrx, X, j X, j, = to M, where, M represent the number of cycles used n the sgnal and j = to N representng columns of the data correspondng to one full cycle of data spannng 7 of crank angle. Although ths s strctly dfferent than the conventonal use of the POD technque, t allows workng wth a sngle dmensonal data. Thus the data matrx s generated as f dfferent cycle veloctes were acqured smultaneously, resultng n veloctes correspondng to same crank-angle to be lsted n a sngle column. Presentng one dmensonal data n an array allows usng each cycle velocty value as a separate coordnate n the POD process. In the present work the mean velocty sgnal was calculated usng the largest egenvalue contanng 9.% of the energetc contrbutons to the sgnal (Fgure 9(a)) and the contrbutons from other egenvalue/egenvector couples were consdered as turbulence contrbutons. The fve egenvalues obtaned n the ncreasng order are 8.6, 5, 36.7, 8.9, and The contrbutons for the egenvalues for a second sgnal results n 5 egenvalues, and the approxmaton mproves wth the smulated sgnal duraton (dscussed n Secton 4.). In the POD method tme dependent velocty s defned as the sum of the mean and the fluctuatng velocty components: pod U t U t u t (5) Statstcal quanttes are defned smlar to other method statstcal quanttes: pod pod U t U t u t u t (6) pod pod u pod N t (7) u t N (8) pod, ave pod Fgure 9(b) shows the dfference between the mean veloctes and the fluctuatng veloctes for the orgnal sgnal and the ones calculated by the POD technque. Fgure 9(c) shows the dfference between the standard devaton calculated usng the POD technque and standard devaton of the orgnal fluctuatng velocty normalzed by the standard devaton of the orgnal sgnal Wavelet Decomposton Wavelet transformaton determnes the correlatons between the orgnal sgnal and the prescrbed wavelet functon at dfferent frequency ranges of the orgnal sgnal [5]. Followng the descrpton used by Farge [5] contnuous wavelet transform s defned as the convoluton of a wavelet () t, wth the orgnal sgnal to determne the wavelet coeffcents, where, ab, C a b f t t d t (9) t b (9) a / ab, t a a denotes the scale of the wavelet and related to the frequency of the sgnal, b s tme value used n translatng the wavelet; ab, t s the daughter wavelets generated by translatng and dlatng (scalng) the mother wavelet. The wavelet coeffcents thus calculated can be used to approxmate the orgnal sgnal at dfferent scales. Reconstructon of the orgnal sgnal s obtaned usng lnear combnaton of the wavelet and the wavelet coeffcent, d d f t C C a b t a a b ab, (3) Copyrght ScRes.

13 8 S. ÖLÇMEN ET AL. U t U t pod (a) U pod, U U - U pod, u' pod - u' tme (s) tme (s) (b) pod / or - (c) Fgure 9. (a) Mean velocty varaton obtaned usng the POD and the smulated sgnal mean velocty varaton (sold lne n red); (b) Varatons of U t U t - pod, and u pod t ut wth tme; (c) Dfference between pod and the or nondmensonalzed by or. where C, s a constant and depends on the wavelet used. The coeffcents wth small scale values correspond to hgh frequency component of the sgnal and the ones wth larger scale values correspond to low frequency values. Durng the reconstructon of the sgnal, length scales correspondng to the hgh frequency content can be omtted to obtan the low frequency (or mean) varaton of the orgnal sgnal. A hstory and a formal descrpton of the wavelet transforms can be found n artcles by Farge [53] and Tropea et al. [5]. Whle the mother wavelet can be presented n a functonal form for a smple wavelet such as the Haar wavelet, n general an algebrac functonal form does not exst. The mother wavelet t, whch can be a real or complex valued functon, s chosen to satsfy multple condtons, such as admssblty condton requrng that the energy contaned wthn the wavelet to be a lmted value, zero mean condton requrng the wavelet to have a zero mean value, and a requrement that ts hgher order moments vansh [53]. In the dscretzed wavelet transform the scale and the poston t are descrbed usng powers of. Matlab [54] gves detals about the dscrete wavelet transform and the reader s referred to ths document for further detals. For a contnuous sgnal f t, the dscrete wavelet transform s gven as: t b Ca, b f ta dt a (3) j j where a, b k*. j,,, k,, and the transformaton gves the wavelet coeffcents, Ca, b,. The orgnal sgnal can be recovered wthout any loss usng the nverse transform: f t C j, k jk, t (3) j k For a fxed value of j, summaton on the k gves the detal at the j th level: Djt C j, k j, k t (34) and thus, k f t Dj t (35) j An approxmaton to the sgnal can be made usng a lmted number of detals. For example f t can be separated nto two components, one usng ndces j J, j J correspondng to the scales, a and one wth j > J. j j J t (36) f t D t D t A D j jj jj jj Copyrght ScRes.

14 S. ÖLÇMEN ET AL. 83 If desred, the reconstructed sgnal could then be expressed only as approxmaton, A, at level J. A relaton exsts between dfferent approxmatons as: A J A J D (37) j J Increasng values of J ndcates that lesser detals are used n generatng the approxmate sgnal, thus the approxmaton becomes closer to sgnal s mean varaton. The value of J should also not be chosen large such that the detals n the mean varaton are also sacrfced. In the current work Daubeches, Symlet 8, and Coflet 5 wavelets were tested and the results obtaned dd not vary wth the use of dfferent wavelets. Thus Daubeches wavelet was chosen for further analyss of the data. Fgure shows the scalng and the wavelet functons for the Daubeches- wavelet. In the wavelet decomposton tme dependent velocty s thus defned as an approxmaton plus the detals: wd U t U t u t (38), J wd j wd jj U t A8 where U t A u t D t, and J = 8. Fgure (a) presents the and the wd orgnal mean velocty varaton n tme. Statstcal quanttes are calculated usng: wd wd wd U t U t u t u t (39) wd u wd t (4) ( N ) u t N (4) wd, ave wd Fgure (a) shows the orgnal mean velocty and the mean velocty calculated by the WD technque. Fgure (b) shows the mean and the fluctuatng velocty dfferences, Fgure (c) shows the dfference between the standard devaton calculated usng the WD technque and standard devaton of the orgnal fluctuatng velocty normalzed by the standard devaton of the orgnal sgnal Prncpal Component Analyss and Wavelet Decomposton In ths analyss method the advantages of the Prncpal Component Analyss and the Wavelet Decomposton methods are used smultaneously [55]. In the technque frst wavelet decomposton s appled on the sgnal to calculate the detals and one approxmaton to the sgnal. Next PCA analyss s employed on each of the detal sgnals and the approxmaton. PCA analyss results are used to determne whch prncpal components to keep n the sgnal usng the Kaser rule. Kaser rule keeps only the egenvalues, whch have values greater than the mean of all egenvalues n re-constructon of the sgnal. As a next step the detals and the approxmaton reconstructed usng the PCA are used n the reconstructon of the sgnal usng the wavelet decomposton. In the detecton of the mean velocty sgnal the wavelet decomposton allows the sgnal to be analyzed at dfferent scales and the prncpal component analyss allows de-nosng of the sgnal at every detal. In the WD/PCA method tme dependent velocty s defned as the sum of the mean and the fluctuatng velocty components, wdpca U t U t u t (4) wdpca Statstcal quanttes are defned usng, wdpca U t U t u t u t (43) wdpca t (44) wdpca u wdpca N u t N (45) wdpca, ave wdpca (a) (b) Fgure. (a) Scalng and (b) The wavelet functons for Daubeches- wavelet. Copyrght ScRes.

15 84 S. ÖLÇMEN ET AL. U t U t wd (a) U wd, U U - U wd, u' wd - u' tme (s) tme (s) (b) wd / or - (c) Fgure. (a) Mean velocty varaton obtaned usng the WD and the smulated sgnal mean velocty varaton (sold lne n red); (b) Varatons of U t U t wd u wd t u t wth tme; (c) Dfference between wd and, and or nondmensonalzed by or. the Fgure (a) shows the orgnal mean velocty and the mean velocty calculated by the WD technque. Fgure (b), shows the mean and the fluctuatng velocty dfferences, Fgure (c) shows the dfference between the standard devaton calculated usng the WD/PCA technque and standard devaton of the orgnal fluctuatng velocty normalzed by the standard devaton of the orgnal sgnal. 4. Results and Further Dscusson A comparson between the Fgures 8(a), 9(a), (a), and (a) show that all the mean velocty varatons calculated by dfferent methods follow the general varatons observed n the smulated sgnal. Fgures 8(b), 9(b), (b), and (b) show the dfferences of the mean and the fluctuatng veloctes between the orgnal velocty components and the components calculated after each method, and gve a sense of how well the methods perform n descrbng both the mean velocty and the fluctuatng velocty vs. the crank angle. If a method could perfectly separate the mean and the fluctuatng velocty values n tme then the plots would be a straght lne at zero. Fgures ndcate that the dfferences between the velocty sgnals are the smallest usng the WD/PCA method, and that the POD method results contan hgh frequency content. POD performance mproves wth ncreased sgnal duraton and the hgh frequency dmnshes. Fgures 7(c), 8(c), 9(c), (c), and (c) show the normalzed dfference between the computed and the orgnal sgnal standard devatons. Fgures 8(c) and (c) ndcate that the hgh-pass flterng and the WD technques result n larger dfferences between the standard devatons than the other methods. The dfferences between the fluctuatng and the mean veloctes obtaned usng the orgnal smulated velocty sgnal and the veloctes computed after each method both contrbute n the calculaton of the fluctuatng velocty standard devaton. Ths s due to the fact that the dfference between the orgnal velocty sgnal and the calculated mean velocty sgnal s consdered as a contrbuton to the fluctuaton velocty calculated by a method. For the second analyss results shown n the fgures, large non-dmensonal dfferences are observed at locatons where the orgnal or values are small such as n the 5-8 and 45-5 ranges. Overall, the WD/PCA technque results follow the orgnal standard values better than the other technques. Copyrght ScRes.

16 S. ÖLÇMEN ET AL. 85 U - U wdpca, u' wdpca - u' U wdpca, U tme (s).5 (b) tme (s) (c) wdpca / or - U t U t Fgure. (a) Mean velocty varaton obtaned usng the WD/PCA and the smulated sgnal mean velocty varaton (n red); (b) Varatons of U t U t wdpca, and u wdpca t ut wth tme; (c) Dfference between wd pca and the or nondmensonalzed by or. wdpca (a) 4.. Smulated and Calculated Sgnal Correlaton Effect of Sgnal Duraton In order to quanttatvely determne the performance of each method n separatng the mean and the fluctuatng velocty varatons, and the sgnal duraton requred to effectvely separate the mean and the fluctuatng velocty sgnal components correlaton coeffcents were calculated between the orgnal smulaton sgnal and the calculated sgnals. Correlaton coeffcents were calculated usng: corr coeff and corr coeff mean fluc * U t U t method U t * U t u t * u t u t * u t method method method (46) (47) A correlaton coeffcent of ndcates a perfect match between the two sgnals. Correlaton coeffcents calculated for sgnal duratons of.6 s to. s wth.6 s ncrements plotted n Fgure 3 ndcate that the WD/PCA method performs better than the other methods n calculatng the mean and the fluctuatng velocty sgnals. Both Fgures 3(a) and (b) ndcate that the performance of the methods get better wth ncreased sgnal duraton. Results do not show apprecable mprovement after the frst 3 seconds, and the changes after 8 seconds are mnmal ndcatng that 3 seconds of data could be suffcent to extract the mean and the fluctuatng velocty sgnals. Plots n Fgures 3(a) and (b) show smlar varatons snce poor mean velocty predcton by a method drectly reflects on the fluctuatng velocty predcton performance of the technque. Results also ndcate that the WD and the hgh-pass flterng methods predct the mean and the fluctuatng velocty varatons poorly even wth ncreased sgnal duraton. Correlaton coeffcents as low as.8 obtaned for the fluctuatng sgnal ndcate that the tme varaton of the fluctuatng velocty calculated by these methods s a poor representaton of the orgnal fluctuatng velocty sgnal. Fgure 3(b) shows that the correlaton coeffcent values calculated even wth the WD/PCA method s about.95 ndcatng to the need for further mproved methods for accurate tme varaton predcton of the fluctuatng velocty sgnal. Fgure 4 shows the rato of the average standard devatons calculated by each method to the orgnal sgnal average standard devaton for a gven duraton of sgnal. Normalzed standard devaton (NSD) was calculated usng: Normalzed standard devaton method, ave or, ave (48) Fgure 4 ndcates that the standard devaton calculated by the WD/PCA method s approxmately the same as the or,ave once a 5 s sgnal duraton s used. All the other methods except the ensemble-averagng method underestmate the ; ensemble averagng method or,ave Copyrght ScRes.

17 86 S. ÖLÇMEN ET AL..995 e a n.99 m - t n.985 e fc e.98 c o n to.975 la re.97 c o Correlaton coeffcent Mean.965 hgh-pass pod wd wdpca A(a) sgnal duraton (second) c flu -.9 t n e fc.85 e c o n.8 to la re.75 c o Correlaton coeffcent fuc.7 B(b) hgh-pass pod wd wdpca sgnal duraton (second) Fgure 3. Correlaton coeffcent values obtaned between the smulated mean and the smulated fluctuatng velocty sgnals and the mean and fluctuatng velocty components obtaned by analyss methods. (a) Mean velocty; (b) Fluctuatng velocty results..5 n to e v a d.95 a rd d n sta d.9 a lze rm o.85 n normalzed standard devaton.8 ensemble hgh-pass pod wd wdpca sgnal duraton (second) Fgure 4. Rato of the average standard devatons calculated by each method to the orgnal sgnal average standard devaton for a gven duraton of sgnal. Copyrght ScRes.