A Hybrid Data-Model Fusion Approach to Calibrate a Flush Air Data Sensing System

Transcription

1 AIAA - Aprl, Atlata, Georga AIAA A Hybrd Data-Model Fuso Approach to Calbrate a Flush Ar Data Sesg System Akur Srvastava Rce Uversty, Housto, Texas, 775 Adrew J. Meade Rce Uversty, Housto, Texas, 775 ad Al Arya Mokhtarzadeh 3 Rce Uversty, Housto, Texas, 775 Abstract A hybrd data-model fuso approach has bee devsed to calbrate a Flush Ar Data Sesg system. Numercal smulato ad artfcal eural etwork based approaches have bee prevously appled to predct ardata state from flush pressure measuremets. However both approaches have ther shortcomgs. Numercal smulatos rely o approxmatos to model the truth whle learg algorthms do ot corporate the physcs of the problem ad ofte eed a large set of data for trag. A prcpled approach has bee devsed to fuse expermetal FADS data ad umercal solutos a optmal maer. The purpose of ths approach s to mprove the predcto accuraces of the ardata state obtaed by a pure eural etwork based approach. Other objectves of ths approach clude better ose tolerace ad a eed for fewer expermetal data. b, b, b = c, c, c = C p = coeffcet of pressure = P/q d = put dmeso f = arbtrary fucto fps = feet per secod F = forward mappg fucto G = verse mappg fucto Nomeclature th bas the approxmato th lear coeffcet the approxmato = umber of bases Pq, = statc gauge pressure ad dyamc pressure r = th stage of the fucto resdual = real coordate space d = d- dmesoal vector space s = umber of trag samples u ξ = target fucto ( ) ( ) u ξ = th stage of the target fucto approxmato a V = freestream wd speed Graduate Studet, Mechacal Egeerg, Mal Stop 3, AIAA Member. Professor, Mechacal Egeerg, Mal Stop 3, AIAA Assocate Fellow. 3 Udergraduate Studet, Mechacal Egeerg, AIAA Member. Copyrght by the Amerca Isttute of Aeroautcs ad Astroautcs, Ic. All rghts reserved.

2 f v = fv = dscrete er product of f ad v, D s ( ) f = absolute value of f f = ( f dξ = L orm of f,d ( Ω ) / / s ( ) ) orm of f f = f = dscrete L Greek Symbols β = yaw agle η = set of olear optmzato parameters ε = objectve fucto θ = polar coordate λ = wdth parameter of Gaussa bass fucto ξ = d-dmesoal put of the target fucto ξ = th sample put of the target fucto * ξ = sample put wth compoet dex j * at the th stage τ = user specfed tolerace φ = bass fucto ν = arbtrary fucto δ = put sestvty ρ = desty Subscrpts = dummy dex j * = compoet dex of r (-) wth the maxmum magtude = assocated wth the umber of bases s = assocated wth the umber of samples = assocated wth freestream codtos Superscrpts a = assocated wth the approxmato d = assocated wth the put dmeso I. Itroducto CCURATE determato of ardata state parameters lke dyamc pressure, agle of attack ad A agle of sdeslp are mportat for arcraft cotrol ad for ordace or wdage correctos for surface vessels. Tradtoal aemometer systems lke ptot probes ad laser velocmetry systems have several dsadvatages cludg degraded performace uder hgh agles of attack for arcraft ad creased radar cross-secto area for surface vessels. Flush Ar Data Sesg (FADS) systems have prove to be a vable alteratve overcomg these shortcomgs may cases. Pressure based FADS systems

3 stall a array of pressure sesors flush to the surface of the vehcle that measure real tme gauge pressure across the vehcle surface. Pressure sesors Fg. Pressure sesors stalled o a) ose of F-8 systems research arcraft ad b) deck perphery of DDG 3. I FADS, the problem of estmatg relatve wd speed ad drecto from the surface pressure at a posto θ ca be framed as ether a forward or verse mappg problem,.e., ( ρ θ β)} (, ρ, θ, β) (,, V, ) P = F, V,, the forward problem V = G P β = G P ρ θ the verse problem. () Here ρ, V, P are freestream desty, speed ad pressure, respectvely. The parameter β represets the yaw agle whle F ad G are the forward ad verse mappg. For arcraft, pressure sesors eed to be stalled oly o the ose whch allows a sem-emprcal fucto to be used for F. Oce F has bee establshed, olear least squares or eural etworks ca be used to estmate the ukow ardata parameters,. Emprcal pressure models ca also be costructed thorough umercal solutos aloe 4. However, estmates of F caot be obtaed for arbtrary geometres. I such problems, t becomes essetal to focus o eural etwork based approaches that have bee prove to be effcet solvg llposed verse problems. I the cotext of FADS problems artfcal eural etwork ad mache learg

4 approaches lear the fuctoal relatoshp betwee pressure ad freestream wd velocty. Ths approxmato s depedet of the complexty of the geometry of the bluff body. However, the predcto accuraces of a eural etwork based approach deped o the followg: a. Sgal to ose rato: Nose could be duced the statc pressure measuremets due to the heret ucertaty of the pressure trasducer. Other sources of ose clude ucertates the atmospherc sesor ad desty measuremets, errors due to thermal stablty, mproper tubg ad sesor stallato. b. Icompleteess: Ths work deals wth calbratg FADS systems for bluff bodes wth arbtrary geometres. Ulke arcrafts, bluff bodes that experece sgfcat cross-flow eed pressure sesors stalled all aroud oe or several plaes of the body surface. However, pressure sesor stallato ca become a ssue because of lack of space, presece of let or exhaust valves that dsturb the local arflow or prohbtve labor requremets. Such costrats ca lmt data collecto ad ca possbly dmsh wd speed ad drecto predcto accuraces. c. Sparsty: Performace of ay algorthm to predct wd speed ad drecto from pressure measuremets drectly depeds o the umber of trag pots. Fer resoluto of trag data would demad legthy, costly ad tedous wd tuel rus. Ay mathematcal techque that helps egeers accelerate through the test matrces s welcome. I a earler work, Srvastava et al. 5 dsplayed the use of a greedy mache learg techque for calbratg the FADS system for a surface vessel. The authors showed that Sequetal Fucto Approxmato (SFA) could be successfully used to ot oly predct wd speed ad drecto from pressure measuremets, but also help egeers accelerate through the test matrx. Fgures ad 3 show the ma results of the paper. Fgures a ad b compare the sestvtes of each of the 5 pressure sesors wth respect to wd speed ad drecto predcto respectvely. These graphs ca form the expermetalst what locatos oe should focus o to stall ew sesors. Fgures 3a ad 3b show the degradato wd speed ad drecto predcto accuracy wth creasg umber of pressure sesors ad decreasg umber of trag pots respectvely. These fgures ca form the expermetalst as to how may wd tuel rus eed to be collected ad how may pressure sesors are suffcet for wd speed ad drecto predcto.

5 7 4 Average Pressure Tap Sestvty Average Pressure Tap Sestvty Pressure Tap Number Pressure Tap Number (a) (b) Fg. Average put sestvtes the predcto of (a) wd speed ad (b) shp bow yaw agle Wd speed Wd drecto 4 3 Wd speed Wd drecto Logarthm of predcto error 3.5 Logarthm of predcto error Number of pressure taps Number of trag pots Fg. 3 Logarthm of predcto error degradato wth a) the umber of pressure taps ad b) the umber of trag pots. The bars show the stadard devato of the errors 5. The objectve of ths work s to study how umercal smulatos ca be used apror wth Sequetal Fucto Approxmato to result better pressure dstrbutos whch would mply better qualty of trag data ad thereby better wd speed ad drecto predcto accuraces. The techque preseted here s smple, yet powerful ad a geerc way of fusg umercal solutos wth smoothess based mathematcal approxmato of expermetal data. The trag ad testg procedure of SFA has bee

6 modfed the past for data-model fuso applcatos 6. I ths problem we use SFA to model the error betwee the wd tuel pressure data ad the pressure dstrbuto approxmated by full three dmesoal Naver-Stokes equatos wth turbulece modelg. Such a approach would result smoother ad more accurate pressure dstrbutos tha that obtaed by wd tuel or umercal expermets. Ths approach could be used wth FADS o arbtrary geometres wth varyg levels of flow complexty but ths work we demostrate the applcato of ths approach o FADS for a Ruway Asssted Ladg Ste (RALS) cotrol tower (Fg. 4). The specfc objectves of ths paper are the followg:. Ivestgate how CFD techques ca be used to compesate for sparsty, ose ad complete wd tuel data.. Use CFD to determe possble locatos of pressure ports that ca mprove wd speed ad drecto predcto accuracy. We beleve that a data-model fuso approach based o eural etworks ca be mmesely helpful reducg materal ad labor costs of expesve wd tuel testg. It ca also be see as a approach that uses physcs based regularzato to reduce ucertaty ad sparsty of the data. Fg. 4 Ruway asssted ladg ste cotrol tower 7 Our paper s orgazed as follows: Secto II we dscuss the wd speed ad drecto estmato schemes. We preset several approaches, lst ther beefts ad drawbacks ad pck the most sutable

7 approach. I Secto III we dscuss the data-model fuso aspect of ths work. I Secto IV we compare the ablty of smoothess ad physcs based regularzato techques to hadle ose, sparsty ad completeess of pressure data. I Secto V we preset the wd speed ad drecto predcto accuraces of the chose estmato techques. Secto VI presets coclusos ad future work. II. Approach A. Forward ad Iverse Problems Borrowg from the flght mechacs lterature ad restrctg our problem ths paper to twodmesos, we ca defe β as the yaw agle. Usg the compressble flow about a rght crcular cylder (Fg. 5), the problem of estmatg relatve wd speed ad drecto from the surface pressure at posto θ ca be framed as ether a forward or verse mappg problem as show Eq. (). For smple geometres, closed-form sem-emprcal equatos ca be developed ad used to solve the forward problem of estmatg the statc pressure coeffcet, C p, gve the freestream wd speed ad drecto 8. Pael methods 9 ad other popular Computatoal Flud Dyamcs (CFD) based approaches solve the forward problem o more complex geometres gve the freestream wd speed ad drecto. However, ths requres guessg the freestream velocty ad solvg for the pressure dstrbuto utl t matches the measured values. To solve the verse problem, look-up tables ca be costructed for the mappg fucto G. Ufortuately, look-up tables suffer from a umber of drawbacks whe ther real-tme use s cosdered cludg the ablty to hadle hgh-dmesoal puts, osy data, ad olear terpolato. Computatoal algorthms lke eural etworks that lear from data ca be very effectve solvg verse problems ad bear potetal advatages the costructo of the mappg fucto G compared to look-up tables. Examples of eural etworks advatages clude hgh-dmesoal mappg, smoother olear cotrol, tellget emprcal learg ad fewer memory requremets,,, 3. For speed of evaluato, reduced complexty, ad the potetal for graceful performace degradato wth the falure of pressure sesors, we have chose to solve the verse problem usg oly eural etworks wth pressures read from all surface sesors, (, d) β (, d ) V = G P, P ad = G P, P, V.

8 The problem of developg the mappg fucto G wth emprcal formato falls uder the category of regresso statstcs ad mache learg research. Popular regresso methods clude sples 4, projecto pursut regresso 5, radal bass fucto etworks 6 ad back-propagato etworks 7. These methods requre the use of user-determed cotrol parameters ad/or kerel hyper-parameters. The user must fd the optmum values of the cotrol parameters for the etre data set ether by cross-valdato or a grd search approach. I such tme cosumg approaches the data must be used to geerate umerous radomly selected subsets for trag ad testg. The values of the cotrol parameters must the be optmzed o each of these testg subsets ad the optmum cotrol parameters averaged. We beleve that for the purpose of ths type of flud-structure teracto problem, a mult-dmesoal learg tool should address the prevously metoed hyper-parameter selecto problem, operate as easly o hgh-dmesoal data as t does o low, ad provde the user a assurace that the tool wll gve ts best performace o a usee test set. The tool should also requre a mmum of storage ad as lttle user teracto as possble. Fally, for the purposes of geeralzato ad data compresso, the scheme should costruct a accurate approxmato that uses as few bass fuctos as possble, preferably less tha the umber of sample pots. I referece [5] we developed just such a learg tool that ca solve the verse problem uder cosderato, whch we call Sequetal Fucto Approxmato (SFA). SFA was orgally troduced to solve dfferetal equatos 8 but was later used to provde kerel based solutos to regresso 9 ad classfcato problems. Fg. 5 Two-dmesoal flow about a rght crcular cylder.

9 C. Sequetal Fucto Approxmato Sequetal Fucto Approxmato (SFA) was developed from mesh-free fte elemet research but shares smlartes wth the Matchg Pursut ad Boostg algorthms that they ca all be classfed as greedy algorthms. We start our approxmato of u utlzg the Gaussa Radal Bass Fucto (RBF)φ. a ξ c = u ( ) = ( φ( ξ, η ) + b ) * * ( ξ ξ ) ( ξ ξ ) * * wth φ( ξ, η) = exp = exp l [ λ]( ξ ξ ) ( ξ ξ ) for λ σ < < () d where ξ represets the d-dmesoal data pot, c, b are the lear coeffcets ad bas, ad * d + η represets the th = { λ, ξ } * d bass fucto parameters, the wdth λ ad the ceter ξ wrte the RBF as Eq. () order to set up the optmzato problem for λ as a bouded olear le search stead of a ucostraed mmzato problem. The basc prcples of our greedy algorthm are motvated by the smlartes betwee the teratve optmzato procedures by Joes 3, 4 ad Barro 5 ad the Method of Weghted Resduals (MWR), specfcally the Galerk method 6. We ca wrte the fucto resdual r at the th stage of approxmato as the followg: ( ) a a r = u( ξ) u( ξ) = u( ξ) u ( ξ) c φ( ξ, η) + b ( φξη ) ( φ ). We = r c (, ) + b = r c + b. (3) Usg the Petrov-Galerk approach, we select a coeffcet c that wll force the fucto resdual to be orthogoal to the bass fucto, ad usg the dscrete er product gve by Eq. (4) b r r, r D r ( + b ) = r + b = r,( + b ) = r, = =, (4) ( )( ( ) ) s φ ξj φ ξj φ D j= c c D whch s equvalet to selectg a value of c that wll mmze r, r.e. D c = ( s φ, r r D φ D D) ( s φ, φ φ φ ) D D D (5) wth

10 b = ( r φ, φ φ φ, r D D D D). ( s φ, r r φ ) D D D (6) The dscrete er product r, r D, whch s equvalet to the square of the dscrete L orm, ca be rewrtte, wth the substtuto of Eq. (5) ad (6), as r φ,,. (7) s s s s D D D D r, φ s r r s s ( ) ( ) D r D D ξj r ξj = r = r, D r = r D r j= s s D φ φ D r D r D D φ, φ r, r Recallg the defto of the cose, whch s gve by Eq. (8) usg arbtrary fuctos f ad v ad the dscrete er product, f, v D cos( θ ) =, (8) / / f, f v, v D D Eq. (7) ca be wrtte as r r r D D D r = r,, r s ( θ) r s ( θ ), D =, D s s s (9) where θ s the agle betwee φ ad r sce r D s ad φ s D are scalars. Wth Eq. (9) we ote that r < as log as θ π /, whch s a very robust codto for covergece. By specto,, D r, D the mmum of Eq. (9) s θ =, mplyg Eq. () r D c ( φξ (, β) + b) = r ( ξ) for =,, s () s Therefore, to force r, D wth as few stages as possble, a low dmesoal fucto approxmato problem must be solved at each stage. Ths volves a bouded olear mmzato of Eq. (7) to * * d determe the varable λ (< λ ) ad dex j ( ξ = ξ ), represetg the bass fucto ceter take from the trag set. The dmesoalty of the olear optmzato problem s kept low sce we are solvg oly oe bass at a tme. j

11 D. Ukow Parameters Ay regresso problem ca be treated as a fucto approxmato problem. I ths work we costructed our approxmato as a lear sum of Gaussa radal bass fuctos gve by Eq. (). The approxmato s costructed va a four-dmesoal optmzato problem for the coeffcets ( ), the bas ( b ), the wdth parameter ( λ ), ad the ceter of the bass fucto ( ξ ) at each stage. Icorporatg the MWR we calculated the optmum value of the coeffcets ad bas gve by Eqs. (5) ad (6). Ths left us wth a two-dmesoal optmzato problem for the ceter ad the wdth at each stage. I order to reduce computatoal tme ad expese we used a heurstc rather tha optmzato techques to locate the ceter of the RBF at each stage. For each bass fucto the ceter was chose to be the trag vector that correspods to the maxmum absolute value of the resdual at that stage. We are the left wth a oe-dmesoal optmzato problem, gve by Eq. (7), for the wdth of the bass fucto whch we solve at each stage the algorthm. A commo stoppg crtero used by greedy sparse approxmato algorthms for regresso problems s to put a threshold o the resdual error. Studyg ad comparg the effectveess of other stoppg crtero lke the mmum descrpto legth crtero 7, the Akake formato crtero 8, or a combato of several dfferet crtera costtutes future work. I ths work we stop addg bass fuctos whe ether the maxmum absolute value of the resdual error falls below a user determed tolerace (τ ) or the umber of bass fuctos exceeds the umber of trag pots. c E. Implemetato of the Algorthm Though our SFA scheme allows the bass ceter ( ξ d ) to be located aywhere, the practcal applcato to problems wth multple puts costras the ceters to the set of sample pots{ ξ,..., ξ }. At each stage we determe ξ such that r ( ) max ξ = r. The remag optmzato varable λ s cotuous ad costraed to < λ <. A beeft to usg RBFs s that practcal applcatos we ca gore the deomator the dscrete er product formulato of Eq. (7). As a result, the determato of η requres oly s

12 r D m r, φ β s φ s D D () The algorthm termates whe max r τ or the umber of bass fuctos exceeds the umber of trag pots. To mplemet the SFA algorthm, the user takes the followg steps:. Itate the algorthm wth the labels of the trag data r = { u( ξ ),..., u( ξs )}.. Search the compoets of r for the maxmum magtude. Record the compoet dex j. 3. ξ ξ j =. 4. Wth φ cetered atξ, mmze Eq. () wth a o-lear le search optmzato techque. j 5. Calculate the coeffcet c ad b from Eq. (5) ad (6) respectvely. 6. Update the resdual vector r = r c ( φ + b ). Repeat steps 3 to 6 utl the termato crtero max r τ has bee met. 7. Use the costructed approxmato to predct o the test set. Our SFA method s lear storage wth respect to s sce t eeds to store oly s + sd vectors to compute the resduals: oe vector of legth s ( ) ad s vectors of legth r ( ) d ξ,..., ξ s where s s the umber of samples ad d s the umber of dmesos. To complete the SFA surrogate requres two vectors of legth ({ c,..., c } ad { b,..., b }) ad vectors of legth + ( η,..., η ). The dmesoalty of the olear optmzato problem s kept low sce we are solvg oly oe bass at a tme. Also, Meade ad Zeld 9 showed that for bell-shaped bass fuctos, the mmum covergece rate s [ κ κ ], exp where, D r r r = () for postve costats ad κ. Equato () shows that for mmum covergece, the logarthm of the κ er product of the resdual as a lear fucto of the umber of bases (), whch establshes a expoetal covergece rate that s depedet from the umber of put dmesos (d).

13 F. Problem Defto The curret problem s to study the feasblty of a FADS system stalled o the Ruway Arrested Ladg Ste (RALS) cotrol tower show Fg. 4. It s located at the Naval Ar Warfare Ceter Lakehurst, New Jersey. Wd tuel tests were coducted at the Flud Mechacs Lab NASA Ames Research Ceter o a /7 scale model of the RALS cotrol tower 7. Oe pressure sesor was mouted o each face of the cotrol tower as show Fg. 6. A four-hole Cobra probe was used to measure the freestream wd speed ad drecto. The FADS system was tested o a varety of wd speeds ragg from 4 fps to fps httg the model from all 36 yaw drectos. Varato of the wd cdece three dmesos wll be a part of the future work. The bow yaw agle β s measured the atclockwse sese wth respect to the ceterle of the model. So wds httg the south port have β = degrees, for the east port β = 9 degrees, for orth port β = 8 degrees ad for the west port β = 7 degrees. I Fg. 7 show below statc pressure s plotted for each pressure port. The pressure port o the south face bears postve pressure values whe wd s cdet head o at the pressure port ad bears egatve values for wds httg the orth face. Smlarly the pressure port o the west face bears postve pressure values whe the yaw agle s postve ad bears egatve values for egatve pressure values for egatve yaw values. Pressure port Fg. 6. Wd tuel model of the RALS tower 7.

14 Fgure 8 shows the varato of coeffcet of pressure for each pressure port. The pressure coeffcet s calculated by ormalzg the statc dfferetal pressure wth the correspodg dyamc pressure. Ths ormalzato elmates the effect of wd speed ad results a coeffcet that s oly a fucto of wd drecto. Fgure 8 shows all pressure coeffcet graphs collapsed to oe graph for each port. It also shows a substatal varato the coeffcet of pressure whch ecourages the use of a tellget algorthm that ca extract formato from pressure measuremets to predct ar data parameters. South face port West face port North face port East face port Speed: fps Speed: fps Speed: 8fps Speed: 6fps Speed: 4fps Fg. 7 Statc pressure varato vs. the yaw agle for dfferet wd speeds. South face port West face port North face port East face port Speed: fps Speed: fps Speed: 8fps Speed: 6fps Speed: 4fps Fg. 8 Pressure coeffcet varato vs. the yaw agle for dfferet wd speeds.

15 G. Trag ad Testg I ths secto we layout the procedure whch we attempt to develop wd speed ad drecto estmato techques. Eve though our problem volves turbulece ad cross-flow about otrval geometres, the exstece of a fuctoal relatoshp betwee P ad q ad betwee C p ad β ca be safely assumed. The wd speed was predcted usg oly the surface pressure data whle the wd drecto was predcted usg the pressure coeffcet data derved from measured pressure ad predcted wd speed values. It s oted that ths couplg of etworks put the burde of hgh accuracy o the wd speed predctor. Freestream ar desty was kept costat the curret problem whch makes predcto of freestream wd speed equvalet to dyamc pressure. Sequetal fucto approxmato was used to costruct oe RBF etwork ad the correspodg wd speed surrogate could be represeted by Eq. (3). q = c exp l[ ]*( p p ) ( p p ) + b (3) ( λ ) pre test test = Oce the wd speed predctor was avalable, the test statc pressure values were dvded by the correspodg predcted dyamc pressure to obta the test coeffcet of pressure values. Several ways exst to predct the yaw agle β. Oe straghtforward way s to costruct oe RBF etwork gve by Eq. (4), where β = c exp l[ ]*( c c ) ( c c ) + b ( λ,,,, ) pre p test p p test p = (4) However, as dscussed a prevous work 5, dvdg the avalable data four dfferet quadrats [, 9], [9,8], [,-9] ad [-9, 8] (Fg. 9) resulted more effcet trag ad predcto of β. (a) (b) Fg. 9 Quadrats depctg dfferet rages of the yaw agle rrespectve of the bluff body geometry (a) DDG (b) RALS tower.

16 However, ths yaw agle estmato techque rus to problems whe the test data has ose. Average pressure each quadrat s a smple way to determe whch quadrat the test pot belogs. Radom ose the test statc pressure ad C p dstorts the average each quadrat sce oly a fte umber of sesors are preset each quadrat ad yaw agle predcto accuracy decreases quckly wth ose. A heret dffculty wth ths verse problem s the strog o-uqueess show by Fg Yaw agle -5 Yaw agle Coeffcet of pressure Coeffcet of pressure (a) (b) Fg. No-uqueess of the yaw agle predcto problem at a) fps ad b) 4fps. Fgure a ad b show the varato of the yaw agle wth bow sde coeffcet of pressure at ad 4fps respectvely whe radom ose of magtude.5 was added to the pressure data. It s evdet from Fg. a that a uque soluto for β does ot exst at ay value of the coeffcet of pressure. Iformato from dfferet pressure sesors helps to take care of ths o-uqueess problem but that has oly a lmted mprovemet. Ths becomes a serous problem whe the umber of sesors are lmted to 4, oe o each sde of the RALS tower, or at lower speeds where sgal to ose rato drops sgfcatly due to lower pressure magtudes show Fg. b. Motvated by ths problem a smple forward problem approach was developed to estmate the yaw agle. I ths approach, frst oe etwork s costructed for each pressure sesor predctg coeffcet of pressure as a fucto of β. Oce a etwork has bee costructed to represet each sesor, the followg objectve fucto Eq. (5) could be mmzed to predct the yaw agle for a vector of test pressure coeffcets β pre 4 ptest () = m c β = qpre p, ( β) (5)

17 III. Data-model fuso Estmatg freestream wd speed ad drecto from statc pressure measuremets s a ll-posed problem. Ill-posedess s duced due to the o-uqueess of the problem whch s worseed the presece of ose the pressure data. The verse problem addressed ths secto s to recostruct a complete pressure sgal gve sparse, osy ad complete expermetal data. There are may regularzato methods avalable for the soluto of ll-posed problems 3. Some are categorzed as ether drect methods, where the solutos are defed by drect computato, or teratve methods that heretly requre teratve solutos. The most popular ad wdely used method s the Tkhoov drect regularzato method 3. A ovel approach to the curret problem was developed utlzg the Tkhoov regularzato method to merge expermetal ad CFD data. Ths approach ca be vewed as a terpolatg ad extrapolatg tool of the expermetal data usg a pror formato from the CFD models. The objectve fuctoal usg the Tkhoov formulato s gve by Eq. (6) ( ( ) ( )) r d fa ξ fcfd ξ ω γ fa, fexp, f CFD = ω ( fexp ( ξ) fa( ξ) ) + dξ r (6) r= dξ where ξ = ( ξ,, ξ,,..., ξ, ad d deotes the dmesoalty of the problem. From ths equato, t ca be d ) see that for ω, f ( ξ) f ( ) ξ, ad the CFD data becomes less relevat tha the expermetal data. a exp Forω, the expermetal data become less relevat to the soluto. Ths work develops a alterate method of fusg expermetal ad computatoal data to approxmate pressure sgals to estmate freestream wd speed ad drecto. The approach uses a eural etwork method as the verse modelg tool ad apples a smplfed Tkhoov-related regularzato scheme to correct for the orgal data error. The purpose of ths secto s to troduce a fusg approach usg the SFA eural etwork that maxmzes the use of expermetal data wth the help of CFD data approxmatg a smooth, cotuous ad accurate pressure dstrbuto. The fusg approach attempts to correct the low-fdelty ad hgh resoluto CFD data wth lmted, yet relable, expermetal data. It s also a method by whch the osy expermetal data ca be codtoed wth the smooth curves of the CFD data.

18 The fusg approach frst volves calculatg the error fucto of the CFD ad expermetal data defed by the followg equato, e CFD EXP ( ξ ) u ( ξ ) u ( ξ ) =, (7) for =,, s, where s s the umber of trag data sets. The error vector, e, s the used to tra the SFA etwork to a predetermed tolerace, τ. The resultg error surface, e(ξ), wll aturally volve some scatter drectly related to the expermetal data ose. Trag the etwork to the gve tolerace allows the SFA to regulate the osy expermetal data wth a pror CFD formato. Assumg the u CFD surface s kow, the the error surface approxmato, e SFA, ca be subtracted from the u CFD (ξ) data to gve the approxmato surface, u SFA = u CFD e SFA (8) The τ value ca be regarded as the regularzato parameter ad cotrols how well the approxmatos ft the expermetal or CFD data. A very hgh tolerace value allows the trag process to ed prematurely wth very few etwork uts. As a result, the etwork uder-lears the trag data ad the majorty of the approxmatos reach a value of zero. For data pots wth a error value of zero, Eq. (8) shows that the approxmato value wll reproduce the CFD data. O the other had, a very small tolerace value wll force the etwork to use too may etwork uts to reach the smallest possble tolerace. I ths case, the etwork over-lears the trag data ad wll ft eve the expermetal ose the error surface. As a result, the approxmatos wll reproduce the expermetal data. The user must carefully choose the tolerace value to best ft the expermetal data usg the CFD formato. IV. Hadlg ose, sparsty ad completeess As metoed before, the objectve of ths work s to mprove the qualty of the trag data set by fusg umercal solutos wth expermetal data. I ths secto we compare the smoothess based regularzed solutos vs. the physcs based regularzed solutos ther ablty to hadle ose, sparsty ad completeess of data. If oly expermetal data pots were used to costruct the wd speed ad drecto surrogates, we call them smoothess based regularzed solutos because the RBF etwork uses just the mathematcal smoothess of Gaussa radal bass fuctos to costruct the hyper-surface.

19 However, f umercal solutos were used as apror formato to costruct the surrogates we refer to them as physcs based regularzed solutos. I the followg graphs, we refer to the smoothess based solutos by SFA ad the physcs based solutos as Fused. The avalable expermetal data had 9 sets of wd tuel rus from 4 fps to fps at cremets of fps. Each set of wd tuel ru had pressure measuremets the rage 8 β 8 at cremets of degrees. To smulate a osy ad sparse data set, a uform radom ose of magtude.5 ps was added to each pressure measuremet. From ths osy data set, pressure measuremets at every degrees were selected to smulate sparsty the trag set. The ablty of smoothess based ad physcs based techques to recover the orgal pressure sgal s tested o ths ew degraded subset of the data set. The umercal smulatos were coducted usg the stadard Star-ccm software. Steady state three-dmesoal flow aroud RALS tower was solved usg Reyolds-Averaged Naver Stokes equatos..5. Sparse Nosy data CFD Fused True.5. Sparse Nosy data SFA True.5.5 Statc pressure, ps -.5 Statc pressure, ps Yaw agle, deg Yaw agle, deg (a) (b) Fg. Comparso of bow sde a) smoothess ad b) physcs based solutos to hadle sparse ad osy pressure data at fps. Fgures ad compare the ablty of smoothess ad physcs based regularzato techques to estmate a clea pressure sgal or dstrbuto at the bow sde sesor. The regularzed solutos show by Fused ad SFA are smooth ad cotuous compared to the sparse, osy expermetal data. However, ths problem both regularzato techques dsplay smlar results. Physcs based regularzato techque wll have sgfcat advatages over smoothess based techques regos where

20 expermetal data s ot measured or s kow to be accurate, for example, due to the presece of let ad exhaust valves. The curret data-model fuso techque could be used to obta fused solutos wth dfferet toleraces dfferet rages of the yaw agle. For example, f the chose umercal model caot properly capture pressure dstrbuto regos wth separated flow, t s possble to defe whch techque, expermetal or computatoal, s more mportat whch regos of the yaw agle. Ths would yeld a more accurate fused pressure dstrbuto developed optmally from the avalable wd tuel data ad umercal solutos...5. Sparse Nosy data CFD Fused True..5. Sparse Nosy data SFA True Statc pressure, ps.5 Statc pressure, ps Yaw agle, deg Yaw agle, deg (a) (b) Fg. Comparso of bow sde a) smoothess ad b) physcs based solutos to hadle sparse ad osy pressure data at 4 fps. Fgure 3 compares the pressure dstrbutos at port, ster ad starboard sde sesors obtaed usg the two regularzato techques at 4 fps. Aga, a radom ose of magtude.5 ps was added to the pressure measuremets. Physcs based Fused solutos look better tha the smoothess based SFA solutos for the ster ad the port sde pressure sesors. Aother qualty that makes ths data-model fuso geerc s that t could be used for fuso of formato/data from ay two sources, two CFD codes, wd tuel data from dfferet expermets to ame a few. A mache learg or eural etwork techque could be used to lear the dffereces betwee the two solutos. Approprate tolerace crtero could be user-defed depedg o the relatve accuracy ad mportace of the two solutos ad by addg the predcted dfferetal hyper-fucto to the less accurate soluto.

21 ..5. Sparse Nosy data CFD Fused True..5. Sparse Nosy data SFA True Statc pressure, ps.5 Statc pressure, ps Yaw agle, deg Yaw agle, deg (a) (b).5. Sparse Nosy data CFD Fused True.5. Sparse Nosy data SFA True Statc pressure, ps Statc pressure, ps Yaw agle, deg Yaw agle, deg (c) (d)..5. Sparse Nosy data CFD Fused True..5. Sparse Nosy data SFA True Statc pressure, ps.5 Statc pressure, ps Yaw agle, deg Yaw agle, deg (e) (f) Fg. 3 Comparso of a) starboard sde smoothess, b) starboard sde physcs, c) ster sde smoothess ad d) ster sde physcs e) port sde smoothess ad f) port sde physcs based solutos to hadle sparse ad osy pressure data at 4 fps.

22 A. Fuso the pressure sesor posto θ I exteral flow past bluff bodes, coeffcet of pressure s a fucto of the yaw agle β ad local pressure sesor postoθ. I the curret problem wd tuel data s avalable at oly four values of θ = [, 9, 8, 7] degrees show by the coordates 4,, ad 3 respectvely Fg. 4. A complete represetato of C p as a fucto of β ad θ s ecessary because that could possbly form the expermetalst what locatos should be chose for pressure sesor stallato to mprove wd drecto predcto accuracy. Coeffcet of pressure Yaw agle - - Pressure sesor posto Fg. 4 Wd tuel coeffcet of pressure as a fucto of yaw agle ad pressure sesor posto. Numercal solutos, however, provde full coeffcet of pressure dstrbuto as a fucto of taw agle ad pressure sesor posto. The data-model fuso techque dscussed Secto III could smlarly be appled to correct low fdelty C p solutos eve at values of θ where o wd tuel data s avalable. Ths

23 could be doe smply by calculatg the dffereces betwee the wd tuel ad CFD solutos at θ = [, 9, 8, 7] degrees. Oce the dffereces have bee calculated, t could be treated as a fucto approxmato problem by SFA wth β ad θ as puts ad C p as output. Oce a surrogate to the hypersurface of dffereces has bee created, t could be added to the CFD C p surface show Fg. 5 to obta a C p dstrbuto more complete tha wd tuel C p dstrbuto ad more accurate tha CFD C p dstrbuto. The umercal solutos were extracted at 56 values of θ for each yaw agle. Numercal measuremets were take a clockwse maer wth beg o the bottom left corer of the square crosssecto of the RALS tower. Coeffcet of pressure Yaw agle - - Pressure sesor posto Fg. 5 CFD coeffcet of pressure as a fucto of yaw agle ad pressure sesor posto.

24 The fused C p dstrbuto was obtaed the maer as metoed above ad s show Fgs. 6 ad 7 for dvdual values of θ to clearly vsualze the valdty of the fused dstrbutos. Fgures 6 a, b, c ad d show the C p dstrbuto for sesors located just ext to the starboard, ster, port ad bow sde respectvely..5 CFD Fused CFD Fused.5 Coeffcet of pressure.5 Coeffcet of pressure Yaw agle Yaw agle (a) (b).5 CFD Fused.5 CFD Fused Coeffcet of pressure Coeffcet of pressure Yaw agle Yaw agle (c) (d) Fg. 6 CFD ad fused coeffcet of pressure as a fucto of yaw agle ad pressure sesor posto at a locato ext to a) Starboard b) Ster c) Port ad d) Bow sde sesor. At these locatos wd tuel measuremets were ot take.

25 V. Wd speed ad drecto predcto I ths secto we preset the freestream wd speed ad drecto predcto accuraces. Wd speed ad drecto estmato techques dscussed Secto II G were used to compute these results ad as dscussed before they are susceptble to ose ad sparsty the trag data. Secto III ad IV dscussed the smoothess ad physcs based regularzato techques that ca geerate a smooth pressure sgal gve sparse ad osy wd tuel data. As preseted Secto IV, a osy ad sparse pressure data was smulated wth a ose magtude of.5 ps ad a yaw agle resoluto of degrees. Ths degraded data set was put to the smoothess ad physcs based regularzato techques to result cleaer ad smoother pressure sgals whch were put to the wd speed ad drecto estmato routes to predct wd speed ad drecto. The ardata estmato techques were tested agast the orgal clea wd tuel data set show Fg. 7. Fgure 7 shows the performace of the ardata estmato techques whe ose-free data at a yaw agle resoluto of 4 degrees was take as put. The error tolerace was ± 3.4 fps ad ± degrees for wd speed ad drecto respectvely. WOD speed error, fps Yaw agle Error, degrees WOD speed, fps Yaw agle, degrees (a) (b) Fg. 7 Predcto accuraces of a) wd speed ad b) yaw agle for ose-free data at a yaw agle resoluto of 4 degrees. It ca be see from Fg. 7a that wd speed errors are well below the tolerace level of 3.4 fps, however, wd drecto errors exceed the desred tolerace level. Ths s because there are oly 4 pressure sesors

26 are stalled, oe o each face of the RALS tower. Sce the sesors are stalled o the mddle of each face, they are uable to capture ay sgfcatly dfferet pressure sgal the rage of 5 β 5, 65 β 5, β > 65, ad 5 β 65 whch s where all the errors show Fg. 7b occur. I these rages of yaw agle three pressure sesors face separated flow ad as show by Fg. 8 the predcted C p values also do ot vary sgfcatly to gve ay useful formato to the drecto estmato techque. Fgure 8 shows the wd speed ad drecto predcto performace whe smoothess based regularzato techques were used to obta pressure sgals from data wth a ose magtude of.5 ps ad a yaw agle resoluto of deg WOD speed error, fps 5-5 Yaw agle Error, degrees WOD speed, fps Yaw agle, degrees Fg. 8 Predcto accuraces of a) wd speed ad b) yaw agle for a ose magtude of.5 ps at a yaw agle resoluto of 4 degrees usg smoothess based regularzato techque to obta trag data. Fgure 9 shows the wd speed ad drecto predcto accuraces for the same set of data whe physcs based regularzato techques were used to obta cleaer pressure sgals for the bow, starboard, ster ad port sde pressure sesors. Sce the pressure sgals obtaed by the physcs based regularzato looked oly slghtly better tha the sgals obtaed by the smoothess based regularzato techques, the estmated speed ad drecto accuraces also look slghtly better. It should be emphaszed here that the physcs based regularzato would have yelded better results f more accurate umercal solutos were used. Also, the mportace of physcs based regularzato would become evdet f the wd tuel pressure data were complete ether β or θ dmeso.

27 5 3 WOD speed error, fps 5-5 Yaw agle Error, degrees WOD speed, fps Yaw agle, degrees (a) (b) Fg. 9 Predcto accuraces of a) wd speed ad b) yaw agle for a ose magtude of.5 ps at a yaw agle resoluto of 4 degrees usg physcs based regularzato techque to obta trag data. Fgure shows wd speed ad drecto predcto accuraces o the same set of data whe fused pressure sgals at all 56 pressure sesors used as trag data. The trag data geerato procedure for ths case was dscussed Secto IV A. 5 WOD speed error, fps 5-5 Yaw agle Error, degrees WOD speed, fps Yaw agle, degrees Fg. Predcto accuraces of a) wd speed ad b) yaw agle for a ose magtude of.5 ps at a yaw agle resoluto of 4 degrees usg physcs based regularzato techque to obta trag data. All 56 fused pressure sgals were used to costruct the trag set.

28 It ca be see from Fg. that both wd drecto accuraces are sgfcatly reduced as expected. I fact, the drecto errors are less tha whe ose free data was used for trag at a yaw agle resoluto of 4 degrees as show Fg. 7b. However, wd speed predcto accuraces have ot reduced sgfcatly because estmato of wd speed does ot deped strogly o the locato of pressure sesors. As log as suffcet resoluto s preset the dyamc pressure ad yaw agle wd speed predcto would ot chage sgfcatly. VI. Coclusos Ths paper has demostrated that scattered data approxmato algorthms ca be used to fuse osy ad scattered wd tuel data wth smooth ad low fdelty umercal solutos. We propose the use of a geerc mache learg based data model fuso techque for hadlg ose, sparsty ad completeess wd tuel pressure data for a Flush Ar Data Sesg system. Freestream wd speed ad drecto estmato techques are dscussed that put statc pressure data to predct wd speed ad yaw agle at a desrable tolerace error. Smooth pressure sgals were obtaed from osy data va two regularzato methods amely smoothess ad physcs based regularzato techques. Physcs based regularzato techques proved to be especally useful whe the wd tuel data was complete. It also helped correct low fdelty umercal pressure solutos eve at locatos where o wd tuel measuremets were take. Ths work provdes a geerc framework to fuse formato from varous expermetal or computatoal sources to provde a more optmzed ad accurate represetato of the doma. VII. Refereces Whtmore, S.A., Davs, R.J., ad Ffe, J.M., I-Flght Demostrato of a Real-Tme Flush Ardata Sesg (RT- FADS) System, NASA Techcal Memoradum 434, October 995. Rohloff, T.J., Whtmore, S.A., ad Catto, I., Fault-Tolerace Neural Network Algorthm for Flush Ar Data Sesg, Joural of Arcraft, vol. 36, No. 3, pp May-Ju. 999.

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