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1 MD- -A UNCLASSIFIED TURBULENT WALL PRESSURE SIMULATION(U) OLD DOMINION UNIV 1/1 NORFOLK Vfl DEPT OF MECHANICAL ENGINEERING AND MECHANICS H R KHORRAMI JUL 8 N00O14-82-K-069 F/G 20/4 NL ll L \. m ^

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3 "^^^-vv ^'., *...»v. ^'«" i ' T " " 1."*. V -P. v ;- r W. - ;,, ^«.-^-^-i-limfll-'-'^'- - -.:* - - ^ - * -. -«-i < O z L* - u_ Oi < H <d LU LU CD < > - 1 DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS SCHOOL OF ENGINEERING OLD DOMINION UNIVERSITY NORFOLK, VIRGINIA TURBULENT WALL PRESSURE SIMULATION By Rbert L. Ash, Prinipal Investigatr and Mehdi R. Khrrami 7«Qj CD C-P Q: LU > B 9 Final Reprt Fr the perid ending June 0, 198 Prepared fr the Department f the Navy Offie f Naval Researh 800 N. Quiny St. Arlingtn, Virginia s Under Researh Grant N K-069 Task N. NR Dr. M. M. Reishman, ONR Sientifi Offier DTIC ELECTE SEP il Apprved fr publi release; distributin unlimited. Reprdutin in whle r in part is permitted fr any purpse f the United States Gvernment. July

4 f ^.». -!l» i- I;M 'i,», ^. * ; -i '_M ;;.»"..»- > ;» :-;: ' ' -J ; '. _? '.»;.T. ".. y -, T "-. 11 '"-.''' * * ' *.' *. '' ' ** * -*".- * V " * *. T "-"-" *.' T" " -" DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS SCHOOL OF ENGINEERING OLD DOMINION UNIVERSITY NORFOLK, VIRGINIA TURBULENT WALL PRESSURE SIMULATION By Rbert L. Ash, Prinipal Investigatr and Mehdi R. Khrrami Final Reprt Fr the perid ending June 0, 198 Prepared fr the Department f the Navy Offie f Naval Researh 800 N. Quiny St. Arlingtn, Virginia Under Researh Grant N K-069 Task N. NR Dr. M. M. Reishman, ONR Sientifi Offier Submitted by the Old Dminin University Researh Fundatin P.O. Bx 669 Nrflk, Virginia 2508 July ; *ra.':- : ".' *.' '.' '..'.' ', '- «".*.*.'-".' -' """-" '»""jfyv«-'.'.'.'"«"">'".-'",-". "."*">."_" V «"- ".*..".' '. - '. '. "." '.". ".** "." ". '."-*.-".

5 1«...'-.'..I A iv.lji' U "*»A'I' ''.»' W -".' -T»*J * -»I»..* «.* -.' Unlassified SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) t. REPORT NUMBER 1 4. TITLE (and Subtitle) REPORT DOCUMENTATION PAGE Turbulent Wall Pressure Simulatin 7. AUTHORf» Rbert L. Ash, Prinipal Investigatr Mehdi R. Khrrami J_ 2. GOVT ACCESSION NO 9. PERFORMING ORGANIZATION NAME AND AOORESS Department f Mehanial Engineering & Mehanis Old Dminin University Nrflk, Virginia 2508 tl. NTROLLING OFFICE NAME ANO AOORESS Mehanis Divisin Offie f Naval Researh Arlingtn, Virginia MONITORING AGENCY NAME ft AOORESSf/f different frm Cntrlling Oltie) ONR Sientifi Offier Dr. M.M. Reishman READ INSTRUCTIONS BEFORE MPLETING FORM. RECIPIENT'S CATALOG NUMBER 5. TYPE OF REPORT 4 PERIOO VERED FINAL July June PERFORMING ORG. REPORT NUMBER 8. NTRACT OR GRANT NUMBERfe.) N K PROGRAM El EMENT, PROJECT, TASK AREA 4 WORK UNIT NUMBERS Mehanis Divisin NR REPORT DATE July NUMBER OF PAGES SECURITY CLASS, (f this reprt) Unlassified 15a. DECLASSIFI CATION DOWNGRADING SCHEDULE 15. DISTRIBUTION STATEMENT (f thle Reprt) This dument has been apprved fr publi release; its distributin is unlimited. 17. DISTRIBUTION STATEMENT (at the. mbettmt entered In Blk 20, If different frm Reprt) 1«. SUPPLEMENTARY NOTES 19. KEY WORDS (Cntinue n reveree eide If neeeeemry end Identity by blk number) 20. ABSTRACT (Cntinue n rereree elde It neeeeemry end Identity by blk number) The rt mean square pressure was alulated with synthetially generated pressure signals fr transduers f different size. The transduer diameters were in the range f 0.02 < d < 1.18 at a Reynlds number, Re, f 12,250. 6* 6 * The lwer limit f the diameter simulated here was muh smaller than diameters reprted previusly by ther wrkers. Calulated rt mean square pressure levels develped in the present wrk were in gd agreement with the measured DO,: "SFn 147 EDITION OF I NOV 65 IS OBSOLETE Unlassified SECURITY CLASSIFICATION OF THIS PAGE (Whtm Dmtm Enfrmd) m f*~s*+'er*m mm * ' -m^me,.-.'- -?- - ' -"»*? *n' *T*I ' '*!-" -' *1 *T* '- ~ ** -' * ---*- - --'» - * lv H-J.*..»*..*.-...» m

6 Unlassified SECURITY CLASSIFICATION OF THIS PAGEflWian Dutm Enlmtmd) '1 : I experimental data. In rder t hek the simulated pressure signals, the pwer spetrum and tw pint rrelatin were alulated as well. The pwer spetra btained with the smaller transduers were higher in energy level at high frequenies. The shape f the pwer spetra were similar and verall agreement with experimental data was reasnable. The tw pint rrelatins were btained fr separatin distanes f X 1.66, and 1.7. The same nvetin and deay effets as 6* " the experimental data were bserved with the alulated rrelatins. Again, verall agreement with experimental result was quite gd. ' I * Unlassified SECURITY CLASSIFICATION OF *" P AGEfWhan Data Enfarad) la '.. _ '-''- ' a i - '-i

7 f#» *»^ "!»J "^i " B* _^* "^B \«,, "Z*." -.,-..-. '.' TABLE OF NTENTS Page LIST OF FIGURES iii LIST OF SYMBOLS iv SUMMARY ix 1. INTRODUCTION 1 2. ANALYSIS Transduers with Finite Size Amplifiatin Fatr Cnvetin Speed Frequeny Cutff Range Amplitude Deay Funtin Wavelength Deay Funtin 19. RESULTS AND DISCUSSION 27.1 Tw Pint Crrelatin 27.2 Pwer Spetra 2. Variatin f P D NCLUSION 44 REFERENCES 46 APPENDIX A. FORCE NTRIBUTION OF A SINGLE PRESSURE EVENT 48 APPENDIX B. MPUTER PROGRAM LISTING 61 Aessin Fr KTIS GRA&I DTIC T\ - Unannuned Justif ;.atin_ =vm Ü By Distributin/ Availability Cdes_ Avnil and/r Speial IA-^_A1.V.:I'.'«!, m.^ M-. m.,. -.% - " - ^ ^ - ±-± - i.; *-. I-' w- ^2

8 '.- V"V'' 1^.'- 1^'*. i»».»:.»'.»-'' T ' f* '-' '»".5V-T r!»'." ''.'". -r. r. " '- r '. ~ ~.»"."» '. '. ^»" " "-". -"- ". -~- -" ~- "-, 1 LIST OF FIGURES Figure Page 1 Shemati diagram f the mdel 12 2 Nminal spanwise rrelatin frm Willmarth and Wldridge (14) 1 Nrmalized amplitude grwth funtin fr a frequeny f H*_ Assumed variatin f individual pressure-event nvetin-velity with initial frequeny 17 5 Assumed variatin f amplitude with distane fr a disturbane with an initial frequeny f OIL Cmbined effet f amplifiatin fatr and the amplitude deay funtin n amplitude 21 7 Assumed variatin f wavelength deay funtin with distane Representative simulated pressure flutuatins f upstream referene latin 24 Ax Simulated pressure flutuatins at ^ = 1.66, shifted in time relative t the referene latin via nvetin speed 25 Ax Simulated pressure flutuatins at g 1. 7, 6 shifted in time relative t the referene latin 26 Cmparisn between the aut-rrelatin funtin resulting frm a simulated wall pressure histry and the experimental measurements f Bull [15] fr different transduer sizes Cmparisn between the tw-pint rrelatin funtin resulting frm a simulated wall pressure histry and the experimental measurements f Bull [15] fr * V 1 Cmparisn between the tw-pint rrelatin funtin resulting frm a simulated wall pressure histry iii t * -.*»-' -' -"'" * *»-'' ^* -v" -O * *. * *' " - ""'. 1«1»* ' - '"..»", ym»\ **. -\»*.. V. «" ** * *.

9 ".*.' >.":+i*\ tt " '" l "- t -- -'"''-' ' ''.' " '"'' r.~* ' --'. -- '. :T^V."- '-.*,' ' ^.'. '.w- -^ ^~ r -^-"T ; ' ~> - -' '..' --- " " ^jn LIST OF FIGURES (ntinued) Figure Page and Che experimental measurements f Bull [15] fr A" J 1.7 fr different transduer sizes Filtered lngitudinal spae-time rrelatins, lw-frequeny band 15 Filtered lngitudinal spae-time rrelatins, high-frequeny band 4 16 Narrw bandwidth spae-time rrelatin at spatial separatin f - = Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [6]. (Large transduer ase) 7 18 Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldrige [14], and Blake [6]. (Intermediate transduer ase) 8 19 Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [6]. (Small transduer ase) 9 20 Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [6]. (Ultra small transduer ase) Variatin f simulated Pj> M with transduer size 4 Al A2 A A4 A5 A piture f the prttype duble sine wave mving ver a transduer 51 Averaged fre as a funtin f time with a time step smaller than simulatin time step (At) 5 Averaged fre as a funtin f time with a time step equal t the simulatin time step (At) 55 Tempral reslutin f a large wavelength duble sine wave with a time step greater than the simulatin time step (At) 56 Tempral reslutin f a large wavelength duble sine wave with a time step equal t the simulatin time step (At) 57 iv

10 V...M^,,...» :.-.;. in -Ji.». _.' -, r. *..»T» ;:., '., '.'.'.'A *.' ' " «" " r '." " " " "V LIST OF FIGURES (nluded) Figure A6 A7 Page Tempral reslutin f a small wavelength duble sine wave with a time step equal t the simulatin time step (At) 59 Tempral reslutin f a small wavelength duble sine wave with a time step smaller than the simulatin time step (At) 60 '-- -'-'-' ' -' ' '-' ' -' -" - -' -'' ,...

11 -j. i» " '.», i i. i"*.< '_ ' ^' 'q,,* [K ',.».'.'.. '. '..-. >'..-. v'vll '-.'-. l-., '--.-lvl-.l^'-:.-l:. Ivlvl-. ^. 1 r- t -i- - r - i..' i '. * ';" * LIST OF SYMBOLS A amplitude funtin A riginal amplitude a amplitude deay nstant C skin fritin effiient d transduer diameter Reynlds number dependent transduer size parameter U R f frequeny (Hz) f Nyquist frequeny *" f Peak frequeny f Sampling frequeny * I L harateristi length m wavelength deay nstant n amplitude deay nstant '. P(t) pressure flutuatins P Rt-Mean-Square Pressure R. M. S. ( q free stream dynami pressure (1 p U 2 ) 2 vi m

12 » i t i I;J M-i"i >. ^»ji»t«.. ': '. 'v y. '. \-'; *,- ^P; 1."«^;''; ' "-" ';:^;v->.. it ;:',v.t-yrt^.-> LIST OF SYMBOLS (ntinued) R Reynlds number based n 6 L R ^ Reynlds number ba&ed n 5 '&) R spae-time rrelatin PP S_ struhal number / xs \ t t time Ö free stream velity U nvetin velity J U fritin velity f V T /p ) T ' W 00 X rdinate in diretin f flw X^ develpment length X^ Mdel length Z transverse rdinate Greek Symbls similarity funtin / mg similarity funtin / um Vll.''^ «V'JV' j'--«'.--j>--'r-i. ->* - '. "--.- J' ".: V '.--V. >."^-V_ i. i '.. ''.I. i.. l.» : '.. :. I

13 .--.- V-.'.-.i... - I. 1...II.J i.,.. t..11 u._ w.. ; i :.» ' '. * *"J *>*. *.» - *"-, " ~~ " *." -'."..^."».' LIST OF SYMBOLS (nluded) Greek Symbls (nluded) r Y 6 * 6 n K X X V Ü T rss-spetral density wavelength deay nstant bundary layer thikness displaement thikness distane in lateral diretin wavelength deay nstant disturbane wavelength riginal wavelength kinemati vissity distane in lngitudinal diretin dimensinless time / U^t \ T W 4»() Ul wall shear stress pwer spetral funtin radian frequeny Vlll *»- -'---''- - - _:. ^ a j-_

14 -» 1 I I. ;.-....«.»».''^^^. <..^.»..'. " "rrr.'t 1 '.',-.'.'... EFFECT OF TRANSDUCER SIZE ON THE STATISTICAL PROPERTIES OF A SIMULATED TURBULENT WALL PRESSURE FIELD. - ] By Rbert L. Ash 1 and Mehdi R. Khrrami 2 SUMMARY?M The rt mean square pressure was alulated with synthetially generated pressure signals fr transduers f different size. The transduer diameters were in the range f 0.02 < d <1.18 at a Reynlds number^^sa-^y f 12,250. The lwer limit f the diameter simulated here was muh smaller than diameters reprted previusly by ther wrkers. Calulated rt mean square pressure levels develped in the present wrk were in gd agreement with the measured experimental data. In rder t hek the simulated pressure signals, the pwer spetrum and tw pint rrelatin were alulated as well. The pwer spetra btained with the smaller transduers were higher in energy level at high frequenies. The shape f the pwer spetra were similar and verall agreement with experimental data was reasnable. The tw pint r- relatins were btained fr separatin distanes f T^ ^ and." "." "- :.\-.".i 1.7. The same nvetin and deay effets as the experimental data were bserved with the alulated rrelatins. Again, verall agreement with experimental result was quite gd. Eminent Prfessr, Department f Mehanial Engineering and Mehanis, Old Dminin University, Nrflk, Virginia Graduate Researh Assistant, Department f Mehanial Engineering and Mehanis, Old Dminin University, Nrflk, Virginia ^- ; Z».. ' -«, " - -r«j * '» '.. '- -".V.'-r. :. i.'.' -". "' -'-" *.*..'-.'-'.'--.*- >'-»'-«'--- 'i** ix -*"' '»' -'* *' -' '-'-'- ' L. J #J

15 -.."'..V.-. _ [ } I. INTRODUCTION Turbulent wall pressure data are required fr the analysis f engineering prblems related t nise prdutin n aerdynami bdies, as well as vibratin and panel flutter studies (Dwell [1])*. In reent years, the identifiatin f mpliant walls whih are reeptive t turbulent pressure flutuatins has been f partiular interest (Bushnell et al. [2] and Bukingham et al. []). It has been speulated that a mpliant surfae might be apable f interating with a turbulent flw t prdue a net drag redutin. When lal instantaneus pressure data are required, the reslutin apability f even the smallest pressure transduers bemes a serius limitatin beause they average the pressure field spatially (Crs [4] and Emmerling, Meir and Dinkelaker [5]). The purpse f this investigatin has been t utilize a syntheti pressure field t study the effet f transduer size. Instantaneus wall pressure flutuatins an be generated either by emplying large arrays f pressure transduers with presribed spatial reslutin in an experiment r, by slving the full three-dimensinal Navier-Stkes equatins. Althugh, bth methds an be mre aurate than a simulatin, they represent a frmidable task in terms f time and expense and are restrited t a partiular tempral and spatial reslu- tin. The Mnte Carl tehnique emplyed in the present wrk is apable f reslving arbitrarily in time and spae. The ntributins frm small sale mtins (with length sales n 9 -f m * 1. «- *. - "\. > * The numbers in brakets indiate referenes. fe

16 t.. i. ^ m».. * v**-1* - *'? *,* *.*'* * *. y. *' "* * '«'.' '.* *T "'."K" *.' the rder f, where U is the fritin velity) t the wall pres- sure annt be measured urrently beause n pressure transduers are suffiiently small. Hwever, during the late sixties and early seven- ties the imprtane f high frequeny pressure flutuatins n the measurements f rt-mean square pressure (P ) was established by K.M.. Blake [6] and Emmerling, Meir and Dinkelaker [5] and a rugh estimate f their ntributin t P n u. was given by Emmerling, Meier and K.M. Dinkelaker [5]. The understanding f the abve features were made pssible by emplying pressure transduers whih were miniature when mpared t earlier experiments. Data generated frm these sensrs indiated that the R.M.S pressure inreased asympttially with dereas- ing transduer size. The present wrk has attempted t larify the abve suggestin by using syntheti pressure flutuatins t study the effet f a ntrlled, repeatable pressure histry n the measure- ments frm different sensr sizes. In rder t validate the syntheti- \ -\ ally generated pressure flutuatins, ther imprtant features f a turbulent bundary layer (suh as pwer spetrun and tw pint rre- latin) have been alulated as well. In reent years, a unified view has evlved nerning the exis- tane f tw different sales f disturbanes in a turbulent bundary layer. Sme disturbanes have sales whih are mparable t displae- * ment thikness (6, whih an be related diretly t bundary layer thikness) and ther disturbanes have muh smaller harateristi v lengths whih are n the rder f. It is nw bvius frm the lg- lg plt f pwer spetrum, that any shift in the high frequeny regin due t the measurements f small sale eddies, an result in an inrease > ***»*,* -".'* W""j/" ***_**" "'" - " *."-.*."» -"-*'»'. ".- ~» V V V ".-» ".".- V " -" * ".".*.* -' " '--* '«-- «- -' ^- «-' t-u' *-' - v -' -' ' -' -' v ' _ - - -'- -' '«-' -". -" -?-> -«-:* -^ -'-= -- -A

17 .4 in R.M.S. pressure (Blake [6]), sine, P M _ is related t the pwer- spetral density by: p2. 2/ * (<*) )düi. Kraihnan (8) first tried t alulate imprtant features f a pressure field by integrating Pissn's equatin 2 U.U, l ^--"SxJx 1. ver a finite regin. In a subsequent paper, Kraihnan [9] assumed lal istrpy in planes parallel t the surfae and used a rude mdel fr mean velity, t estimate the ntributin frm turbulent mean shear interatins (whih he nluded were the dminant terms). He als assumed that any lumn f fluid with a length sale mparable t the bundary layer thikness was statistially independent f the adjaent lumns. The estimated R.M.S pressure was fund t be abut six times the wall shear stress. Reent experiments by Emmerling, Meier and Dinkelaker [5] and Bull and Thmas [10] have shwn that Kraihnan's P_ R. M. S. estimate was in gd agreement with their measure- nsidering the degree f apprximatin he had emplyed. Crs [4] integrated a rreted pwer-spetrum (his methd will be disussed later) and was able nly t give the rati V where P 2 is the integrated pwer spetrum f Willmarth and Wldridge m ^^^^^^^^H

18 ,...,.,.., i!,..,.t t..«..j..... t, «.._... n'i',».».".». i tt ' '.«.',' ' ".».». ^. [14]. The integratin was up t radian frequenies given by )6* 14., whih was the highest measured frequeny by Willmarth and Wldridge [14]. He indiated that the ntributin frm higher frequeny flutuatins might be signifiant. Emmerling, Meier and Dinkelaker [5] have emplyed an ptial methd t evaluate the flutuatin field at the wall. Fr the sake f mparisn, they measured the R.M.S. pressure with mirphnes munted flush with the wall. They pltted the variatin f nn-dimensinalized P with nn-dimensinalized transduer R.M.S. diameter and mpared them with measurements f several ther experi- ments. Their variables were made dimensinless with respet t free v stream dynami pressure, q, and inner variables, respetively. -'~Vf T : > Their plt shwed a dramati inrease in R.M.S. level as the nn-dimen- sinalized diameter was dereased belw T 150. Emmerling, Meier and v _m., f, ^jkt Dinkelaker [5] have indiated that when the saling f transduer diameter was with respet t the displaement thikness, 6*, the data did nt llapse nt a single urve. This suggested that small sale eddies were majr ntributrs t the P D M (r pwer spetrum) f a turbulent bundary layer. In rder t understand the results given by Emmerling, Meier and Dinkelaker [5], Bull and Thmas [10] nduted an experiment where diret pwer-spetral measurements were made using pinhle mirphnes and piezeletri transduers with the same diameter. With a similar plt t that f Emmerling, Meir and Dinkelaker [5], they have shwn that althugh there is an inrease in P_ e with dereasing dianeter, R.M.8.. i ^ -: > > ^ * «: *-.«M.i

19 m-v.-»:t.-..-v'.in : «:-'. «.- w«.»:*'.-»v:'*'.j*_ :- v:» v- v*:-': '- *.* *: : "^-vt-*: 4 T.TTrrrrr?.'-'.".:: ' " " -" " the inrease is nt as signifiant as reprted earlier. Bull and Thmas [10] have attributed the high values f P measured by Blake (6) R M.S and Emmerling, Meier and Dinkelaker [5] t the lal flw disturbanes reated by the pinhle-mirphnes. Reently, Bull and Langeheineken [11] reprdued the experiment f Bull and Thmas [10]. They reinterpreted the transduer size parameter + d, whih was Reynlds number dependent and given by: U.+ d _1 Re. d L L vt where L is the harateristi length f the flw field (pipe radius r bundary layer thikness). Their plts f*p^7q vs d and Re shw similar trends fr ndensr mirphnes and piezeletri transduers. Therefre, they nluded that ndenser mirphnes and piezeletri transduers give essentially idential results (whih is ntrary t the explanatin f Bull and Thmas [10]). Hwever, the abve nlusins have prdued ther questins. Firstly, sine the transduer diameters were nt equal, diret mparisn with Bull and Thmas' [10] results : : : : : : :- : were nt pssible. Sendly, their experiment was nduted in a pipe flw whih raises questins nerning hw different flw fields affet the turbulent wall pressure field. Speifially, the questin f whether the pressure gradient impsed in a pipe flw alters the statistial prperties f the pressure flutuatins must be addressed. "-. % / ' " -. ' ".' ,. _.- - ^..: i.; -.1.1,,>,

20 I»< ' 1.».' '. '..'. ".. ' t.«j'*.,'., ".,.'». «:*. ^, ** t.*'.'.* f "* i^" ~» :,'» ' * ' -.*.*-*-.*." Mst f the semi-empirial wrk nerned with the rretin fr transduer size has fllwed the methd first utlined by Uberi and Kvasznay [12]. They develped a mapping predure based n a knwledge f the instrument respnse whih enabled them t frm a respnse kernel fr the measuring system. The riginal signal (r the instantaneus signal, sine they have assumed the instrument respnse is perfet) uld be retained. But, the limitatin f their methd was due primari- ly t the assumptin f a ttal randm field whih was istrpi and hmgeneus. The established fat that turbulene has a "fading memry" implies that it is neither hmgeneus nr istrpi. Crs [4] util- ized the wrk f Uberi and Kvasznay [12] t develp a similarity mdel fr rss-spetral density in the frm: -iuc/u rk.n) - (») "(fwr) 6 where a I n \ and P 0J_n_ \ are similarity funtins (based n the ex- perimental data f Willmarth and Wldridge [14]) fr the lngitudinal and lateral diretins, respetively. He assumed a unifrm respnse ver the transduer surfae whih had a kernel funtin equal t the inverse f its area. That frm is nsistent with the earlier wrk f Uberi and Kvasznay [12]. Crs was then able t shw hw sensr size influened reslutin fr high frequeny flutuatins. Hwever, he was nt able t give a quantitative estimate fr the errr. Willmarth and Rs [1] used an idential predure with Crs [4] t shw that Crs's similarity rule was nt valid fr the high frequeny range when the spatial separatin between transduers is less -"»-' '..'-."-*.*-.-*«'-..*.' -.' - * » -» -'*. ---.^. ". i'. ".»-.»*».^...

21 ^,.. ;.,.".».» : y^v :«.» -....yi* a". A.«". - '.JT".-". J^1"-- T»*. i..' -.* * ;'-.' *"* r '.' :.*_"V". '*' ".' ' *> ~~-~~ T^J than *. Willmarth and Wldridge [14], in their measurements, disarded flutuatins with frequenies less than 0.14, beause f pr reprduibility f the pwer spetrum in that range. They have attributed the prblem t wind tunnel nise, and prpagatin f disturbanes upstream. Bull's [15] pwer spetral density has the same drp ff and peak frequeny as Willmarth and Wldridge [14] but he was able t make reprduible measurements at frequenies as lw as Blake [6] used a very small "pinhle" mirphne t study the n- tributin f small sale flutuatins t the pwer-spetrum. He pinted * ut that at high frequenies the saling n uter variables (6 and U ) was pr. As has been disussed earlier, Bull and Thmas [10] made spetral measurements with tw different types f transduers. They fund that the high frequeny energy ntent was fur times greater when measured with pinhle mirphnes than when measured with piezeletri transduers. Hwever, by emplying a speially fitted ap, the piez- M eletri transduer was nverted int a pinhle mirphne and the..' measured pwer-spetrim was then in lse agreement with the spetrum prdued by the riginal pinhle mirphne. Bull and Langeheineken [11] have pinted ut the lak f experi- mental pressure data in lw Reynlds number turbulent flws. Their wrk vered fr the first time a wide range f Reynlds numbers in a single experiment. They have shwn indeed that the pwer-spetrum--partiular- ly at high frequenies is highly Reynlds number dependent. Als, their nn-dimensinalizatin f pwer-spetra (they have used a variety f dimensinless rrelatins) suggest that there is n single similar-

22 1 -,!. ' V " : ""*.''' '.""'- '.' "> "." "*" ". " " T - ".' '"."." -T"."»."."."-" T ity variable whih auses the spetral urves t llapse nt a single urve. Calulatin f aut-rrelatin, R (T), is very useful in deter- PP minatin f the time sale ver whih the pressure flutuatins rrelate (alng with determinatin f nvetin speed and deay f the pressure field). That is, the time sale is indiative f the time interval ver whih flutuatins "remember" their past; and the pressure rrelatin is apprximately the same as the velity rrelatins, beause they are determined by the same physial presses. Theretially (statistially is a better wrd) it has been understd that any peak in the pwer spetrum (<)>(ü))) away frm the rigin wuld result in negative regin f R (x). Hwever, the reverse des nt hld PP true. Experimentally (Bull [15] and Blake [6]), it has been prven that there is a peak away frm the rigin in the pwer-spetral density; therefre, ne must expet ertain negative regins fr the aut-rre- latin. Based n his integratin, Kraihnan [8] was able t predit that, unlike istrpi turbulene, any turbulent flw with a slight anistrpy will have negative rrelatin regins fr lng separatin distanes. The abve suggestin was later nfirmed by Bull [15] and Blake [6]. Mnte Carl methds have been used in this study t nstrut the syntheti pressure distributins. This wrk is an extensin f earlier wrk by Ash [16] and Ash and Khrrami [17] where detailed aunts f the nstrutin f the mputer prgrams an be fund. Cnsequently, nly the majr mdifiatins t the previus wrk have been disussed here. '."-.' -" «'- t"«."»."".'.' ".»»«>", V '.' ' -.' - " "." '- V V.- V *- ". " " -T.'-" ".»'. -\ ". *, ". ". -'. ".' ' >* '- '.!

23 <<m '. /^^^r^» 1! iini'ni i,i i '. \ ' '- -. "^..'' - i. The wrk whih fllws disusses simulatin imprvements related t individual event nstrutin, amplifiatin fatrs, amplitude deay, and wavelength deay funtins. Subsequently, the statistial measures f the quality f the simulatin are presented and disussed. Finally, an interpretatin f the transduer size effet n rt-mean-square pressure alulatins is presented and nlusins have been made.." '- --.*'.. -»:.. ^ ^L^ T r *-'-*'*.' Tr'i"^-*'"**^'^* ~-' 1 '

24 * : -". «: ": " *. '- ' ". -". " --.- " v \- w-.--.» » ] 2. ANALYSIS 2.1 Transduers with Finite Size The riginal mputer prgram (Ash [16]) generated pressure as an instantaneus pint funtin rather than the average fre whih is measured by a transduer f finite area. The distane between adjaent simulatin pints and the time reslutin uld be speified by the user, whih enabled arbitrary spatial reslvability. Hwever, in rder t hek the simulatin and prdue results whih uld be mpared with experimental data, it was neessary t intrdue features whih allwed the influene f surfae area t be studied. As desribed in Chap. 1, the finite size and the type f a pressure transduer an reate several size effets (Crs [4] and Bull and Thmas [10]). The frequeny reslutin f any sensr is related diretly t its harateristi diameter, as well as its nstrutin and its sampling time. As the size f a transduer inreases, its frequeny respnse dereases. Furthermre, a large sensr auses a severe averaging effet (see appendix A) whih an eliminate sme pressure features. The averaging effet results in errneus measurements, and auses an inaurate representatin f the high frequeny flutuatins. With the previus disussin in mind, an array f strage latins (with arbitrarily presribed size) were substituted fr the pint latins used in previus simulatins. The prgram has been mdified t permit arbitrary speifiatin f "sensr" latins and sizes. The span f the "flat plate" frm the frnt f the first sensr t the bak f the last sensr is alled the mdel length. A shemati diagram f 10 ^ ' / W.-:/.-V.V.V.J:.;VV..:... -._'-... *....ii > ii.»i ii.i i ~.i i.

25 ii ^i. n i!»; «i» iimii, n, \m_ ij.. n.. i.'» ". '..< '. I'," "i " '. -i ' ".»"'»- ;. ' *^ * '"»".,' *. * the mdel is shwn in Fig. 1. The develpment length is the segment ahead f the first sensr latin whih des nt ntain a mplete simulatin histry. The width f the transduer is assumed t be equal t its length (square area). The range f presribed diameters (here we use the wrd "diameter" fr nveniene) is < d < Figure 2 whih 6 represents the spanwise r lateral rrelatin data measured by Willmarth and Wldridge [14], indiates that the lateral rrelatin effiient remains near unity ver that diameter range. It has been assumed subsequently that the pressure flutuatins at the enter f the sensing element are nt affeted appreiably by adjaent unrrelated spanwise flutuatins. 2.2 Amplifiatin Fatr When a nstant nvetin speed f 0.8 U was used in the prgram, it was fund that nvetin speed alulated frm the spae-time rrelatin urves exeeded the free stream velity; this is bviusly innsistent with every experimental finding. Several different rretins were tested befre it was nluded that the prblem was aused by the instantaneus reatin f lw frequeny (large wavelength) disturbanes. Sme f thse individual disturbane events were reated whih spanned mre than ne simulatin latin at the instant f reatin. These large eddy events wuld thus have an infinite nvetin speed. Of urse, the number f suh disturbanes is small mpared t the ttal but, sine the lw frequeny flutuatins deay slwly, the nvetin speed effet an be prnuned t _. # - t *. '. *. -* -, * '". ".-"." * '

26 * '.».*-: -. -T«-. r =. ' ' I -I I ' ' ' ':' '.' >. I 'A > '% * '. «.,.!.. Ill, I. J. fm. a z H g 1^ B S x Q <u a /- * e u M 0) J= 4J 00 <4-l O H a 0 a) h 00 a) M H T 0 <u H M 4-1 a (0 6 1 H u Vi -H <v H 60 a ^^ 12 «ii «i.. i i i 'i 'n -., - v %.».

27 '-. <r«"-v-". '.*«" 0) H O T C 50 u *m N <O >- u-l u <y t H Q. tt) H I M s ddy ',.',^»"»':,. v.-. "V *., «..«...»_.»...

28 . -j An amplifiatin time interval has been emplyed that auses the disturbane amplitude t grw quikly frm a small "zer" value t full amplitude, in rder t eliminate the infinite nvetin speed effet. An equatin f the frm t 2 (5/U ) 2 P(x,t) =- P (x,t) [l - e ] m. has been used where P (x,t) is the randmly generated event amplitude, * U is the disturbane nvetin speed and <5 is the displaement thikness, while t is the relative time (0 at the instant f gener- atin). Sine nvetin speed is frequeny dependent the abve grwth time interval bemes a larger fratin f the disturbane lifetime fr high frequeny flutuatins. In fat, at very high frequenies the amplifiatin time an be larger than the deay time, preventing the flutuatin frm beming signifiant. Figure shws the plt f the amplifiatin funtin fr a fre- queny f whih is near the nminal peak frequeny. In earlier simulatin prgrams, the amplifiatin funtin had been nn-di- mensinalized with respet t bundary layer thikness, 6. Hwever, further testing has shwn that 6 was nt a prper hie. Rather, the * displaement thikness, <5, sales the simulatin results adequately t enable lse apprximatin f a variety f experimental results. Al- thugh the amplifiatin predure aused the simulated nvetin ve- lity t derease belw the free stream velity, the spanning effet due t lw frequeny disturbanes still remained. That is, fr shrt 14

29 * > > «> :\«.i«"."r,^". v " '. :',,.,.., ;.-». i J «... IT. J, '. 4».'. >. * ' ": ". '. ^...*.--/T ' ' - -.-^r" *" T -77TT^:T - u-l O CD 00 II * 1 l ta 14-1 O 0 in >< (N <N Lri «1 u- ID u 14-1 Cfl u U-l a H 4J CJ 8 * IM u 60 <u "O 4-1 H 0) N H H 8 ir> 2 (1) ä M aivj NOiivundwv 15 :! :. ' '.- ' '.-' '..' *.* ' :V. ', " b»i«ri J

30 i i., «i m in»i i :.»,».'»»-»*,«: '». [ Ti'.n.'.'.'.'.'V. '-TV W V*"**"* - " v : l separatin distanes the nvetin speed is f rder 0.9 U mpared t an experimental value whih is f rder 0.6 U (even thugh the assumed 00 event nvetin speed is n the rder f 0.6 U ). 2. Cnvetin Speed Althugh the nstant nvetin speed f 0.8 U^ gave reasnable results, experiments have shwn that wall pressure flutuatins travel dwnstream with frequeny dependent nvetin speed (5, 15). In rder t represent the experimental results mre realistially it was neessary t intrdue a variable nvetin speed int the simulatin. A prblem still remained, beause virtually every experimenter has fund a different range f nvetin velities (see Emmerling, Meier and Dinkelaker [5], Bull [15]). This disrepany between individual event nvetin speed and apparent nvetin speed was mst prnuned fr the high frequeny flutuatins whih are nrmally attributed t near wall effets. Initially the semi-theretial nvetin speed mdel f Crs [4] was used t represent the nvetin velity. His mdel assumed an expnential variatin in nvetin speed. Hwever, in rder t avid mputatinal mplexity, a linear variatin f nvetin speed with frequeny was preferred. Sine experimental data were s sattered, the linear assumptin was a reasnable mprmise between the reent experi- ments and a simple funtinal frm. The linearized nvetin speed was assumed t derease linearly frm 78 perent f the free stream velity at "zer" frequeny t a nstant 46 perent f the free stream velity at frequenies greater than r equal t 4 times peak frequeny in the 1 U ;yi pwer spetrum. The peak frequeny has been taken here as f = 0. 16

31 «.... «' ' '.'T :»;-H" ' ;!.; i. ".»: '. :».»> "i-»-r«-r~- -- Hz frm Bull [15]. The assumed variatin f nvetin speed with frequeny is shwn in Fig. 4 alng with the limited experimental data I frm Willmarth and Wbldridge [14]. ^.; 2.4 Frequeny Cutff Range SS The upper limit f the frequeny range fr individual events is V- determined by enmis (in terms f mputer time, and memry) and the reslving time step, whih in turn is related diretly t the peak ^ frequeny. Hwever, the lwer limit f the frequeny was determined frm ttally different nsideratins. First, in every experiment the lwer frequeny flutuatins have been disregarded beause f external ** unrrelated nise generatin (due t vibratin f the wind tunnel r -'. the fan). Sine the mputer des nt have this prblem, the lw frequeny limit uld be adjusted in agreement with flw physis. The «? lwer limiting frequenies whih an be emplyed in this study reate disturbanes whih have wavelengths n the rder f 50 6, and imply signifiant rrelatins ver very lng distanes. This is unrealisti in a bundary layer flw. Fr the partiular ase nsidered here, the simulatin generates frequenies in the Struhal number range f t 15.78, where the lwest frequeny utilized rrespnds t a M* wavelength f apprximately 8 6..:. 2.5 Amplitude Deay Funtin The earlier experiments nerning the pressure flutuatins have revealed that, ntrary t the Taylr hypthesis, eddies mist deay while they are nveted dwnstream. Experiments by Willmarth and Wldridge [14] and later by Bull [15] have shwn that eah disturbane I» Ml - ' v «17 V.-..* -".*;.'.' -' -' -* ' ' -' -' -' -" ' ' -' ^...,..-...,...,.- >.... '».«. ',»... ' t -H»»

32 ?.". : ;A : ; ^ v \r; *".'. ir> >> 4-1 H O O r-i 0) > 0) % a 4-1 > 01 M Cfl 0) ai S-i a fl T (N»^ 4-1 C O 01 a* 01 h Cfl U fl fl -H > 4J H 1 C 01 «H i 4= 05 4J Cfl i-l < 01 u ÖC 00 Ö (0 00 n 'n i 1 Ö 18».,.... >... -., i i '- '»*' *"-»'-'.'-'»-. -

33 -..,i...j..^...»t^.j.,.,....,- 7 -^.- -.«i ;,,-,.,^,...i, I.,,, :'._y_?;_r:.-:- -.«: "-.,'\" *:; ^', ; '^'. travels a distane whih sales with its wavelength. funtin f the frm: An amplitude deay A(x, X) -A e-*<*a> n has been emplyed where A is the riginal amplitude, X is the disturbane wavelength, x is the distane frm the pint f generatin, and a and n are tw arbitrary nstants, whih were determined by trial and errr. The nstants were speified ultimately by assuming that eah disturbane travelled dwnstream a distane f five wavelengths befre it had deayed t ne perent f its riginal amplitude. Figure 5 shws the variatin f the deay funtin with distane fr a disturbane with a frequeny f The funtinal frm used here is nsistent with the experimental findings f Pantn et al. [18] _ a whih indiated the initial deay rate is mst rapid. Als Fig. 6 shws the mbined effet f amplifiatin fatr and the deay funtin n the amplitude. 2.6 Wavelength Deay Funtin The three dimensinality f a turbulent bundary layer suggests that the wavelength f a disturbane may hange alng with its amplitude. This is a very imprtant nept when it is thught f in terras f the pwer-spetrum and aut rrelatin. As eah disturbane deays in wavelength, the frequeny f the disturbane will inrease whih in turn shifts the peak frequeny, alng with the pwer spetrum urve, tward higher frequenies. : ;: 19, *"t *.>.» <.*..: «_' ',* /..'-*. ' ' ' -j - '- *--' *-. -.' «. «..:. s.'.. N-^'^'^'^^CfN'Avji?C\\vl'j

34 ;" -'.».. ".'.: '." : *: v...v «1. I. 1 1 O Ö CM O tb ^ 00 ^ *fe> 4-1 H «a U 4J H T u O y-i 0) 4-1 H -a a= 4J i-( f amplitude = variatin y f )6* u <1) OJ n D* 5. Ass fre LT> O Lf> " '"! O ^ LO (N '.. O Ö Ö t AVQ aanindwv <D M 00 H Pi J.>j s * " * Ö * - - ] i './*, V**' M * ' ^-i^_ nv.,-v < - -'* -** -" -" -. -

35 i -i i.».». 171,1'.«, 1... : ". '. '.' " '. "7 V -.-" " dl a QJ "O 4-1 a. E 1 M O u a «C * 0 a u QJ T 0) C H e u a) 4-1 H.H a. e J- 1 LO LO OJ h 00 H AVQ QBNiaWOO

36 r^t» « r.. JT, v;.-, v *. -. -*..,-?: ~.-^?r : '.- : " T '." : '."? T"-'."^1."-',' '.': v J; :- : -:. -."-.- ~-*.v-v-'.~.-. - v >i A wavelength deay funtin f the frm X(x) - X [K D e" b(x)m u ] J has been used where X ehe riginal, randmly generated wavelength, _ K, D, b and ra are arbitrary nstants whih have been adjusted by trial and errr. The wavelength deay funtin allwed the wavelength t be redued t a speified fratin f its riginal wavelength at the m time f ttal deay f the disturbane. The wavelength funtin has imprved the simulated aut-rrelatin but it annt be nsidered an ptimum funtin. Figure 7 shws the variatin f the wavelength deay - I funtin with distane. In a few ases a frequeny dependent deay funtin similar t the ne used fr amplitude (see Se. 2.5) was emplyed. Hwever, that frequeny dependent funtinal frm did nt prvide any majr imprve- ment in simulated statistis but did reate sme prgramming diffiulty.." Therefre, all the runs presented in this wrk have used a wavelength deay f the frm given abve. T make sure that a realisti signal was generated by the simulatin, shrt rerds f the pressure histries at three different spatial latins have been prdued. Thse histries are shwn in Figs. 8 thrugh 10, and it is bvius that there are regnizable events whih have been nveted ver lng distanes. Hwever, there are als substantial features whih represent "unrrelated" infrmatin. Therefre, it was nluded that the simulated pressure rerds were similar t atual experimental rerds. The prgram inrprating the mdifiatins desribed herein is listed in Appendix B 22

37 .".'i'.'".'-. "." ""» r-.."-.-- «TTTT' v :.- v v O O Ö CN 0 W 4J..: O O H 4J O :>> t <ii a «I 01 B O 00 O rl u t H (0 > I 05 - I CM ö AVQ H19N1AVM 0) I I 2 - «I - *. «-. - * -. *. >.., ,-.:

38 1,1. l'.l» J1. 1. I I.» [''.". I. I i -1" ' *» *.J.vV\- - ' -.-_. _, J» r.. T ' «Li >»» Ini» i».., -.. ' V V * B u y <u h 01 <u C I V* u a ;- a 4-1 a ' I u u a. T I tn u > (0 4J 0) en u i«a. I - - _ ~% <u u SWUd/UHd 24

39 . '- - '. : -..»-.» L"i - ^. i -» -'.«7-.» «. :., HI > H 4-1 r-l CD u 1-1 «4-1 Vl-I H.e t ft vo vo ^* II "O <U xl* J < a. en 4-1 H 4-1 CJ H 1) 4-1 > 4J CJ O «J t-h H «M > OJ h f-t 4-1 t 0) u ^ a. r-l T 0) tu u 4-1 fl 0).-1 u a fi 'j- H H) C/l h 0), LT> O in SWdd/UlZd 1 1 i 25 w.;... '-»' " ' '._ ^_ i_ :» : J : - _ _ - 'y.ih'.v.v.v.y...

40 **!^^^^^'*^^^*r mmm '» m ".' ^»i... «,. m... i ' " ^ i S".'»"..l"r ' -'>. Vi- - '--.".- ' -. ^ '., I 01 en X * < < 4-1 a H x 4-1 «u r» 4J i-h 9 a 01 j-j CJ u 01 i > u IM 0) S~ h 9 en 01 X "j 4-i u 0 4J ~ u > «w i-h 4J 9 a 1 rh H 01 U 01 IM OC fa swad/wed O LO I _i in i 26

41 «. RESULTS AND DISCUSSION.1 Tw Pint Crrelatin Tw pint rrelatins have been alulated fr separatin dis tanes, x, between 1.66 and 1.7 fr a wide range f transduer diameters. 6 The aut-rrelatin and bunding ases (1.66 and 1.7) are shwn in Figs. 11 t 1 alng with the experimental data f Bull [15]. Figure 11 shws the aut-rrelatin at = 0. 6 It shuld be nted that the widths f the psitive rrelatin urves beme narrwer as the transduer diameter is dereased. Nte als, that the negative prtin f aut-rrelatin dereases as the diameter is redued. At the pint where - = , there is an 6* infletin in the aut-rrelatin (at a dimensinless time delay T*0.). step. The infletin pint has been aused by the reslutin time d.. That is, at the pint where is , the time required fr 6 many f the simulated pressure events t rss the transduer is smaller than the user speified time step. Hene, the regin f large psitive rrelatin is diminished due t the absene f thse events. A phenmenn whih is similar t the infletin in the aut-rre- latin urve an be bserved in the tw-pint rrelatins fr the small separatin distane ase [ y 1.66, Fig. 12] and fr the large separa- 6 tin distane ase [Fig. 1]. As the transduer size is redued, it is apable f reslving smaller wavelength disturbanes, but thse disturbanes deay s rapidly that the" derrelate adjaent 27 1.»,, «-'"«- "«- «-'".... I*.. <!.-'- '- '- «"- '. *- '-. '' i' '. ' "... _..:_...»-. ^:. *. _... -.,v»,. m... _... _

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44 M»Vy VC..^ ^«f».-'.['./.. 1.'./ t.«..' ', *' ' ' '.* *"" ' "'.' '.' ' v^«. 1 *' ''-- " ' " ~--~-. " *"- '" s I. 0) in u) *- 0), «Ö CD 0) I 2 CD 2 ^ & ^ «- *- ö d ö ö ii 4J r-» m.h PI - H II d XI* < < e O ^ ^ m ih 05.-I OJ >J pa u - u M-i en > e 0) kl Ö H 4-1 0) OJ 0 u 0) C -H H S-J 0 CU. 1 X Oi <u 01 N s 4J 4J "O <u 4-1 0) u <u U l-i 4J u A O 4J u e H h OJ P I* u D g II»- O U -H u au i 0) l-i i 0 iv..-.»-;. b«l»t - - *. ' --' ' _.

45 : *. y *. '. '. '. ' ^. ^ ^ ^ " T^V 7 -^ " " pints. Cnsequently, tw-pint rrelatins are redued as d_ is dereased belw A mre subtle effet an als be bserved in the tw-pint rrela- tin data whih is due t imprved reslutin f the lw frequeny, lng wavelength events. Fr large transduer sizes, a filtering effet is reated in the ppsite sense t the high frequeny ase just disussed. The larger transduers annt reslve and, therefre, are nt influened by the high frequeny disturbanes. Cnsequently, the lw frequeny infrmatin is less ntaminated and prdues a larger rrelatin ef- fiient than is atually present. This inreased rrelatin effet is manifested bth in an inrease in maximum rrelatin value and an in- rease in rrelatin width at a given separatin distane. Finally, fr suffiiently large transduer sizes, the measured pressure signal bemes ntaminated by the splitting effet disussed in Appendix A. That is, a ntinuus pressure event is split int tw events separated by a "dead band" when the partiular pressure event re- sides entirely upn the transduer. This splitting effet auses a de- rrelatin t ur between adjaent pints whih evidently begins fr transduer sizes n the rder f g in this ase. 6 Filtered rrelatin data have been reprted by Willmarth and Wldridge (14) fr pressure signals in the 0.41 < < 0.95 (lw band) * and 4.1 < < 6.8 (high band) frequeny ranges. In rder t mpare the rrelatin data prdued by the simulatin with these results, a pseud band pass filter has been emplyed. A mparisn an be made by * 6 1 " -' -* -' - -'-»' * ' ,.,-,:,. 1,; ti f. i".. '..'. 1 iii'i.'iiiiri'.i..

46 »>».. m< m'fif ^'F;'r,'t,t;'»t»n i',".' 1.,.'». ":* ' '-.* '..', ' ".'" " ;. J '>.' *. *., * '.,^r., t '' 7 r-;- ' emplying IF STATEMENTS in the simulatin prgram rather than digital filters. That is, nly thse randmly generated pressure events whih had frequenies in the presribed intervals were allwed t ntribute t the transduer histries. Sine there are higher harmnis assiated with the disrete pressure events, (whih were allwed t aumulate using this apprah), the "filtering" was nt perfet. Hwever, the atual experiments did nt emply perfet filters either. The lw band and high band data are shwn in Figs. 14 and 15 alng with the experimental data f Willmarth and Wldridge (14). In Fig. 15, nly the lus urve f the spae-time rrelatin peaks is shwn as a slid line. The apparent nvetin speeds fr the simulated ase are apprximately 0.6 U fr the high band and 0.8 U fr the lw band. Narrw band rrelatin data have als been prdued using a x similar pseud filter fr a spatial separatin f = 5.0. The fre- 5 queny bands were defined as 16.0 perent n either side f entered * frequenies f = = 0., 0.62, 1.15 and 2.9 and are shwn in Fig. ;-J 16. The narrw band nvetin speeds shw the prper elerity de- rease with inreasing frequeny. A enter frequeny f 0.14 _^ was * 6 als examined but the rrelatin results were unsatisfatry fr rea- sns whih will be disussed in the next setin..2 Pwer Spetra Pwer spetra fr the different sized transduer simulatins an be develped diretly frm the aut-rrelatin data. Hwever, interpreta- 2.«.

47 1 T C ID C 01 O" > h I 0 H 4-1 <d H 01 M 0 <J <U S u I 01 D. H t>0 T 0) h I ^^; * >"* I'-.V*. «*-. «V«- V- v^ '_ *- «"_ *'~ '- «r_.*. -' - - -*~. * »i

48 - - i "'.". -'..".''-". \ - _"-". - ". -. ". ". -". -". -"_ ". ". -" Ej 1 **~.2 band rm i C f high peaks 1 in + 1 a E ;J 00 ;: 10 ix> u i I \s \/ Lus retatin 1 b* &, > j V > n * «-.< >»»* i 1 ; eg en ii 1 Ja» «I O" a h u- I US 00 C u I w I tu a '*.'.:* V.v.i 60 'S MM in. - M,." 4 '-«-«- ^- J-A^- S -'.'-, L.J.'.J^«.I

49 »»*"» II II I* O (N m > < «- r> Ö»^ <N ti ii ii ii 8 n Ö II t u Cfl D. CD. en u fl n u u i I 01 a n H I I M h SB «8 0) fa > L. O _L_ 00 Ö CD ddy Ö i Ö I 5

50 r IF' i 1 '<:. '.. '! *,' ',* *-'* '.' ".* *> V *.'*'* V '"' * -J: '" *"".-"** '" *. (i.-- _~ tin f pwer spetral data is limited t a frequeny interval whih is ntrlled by the sampling data (speified time step) and the ttal time rerd. Sine either a Furier transfrm r a fast Furier transfrm must be used t extrat the pwer spetrum frm the aut-rrelatin, nsideratin must be given t the upper frequeny limit (Nyquist frequeny) where the inverted Furier transfrm will begin repeating itself. The sampling frequeny f must be greater than r equal t 2 f N (where f is the Nyquist frequeny) in rder t be able t rever the riginal signal frm the Furier transfrmatin. Figures 17 thrugh 20 represent the variatin f pwer spetra with transduer diameter, where the shaded area is the range f experimental data given by Bull [15], Willmarth and Wldridge [14], and Blake [6]. The maximum and minimum frequenies generated in the simulatin are f < 1 max where At is the reslving time step and f. > 1 mm where T is the ttal rerd time. The abve maximum frequeny gener- ated by the simulatin is smaller than the sampling frequeny f assis ated with Nyquist frequeny f f turbulent wall pressure flutuatins. N Therefre, the disrepanies between the experimental data and the simu- lated spetra at the high- and lw-frequeny extremes are due t the time step reslutin and the shrtness f the simulated rerd, respet- ively. Hwever, the results are in lse agreement with the experimental data in the raid-frequeny range fr a wide variety f diameters. The abve figures indiate a small inrease in the pwer level at high frequenies as the diameter apprahes zer. Cnsidering w

51 ^- - -,. i. - - ^ v. -.».' - "V ~. - J - *;J ' -TTT.-:":" ". -".""- ^V-*-..-_... M_,.... ;-.-;« *. *. /// Experimental Data Simulatin ~=0.784 (J0Ö* U 00 Figure 17. Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [141, and Blake [161. (Large transduer ase) 7 -_^L^. _^.

52 r; Vj y; ^T»W *".* :* '.*.* -." V.'.'.*.*.*..*.*-.^-?* -* ' ;.''- w _. ' *. '. "* ": V * -'/:.» "y ~~. i _ p» i^ - ", Experimental Data Simulatin - r = : * < < > w * 1 <- ^M» CN8 el - 'Oft (JUÖ U Figure 18. Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [61. (Intermediate transduer _ase) _: _ :"-.'.'. -- 'v>.t'f--->,._.\v_-.v.v "-v. ^a -« "

53 I'J-...*.. - V 7* "7 "7 * " - -"' - V*"» 1 ;'V -1 /7/7 Experimental Data Simulatin = * W* w* U 00 Figure 19. Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [6]. (Small transduer ase) 9 *_^_d s -» -» i - - -a ; i hi

54 ."..".."'." »» " _ ii» i i ' N i,, ;, ;. ' ;. *i ". ;. ':". ; iv't ':»,'.' v. '.yn W //// Experimental Data Simulatin -jr* (N «4- ~8 a-..!*# & - (Jüö U 00 Figure 20. Cmparisn between the pwer spetrum f a simulated wall pressure histry and the experimental measurements f Bull [15], Willmarth and Wldridge [14], and Blake [61. (Ultra small transduer ase) 40 -* ^ -^ -^ -^ ' r "-» '- '-r *-*> «--. _ , -.« ^..-. jh..*., JL^..*.*,,r..*».,m.j* JU..a..J,,.».,'

55 ....».._».._...,, _ «_.. «_... ^ ;.«.,.-. im -^..,-r-wr-^--w--r ty, ;* *-.-<.- the degree f apprximatin used in the present wrk and the sattering f experimental data the agreement between the simulatin and experimental results is nsidered satisfatry.. Variatin f P R.M.S. Fr many years experimenters saled transduer diameter data with * the bundary layer displaement thikness 6 [1-15] and shwed nly small variatins f P as the parameter d R.M.S.. apprahed zer. Hw- 6 ever, mre reent results suggest that small sale pressure flutuatins in turbulent bundary layers may have a signifiant effet n the P M value (Blake [6], Emerling, Meier and Dinkelaker [5], and Bull K.M.a. and Thmas [10]). Sine mst f these high frequeny flutuatins are prdued in the wall regin, it was suggested by Emmerling, Meier and Dinkelaker [5] that the saling f transduer diameter shuld be with respet t the inner variable _v whih expands the lwer limiting regin signifiantly. They als suggested that instead f wall shear stress, free stream dynami pressure shuld be used in nrmalizing the R.M.S. pressure. Fr the present study, it was deided t nentrate numerial U d testing in the range T < 200, in rder t verify the abve suggesv U d tin. Therefre, diameters with the value = 9, 18, 5, 54, 72, 90, 15, 180, 271, 60, 451, and 541 have been used fr testing. The ampli- tude f P R.M.S. an be adjusted in this simulatin by simply hanging a saling parameter in the prgram. Hwever, an adjustment was made nly fr ne large diameter ase t establish a realisti level fr use with 41 "-f "-- *-f l*j* "-I '- '-' "-A*_- "-T V '-- '-" " m J> _*- '- '-' - *_ _. '- '- - '- '^ - ' -«' '.,T '-,.! I J 1 1 ' I *.<» m\ * ml\ -*.1 -

56 -,, ;,'.,,',.;, -. ' -, -.. :- -, ^~:^".~-:^ *. ' -- > all ther ases. Hene, eah subsequent ase was subjeted t an idential simulated pressure histry. Figure 21 shws the variatin f Jl P w s with the transduer size as measured by Emmerling, Meier and Dinkelaker [5] and Bull and Thmas [10] alng with the present simula- tin results. Althugh 6 there is a flutuatin f P, values fr _ R.M.S.' *! U d K T > 200, the sillatin is small and P an be represented by a "v" R.M.S. straight line fr large diameters. The nstant simulated P R.M.S. regin is als fllwed by a sharp inrease in the R.M.S. value as the diameter shrinks. The simulated variatin apprximates the urve given by Bull and Thmas [10]. Althugh the inrease in P D M. is signifiant, it was nt lear ^ K.M. whether the jump was aused by the inlusin f high frequeny disturbanes r by ahieving a better reslvability f large distur r- anes at small diameters. Therefre, a simulatin was nduted in U X T whih all the disturbane events with wavelengths smaller than = 180.-* ; were exluded frm the simulatin. The remaining flutuatins were <"-.-* reslved with a diameter f U d = 18 whih is in the range f inreased P w That test prdued the same P and has indiated that R.M.S. r R.M.S. very high frequeny pressure flutuatins d nt ntribute measurably t the R.M.S. level. 42 -Jf --- '- -"- -* - -.' *. «. ' * : i,.,» i '«.'«*» '. j. I»,». 1. v «.', i."»»- -»-

57 ,,'..».. j j". - j» \ m r'v"m m "" V ; L'A.'V.TS-««.»s.«.^ * ^- y.vw LO O N O O LO 8 >4" -> U tu V (0 O O T 0) CN a I O (0 >. ' «b C 0lx Td ^ CN 60 -: : 4 : - - -* --»*- - ' Hi

58 rr» im- J~ v_«..».'....*..*»; - -J -*,.*. *. *. *.- w. rrr ~'-'''' ''.' ' ' -" '.-".- * " " " " " - ' 4. NCLUSION A wide range f transduer diameters have been used in the present simulatin t study the variatin f R.M.S. pressure with sensr size when the transduers are expsed t idential signals. It has been shwn that the P D level inreases mderately at small values f K.M.b. U d whih is nsistent with mre reent experimental data. The simulated result indiates that P _ inreases frm a nminally R.M.S. } U d nstant value fr sensr sizes smaller than = 40 whih differs frm v U d the departure pint f * 170 suggested by the experimental data. The simulated urve lsely apprximates the result given by Bull and Thmas [10] in terms f the magnitude f inrease. Althugh the rise in the R.M.S. level is signifiant, the result- ing urve suggests that the inrease is nt as large as that reprted earlier by Emmerling, Meier and Dinkelaker [5]. It is knwn that the abve ntributin t P R _ has been attributed t small sale strutures in the inner regin f the turbulent bundary layer by several wrkers in the field. Hwever, the R.M.S. level f the simulat- ed signal whih deleted high frequeny events indiated that the ntri- butin f small sale flutuatins t the P is insignifiant. R.M.. Pwer-spetra have been alulated fr several different transduer diameters. The pwer spetra shwed slight inreases in energy ntent at high frequenies and a slight inrease in the peak frequeny as the diameter was redued. The shapes f the spetra are very similar, and the agreement with experimental data is reasnably gd. 44 * '- ' - ' ^' -' <- " - -%-.'-' - --L -" - - " "-' - -" -"-.- 1'.^A--.-'.J...A-I.,'.». -,--.'.. '.. V. -.»*-...-,»'

59 TT '» i J.' ' ". ' II I ', I _ I ' ' I - P^ F TV Tw pint rrelatins have been alulated fr separatin dis- tanes f = 1.66, and 1.7. The rrelatins shw learly the 5 nvetin and deay patterns f the pressure field. Fr large trans- duer diameters the width f the rrelatin urves inreases. This is in ardane with the expetatin that higher rrelatin is btained when the reslvability f small sale strutures is pr. As the diameter is redued the rrelatin urves beme narrwer. Again this is expeted sine, the inlusin f small pressure flutuatins whih d.- nt rrelate ver large separatin distanes r time delays, have a signifiant derrelatin effet. Lw- and high-frequeny band rrelatin f the generated pressure flutuatins als shwed the rret nvetin speeds. The narrw band, tw pint rrelatins shwed a dereasing nvetin speed as the enter frequeny was inreased. Overall the agreement between the simulated urves and the experimental data f Bull [IS] and Willmarth and Wldridge [14] is quite gd. In summary, we have shwn that a simulated turbulent wall pressure signal with prperties similar t experimental measurements prdues rt mean square pressure levels whih vary with sensr size in a manner whih is similar t experiment. The size effet is nt prnuned fr the pwer spetra, but the interesting result that the maximum values f tw-pint rrelatins derease with dereasing size was bserved. This effet an be attributed t the inreasing ntributin frm small wavelength pressure flutuatins. 45

60 -r-r" ^, Mf, t..t. ^,-, REFERENCES 1. Dwell, E. HJ., "Panel Flutter: A Review f the Aereletri Stability f Plates and Shells," AIAA Jurnal, Vl. 8, N., pp , Marh Bushnell, D. M., Hefner, H. N., and Ash, R. L., "Cmpliant Wall Drag Redutin fr Turbulent Bundary Layers," Physis f Fluids, Vl. 20, N. 10, Pt. 2, 1977, pp Bukingham, A. C, Chun, R. C, Ash, R. I., and Khrrami, M. M., "Cmpliant Material Cating Respnse t a Turbulent Bundary Layer," AIAA Paper N , June Crs, G. M., "Reslutin f Pressure in Turbulene," Jurnal f Austial Siety f Ameria, Vl. 5, N. 2, February 196, pp Emmerling, R., Meier, G. E., and Dinkelaker, A., "Investigatin f the Instantaneus Struture f the Wall Pressure Under a Turbulent Bundary Layer Flw," AGARD Cnferene Preedings, N. 11 n Nise Mehanisms, Paper N. 24, Blake, W. K., "Turbulent Bundary Layer Wall Pressure Flutuatins n Smth and Rugh Walls," Jurnal f Fluid Mehanis, Vl. 44, Pt. 4, Deember 1970, pp Bradshaw, P., "Inative Mtin and Pressure Flutuatins in Turbulent Bundary Layers," Jurnal f Fluid Mehanis, Vl. 0, Pt. 2, Nvember 1967, pp Kraihnan, R. H., "Pressure Field within Hmgeneus Anistrpie Turbulene," Jurnal f Austial Siety f Ameria, Vl. 28, N. 1, January 1956, pp Kraihnan, R. H., "Pressure Flutuatins in Turbulent Flw Over a Flat Plate," Jurnal f Austial Siety f Ameria, Vl. 28, N., May 1956, pp Bull, M. K. and Thmas, A. S. W., "High Frequeny Wall Pressure Flutuatins in Turbulent Bundary Layers," Physis f Fluids, Vl. 19, N. 4, April 1976, pp Bull, M. K. and Langeheineken, T., "On the Wall Pressure Field in Turbulent Pipe Flw," Max-Plank-fr stromungsfrshung, Göttingen, Reprt N. MPIS-MITT-7, Uberi, M. S. and Kvasznay, L. S. G., "On Mapping and Measurement f Randm Fields," Quarterly f Applied Mathematis, Vl. 10, N. 4, January 195, pp ','«" ' -.. -". -\ -'. -'. '.'.-' "-". '"- '"«"> " " '* - "" "\.""-."' :'".' *.'. "' ' '.' "." "-" ".' -" ' -." ' "." ' ' -> '.. T.» '....:.» -.'.. -.*» ' -«'-f "-»*-- '!! -T -.'.--.*...'., ^1», M*. «"-.*- ^-.V.*-»* «*. «'.. -»,. l'.l'.l

61 . ' ":'.v.-.r.'...-.t-.'.v-".^. ". v , * ^.'..-.^..y._.-._r ^~-.--;.--.w/vt.'. - ">': ' -" - "1 1. Willmarth, W. W. and Rs, F. W., "Reslutin and Struture f the Wall Pressure Field Beneath a Turbulent Bundary Layer," Jurnal f Fluid Mehanis, Vl. 22, Pt. 1, May 1965, pp Willmarth, W. W. and Wldridge, C. E., "Measurements f the Flutuating Pressure at the Wall Beneath a Thik Turbulent Bundary Layer," Jurnal f Fluid Mehanis, Vl. 14, Pt. 2, Otber 1962, pp Bull, M. K., "Wall Pressure Flutuatins Assiated with Sub-Sni Turbulent Bundary Layer Flw," Jurnal f Fluid Mehanis, Vl. 28, Pt. 4, June 1967, pp ^ 16. Ash, R. L., "Simulatin f Turbulent Wall Pressure," NASA Cntratr Reprt 2958, May Ash, R. L and Khrrami, M. M., "Simulatin f Turbulent Wall Pressure Flutuatins fr Flexible Surfae Respnse Studies," AIAA Paper N , January Pantn, R. L., Gldman, A. L., Lwery, R. L., and Reishman, M. M., "Lw-Frequeny Pressure Flutuatins in Axisymmetri Turbulent Bundary Layers," Jurnal f Fluid Mehanis, Vl. 97, Pt. 2, Marh 1980, pp : 47 J

62 -~." -"."." -~."." " " " "»" _ * APPENDIX A FORCE NTRIBUTION OF A SINGLE PRESSURE EVENT The streamwise Length f the transduer (size) and the sampling rate (time step) bth distrt the rerded pressure signal. Depending n the wavelength (and frequeny) f the partiular flutuatin event, the net pressure ntributin sensed by a transduer an vanish even thugh the disturbane is ver the transduer. Als, if sampling rates are slw, an event may be nveted mpletely arss the transduer in less than ne time step. In rder t understand the distrtin prdued by finite pressure transduers and sampling rates, alulatins have been made fr a variety f prttype pressure events, AP(x,t), given by: 0, x < t-l AP(x,t) sin 2ir(x-t) - 1/2 sin 4ir(x-t), t-l L L < x < t 0 (t > 0) x > t 'I"-' where L is the wavelength and is the nvetin speed. Eah event was assumed t have unit magnitude and n deay f any type was assumed in rder t shw learly the spatial and tempral reslutin effets. Sine the wave frnt fr AP(x,t), designated X f (t), is lated by: X f (t) - t *, 48 ;: :<..*"»"..

63 "I "1 L. I ^ I J»!. - l. -. u - I l T»".".T^«^"». : ;. ". _.. - _' "._-*. ':_ ': ". '"..'- z-~!-:~.-=- -. a ntributin t the pressure fre n a transduer lated n the in- terval, Xj < x< X2, will be made when X f (t) > Xj and X f (t) _ L < x 2. where the rigin f the rdinate is sitting n the wavefrant and is mving with the event nvetin speed,. Assuming the spanwise (Z) dimensin f the transduer is unity, the inremental fre ntributin due t this pressure event at sme time, t, whih satisfies the wave frnt riteria (x\ < t < X2+L) is given by: (t n ) - J* [sin 6-1/2 sin 26]d6 s s 2e s 9i - s 02-2TTF where 2TTX1 when X- - L < xl Er A N p. «^ 9 1-2ir (X -L) f when X, - L > x, L. ' - 'v! and 2TTX, f, when X < x 2 L '. 1 -* «-" TTX,, when X, > x,. «1 f.v 49. v.. ' ]-. ' -- '.-V--. '«** '" ' "" " * " -"

64 ;. \i '_. 'y V. "!"«". ' J*.' <-' -T ^.' r. w ' ' * * - * -.* V *-* *-*.* ".* -* ^ T* - -*' * ', w-^-w T~- " " ""." '." -" '. ' ~~ " r ' _' Cnsequently, the average inremental pressure ntributin, AP, an be written: AP X.-X. 2ir (x 2 - x.!l ^ 1/4 [s 26 -s 26 ] + s 8 - s 8 The time step disussin will be deferred temprarily and attentin will be restrited t sme time, t, when the disturbane is ver the n transduer. Fr very early times, the wave frnt will be ver the transduer, while the trailing edge f the wave has nt yet rssed the bundary, xi. (See Fig. Al). Then,. v while, 2irt n 2nx s that AP. w^qr (»* I s 4irt 4ir: 2TTX. 2irt li l n - s = J + s s - Fr very late times, while, 9 1 " 6, - 2*(t -L) n L 2TTX2 1 Integratin an be heked by taking the limit as Xj > x?» whih restrits time s that 8 2 * 6^. 50 ' - * - - ' -.' -.».'..-.: :- - iiri*n>* vwftfaflr\> T> -1 " 'fr - ^ ^ *-"* -" -"* -" F -'»*-" - *->*.* -.^i» «-.'^

65 " " u <u V u 4-1 u S 60 5 > i > H -a u. 5^ O U a. u 0) I- u H a. a) M 51 ^ w«^ :. :..,* v.... >'_ «-. -,

66 i PH II -i - i -n m y mt»^~^ >T- I - r»-i»t i y» -^ ^ *^i -T». -i. L -^t -. "^i ^-TT - -*-*^ -;- -j leaving ip 2n (x--x. )... r 4TTX- 2iit 1/4 L s 2 - s 1 2irt 2ir x. nj + s n - s 2 Tw types f wavelength ases are delineated by the ase when x -x. * L. When the wavelength is greater than the transduer length, L > x_-xj, there will be times, t, when neither the wave frnt nr its trailing edge will be ver the transduer, in whih ase - T... r 4Tr(t -x ) 4TT(X -L-t ) 1 & P L! 1/4 [ s n 2 + s 1 n_j 2n(x 2 -x 1 ) I L L 2ir(x, -L-t ) 2ir(t -x.) + s 1 n - s n 2 When the wavelength is less than the transduer length, L < x -x, fr, V L < t n < ^2_, AP = 0. n The send ase in whih L < x -x, is disussed next. Figure A2 represents the averaged fre as a funtin f time fr a large transduer (L < x -x ). That figure shws hw the shape f the duble sine wave is reinterpreted suh that the negative and the psitive pressure regins are separated by a dead band. This dead band is reated during the time interval when the entire wave is ver the trans- 52 : *: :. - '* _ ,....., '..» m

67 -'"*. ' -' " - " * i.»".» a w u t - ->- -." *-* *. ih ** ** 1 v;/1 e l" ". i H.* *.." -, V» *,* *] A 4J < 4J! -; --;! M "" <U * "7. *' H H a :- : : : :.'" ^" j A \* a m.'." < <J 1 * V ' OJ V '-*."< a "- "**- * 4-1 : ibäj S v^. -."-*. 1 u.' -* «H '.*""- '.! *,^r\*-j <U '-- * " a 4J - ".*-*.* 14-1 & 0 ' '.*"*.** v a *-/"/-*<, '"- * H - * - - * V *«4-> O ' '-* *»" *. *\ -*.* IM - i " -' -""' * ~ -"" *i & /-N '-V^ O 4J "V *.* 1 u <»-. -r«o ' H. * - -. a a u CD u u <U 6 < 4-1 CN < 01 AVM NIS CBZIlBdOSICI» \<<<*-i<.>:' ^ ^s-i--.-.-^ -_«. -. -j--.» '.' V-'."!.' _ ".-, _-' _ * 5

68 7VT CV.Tv,'' «T'L"«^':* -' v* -* -.' * "* -.'."' ' :.' -.' -1* ' ' *' ' '. *'-' r ' ' ' "' '' :. r.'"", ';', V-~>7 r: ' -'" " " " : " '" r " " -" " " ' " -" -" " -V.. -.-: duer. The same effet is als respnsible fr a signifiant hange in the wavelength f the averaged transient fre. The indiated wave- length will be twie that f the event when the transduer length, x -x., is equal t L. When x? -x is greater than L, the apparent wave- length is inreased by mre than a fatr f tw, as shwn in the fig- ure. Als nte that the averaging aused by the transduer has a severe effet n the amplitude f the wave. At this pint, it shuld be pinted ut that, in rder t shw the abve effets as learly as pssible, the smth urve f Fig. A2 has been btained by emplying a reslving time step whih was an rder f magnitude smaller than the time step used in the simulatin. Therefre, the same ase was repeated emplying the nminal simulatin time inrement used in this study. The resulting urve is shwn in Fig. A. Althugh it is nt a smth urve anymre, the majr features desribed abve still are detetable. A series f tests have been run fr the ase in whih L > x ~x. In thse tests the width f the sensing area has been kept nstant at a value f ^ 0.176, while the frequeny f the wave and the reslving 6 time step have been varied. Figure A4 shws a typial wave with a frequeny f 00 Hz (S 0.287) reslved by a time step whih is an rder f magnitude larger than the time step used in the simulatin. The slid lines are the numerially integrated value««, while the squares represent the theretial wave at a pint latin. Althugh, the agreement between the tw types f alulatins is gd, the figure des nt arry suffiient infrmatin t represent the prttype duble sine wave. Figure A5 shws the same wave reslved by a time step equal t 54 '*- - '- ' **.»*. '-«*. *»**. *..-'.<'.'r^» /«:. i.'. -' ' _-..' J -: L -1 >! '. 1.?.. J^-i-

69 ". -\ - : -.' -'.-'.-*. :*.-*.!".'i"i i". i". ' ". "". ' '- " " " " f..'\- " % - * -".-:- 0) u O 4-1 Bj a- 0) a. < B ea ' i i LÜ Q 0 0 H 4-1 O 00 (0 0) u / *U 4_) ai < 00--' TO V4 Q. m n > 4-1 in J n *r r AVM NIS QZI1USIC] C1 < h ä H ".-- - '- '- - *- - '.. '--'- - - ' --'. ' " - ' a. -' - - «. i r "-

70 -.'.?:.-..- ' *. " * ^^ J. ^^^.^^-^-*' "PW TT»"» j... J^ll^^ u^^ _i^.^_. m V "J I* UJ F 5 Q n u 4J ni V i- M a u u eg i H U x: 4J H 0J > r 2 CU C H CU t-h XI O - X u Of, B u ih u > ^ Cfl 4J < *^ u n a U u a) u i-i t u i u_ H O U H H 4J i-j 9 0 H H O 9 i <u H u ih eg d X M U a 1= u x: H 4-J "'Ä *, 9«-.*-.I -' -", < u t-i AVM NIS QZI1U0SIC] * * " 56 ft

71 WH.-r.v., -'».-v.-^_- %"""l! " I * *". «V '. > r OJ a a; IU LU e < Q O LU LU LU r LU O 42 0) > ID a) O T 00 C <u 01 > 4-1 <U < h «D..H <U 4-1 Cfl U-i u e H C -M H C 4-1 O -H U a).e a 4-i E CU H 4J u SO AVM NIS iazuaysii 57 is,v '"- '"- '- *'- *- ' -»*- **- ' * rr'n - -*-*- - t-m.-f ^ _. -» '-.^_

72 ^ -1 "^ -" ""I f " * " * -L. ""-I»1 --^ the simulatin time step. The agreement between the tw methds is exellent and the shape f a duble sine wave is retained. Subsequently, the frequeny f the wave was inreased t 2500 Hz (S =2.9) while the reslving time step was kept nstant. Figure A6 suggests that as the frequeny f the wave is inreased, the tempral reslutin bemes a prblem. Als, nte that there is a phase shift between the tw types f alulatins. Figure A7 represents a wave with a frequeny f 2500 Hz reslved by a time step whih is an rder f magnitude smaller than the time step used in the simulatin. The abve figure indiates that the reslutin prblem disappears as the time step apprahes zer. In summary, signifiant alteratins f prttype high frequeny pressure flutuatin events an ur due t transduer size and reslving time inrement. But, as the wavelength f the individual event inreases, the effets f transduer size and tempral reslutin diminishes. 58 '.V.'.-.»... -_._.

73 .*."... : '..v..*. T -.' : T.J."V.--_-" ' ;.* ;:-.- :.--;.--.» - -." " -, LÜ «CT u a 0) O < Q UJ i ÜJ (T O UJ LU X H 4J 4) I CU ^1 O L5 60 C CU <-i Hi > 4-1 i < en w a es en H C en i-( u e 1-1 H «6 tn a. 6 u u j= H 4J u I H AVM NIS QZUyOSia * - -= ^_^L_

74 *-^ry*7 l»lt^tv»k"? ' m K m : r- m "- -"»^"»T"1 '^T r*-z^r: i-. vv +\ -"V "- " "^^ =-: " -\ -". a OJ LÜ LÜ _l < _I > < _l > < Q O LU 1- H LÜ < (Z r O LU LU \- (N H 4J OJ > 0 01 C U 0 T u < C (0 6 C en t O 6 0 -H O 0 H 4-1 II JS 1-1 4J O CD C tlj U JZ 4-1.H U U 0) O H a. >H 6 t Ä e H < a; u AVM NIS QBZiiaasia - mt 60 ~»' - -*-**-*.'. -» -^ * i. - * i -' -' -' i v " *-' *-' «-*«-* - - '-"-.,.. _ J

75 " - '«- "--'" ' ' '». '., *J.---V.'- '- '. '-' '. _,. -/ J''-«'"---''-«''.»'.'': "_v- _v......_ '.. '. * <.! >*.'*A'.N'A'.S'A ' -.". APPENDIX B MPUTER PROGRAM LISTING C C 157 PROG RAM SIMU<INPUT»OUTPUT»TAPF5»INPUT»TAPE6«0UTPUT»TAPE 1»TAPE 2 1»TAP E7»TAPE8) DIME NSION P(5i»»16500) DIME NSIOM SUM) <X(8000)»SUMY<8000) EOUI VALENCE«Pi (60000)»SUMX)»(P(74000)»SUMY) DIME NSION PA<( 6)»PR(6)»STRT(6)»STP(6) NR«5 NW»6 NDIM MAXIMUM TIME DIMENSION NDIM N2C «N0IM/500 CDIH «NDIM R RX»1.5 RT IS THE T IME ADJUSTMENT FOR DT-STREAK RT»2. XD»D EVELOPMEN T LENGTH (M) XM«M ODEL LENG TH (M) PAR IS THE DE CAY ADJUSTMENT ACUNTING FOR DISTURBANCE SIZE PAR» 10. XD«4.1*5 DXT IS THE DISTANCE BETWEEN STORAGE LOCATIONS DXT«0.008 DWT OXT DWT IS THE WIDTH Of THE PRESSURE STORAGE LOCATION DWT IMJ 0 XM»0.0 XD«X D + XM XT«X D+XM DXT«NODE SPAC ING ON MODEL(M) NXM NXM» STRT STP( STRT STP( STRT STP( XSMX XSMX US«BLT«UFRI CNU» PHO«DPT«X IS THE IF MORE IS THE NU NUMBERS <1)«XD 1)«STRT(1 <2)«XD+0. 2)«STRT(2 ()«XD*0. )»STRT( 2.0*STP( IS THE R C BLT/8 ADJUSTMENT FOR DX STREAK. THAN ONE BURST PER STREAK» RX IS NOT UNITY MBER OF STORAGE LOCAT IONS...YOU MUST CHANGE THE IN THE DIMENSION STATEMENTS AND IN THE TAPE OUTPUT» DWT )*DWT )*OWT NXM) IGHT-MOST LOCATION STORING PRESSURE r-' -> fe»fl 61

76 ; - _.".,-j.'.'._»...." -.' " K'.'".' ".,'i V.".'-.."."'.' ',..-_.-_ J- /».»,».»." T- - 1^ ^«1. A.' ". '" CJ C C ENTER OTHER DIMENSIONLES WDMmOWT/DPT WRITE(NW,205 FORMAT!111,5 WRITE(NW,206 F0RMAT(5X,*L OCATION NO. DIM. LOC.*»/) DO 0 I»1,NX M XDM«STRT(I)/ WCITE(NW,210 F0RMAT(10X,I NTINUE CALCULATION TW»RHO*UFRIC CALCULATION PRMS»2.0*Ty CALCULATION TWV IS TIME TWV«DPT/US LENGTH OF SM ALL DISTURBANCE CLM«0.7*US*T WV DECAY BIAS P ARAMETER CWV«PAR*CLM CWV MISCELLANEOU S NSTANTS PI« TPI-2.*PI HPI-PI/2. C0» C P5 C2» D1> » D DISPLACEMENT THICKNESS HERE IF DESIRED (CM) S STORAGE LOC. WIDTH--W/OISP. THICK. ) WDM X,*DIM. STORAGE WIDTH EO. *»E16.6W/) ) DPT ) I»XDM 4,10X*E16.8) OF WALL SHEAR STRESS* TW *UFRIC OF RMS PRESSURE FLUCTUATION OF MINIMUM RELEVANT DISTURBANCE LENGTH DURATION OF DISTURBANCE C STARTER FOR RANDOM NUMBER GENERATOR XSTART RNM»URAN(XST ART) XSTART-0.0 C CALCULATION OF NOMINAL PEAK FREQUENCY FPEAK» *US/ITPI*0PT) FMAX«10.*FPE AK C MPATABLE T IME STEP DTT-l./FMAX/ 10. C A USER SPECI FIED TIME STEP CAN BE ENTERED HERE C THIS TIME ST EP IS THE TIME INCREMENT USED IN THE RESOLUTION OF THE C* OUTPUT- -THE INTERNAL FLUCTUATION TIME STEP IS RANDO" C MAXIMUM TIMF IS NSTRAINED BY MPUTER STORAGE. SET THE NUMBER C OF ALLO WABLE TIMESTEPS NTM JM 62..."«,".'"»'."." -'".' -".' " <" if.'»""."."."* "" -''j-'" -'" -"" - " -.» '.- '» ' '-. <

77 'W "T - I 1 " i. «-.» -. «.» *»".-r-v-.-j-.. S'^v " 1 N * NTM INITIAL TIME IS TO(SEC) T0»0. DTSUM.O NCT-1 PREFIXF S FOR RANDOM NUMBER CALCULATIONS PX»DPT* UF RIC/US PT-DPT/ US PW-US/D PT CALCULA TI ON OF REQUIRED START UP TIME FOR SIMULATION TSO-1.8 *X T/US NMIN-TS 0/ DTT TMAX1»N MI IF NMIN I IF(NMIN -N TM)201,201,202 NTM-NMI N NTINU E TMAX»NT M* OTT WRITE16,1 FORMAT«5X THAXS-C DI M*DTT TMAX-TM AX TMAX1 NTM2-ND IM -NMIN+2 TREF-TM AX 1 NREL«0 INITIAL IZE PRFSSURE ARRAY N X«1,NXM DO 52 N T«1,NDIM P(NX,NT )«0.0 NFLG«0 TSUB»0. INITIAL X-0. DTAVG-D NCT-0 DTSUM-0 TO-TO+D TAVG IF(TO-T MAX1)2,150»150 IFJNFLG ) 151,151,15 NFLG-1 TMAX1-T P(I,J ) NTINU TO-0. NTINU TSUB-TR TMAXS'T NREL'NM N*DTT S GREATER THAN NTM, NTM IS OVERRIDDEN 00) DTT,TMAX,*DT«*,F10.8,10X,*TMAX«*,F10.4,//) IZE LOCATION AND TIME BASE» ETC. TSUM/NCT MAX J»NTM2,NDIM 0.0 F E EF MAXS+TREF IN ' i *m m _ AC-

78 .»...'-..".. " - '-*. ' ', ' '-» «_).- Ik -»- i _ -_. - V -_4 * - -J. -_ » - -] ;.-j C 2 RNM»URAN(XSTART) PNM»0.005*0.99*RNM CALCULATION OF OX USING RANDOM NUMBER RNM HPII«HPI*RNM DX«PX*(2.2-2/<RNM )+72.*RNM**2*0.6*TAN(HPI I)) DX«RX*DX X«X+DX RNM«URAN(XSTART) RNM» *RNM CALCULATION OF RADIAN FREQ. FROM NEW RNM SRNM«RNM** RRNM«(1.-RNM)**0.74 FRNM«1/RRNM-1 W»PW*(0.52*FRNM+0.799*SRNM-0.785*RNM) F-W/TPI IF(F.GT.700.)G0 TO 85 DC«0.78-F/(105.*FPEAK) UC«DC*US GO TO UC»0.46*US 87 TP-l/F B0BFAC«4.605/((DPT/UC)**2> DXE»UC*TP XQ«X-OXE XO IS THE ORIGIN OF THE SINE WAVE FLUCTUATION X IS THE FRONT OF THE SINE WAVE XS IS THE FIRST STATION AT WHICH P IS RERDED. XS AND NXI WILL BE TAKEN AS THE FIRST STATION VALUES THEN OVERIDDE XS«XD NXI«1 CHECK TO SEE IF THE DISTURBANCE IS OVER THE MODEL IF(XD-X),,5 IF THE DISTURBANCE IS OVER THE MODEL* HAS IT PASSED THE LAST DATA STATION NTINUE IFU0-XS»«X>971,1,1 971 NTINUE IF(XO.LT.O.)GO TO 96* DO 4 I«1,NXM NNP-XO/STPJI) IF(NNP)961#962, NXI-1 GO TO 96* 962 NXI«I GO TO NTINUE 4 NTINUE 964 NTINUE I NXI IS THE NUMBER OF THE FIRST STORAGE LOCATION ^ 5^ 64 V. « V <. «-» »

79 .. i. Vn v ^. i. i.,., y.:'». ^i,»..».. i/. ;T;-;. '. /.,-.,*.'.--.*-'. '.'.'-.~-"7~^ ~~-."" ""."..' -'.- - CXNI-NXI 5 NTINUE C GENERATION OF RANDOM TIME STEP 6 RNM»URAN(XSTART) RNM-O *RNM HPH-HPI*RNM DT»PT*(2.2-2./ (RNM «5)+72.*RNM**2+0.6*TAN(HPII) ) DT-RT*DT T-TO+DT NCT-NCT+1 DTSUM-DTSUM+DT IF(XO.GT.STP(NXM))GO TO 2 C C C C C C C GENERATION OF GAUSSIAN RANDOM PRESSURE AMPLITUDE RNM«URAN(XSTART) CIND»RNM*0.5 IND-CIND CIND-IND PPP«2.*(1.-CIND)-1. ARGR"RNM/(1.+CIND) ARG-1./(ARGR*ARGR) CT«ALOG(ARG) CM«SQRT(CT) PMG«CM-(C0+CM*(C1*CM*C2))/(1+CM*(D1 + CM*(D2*CM*D)) ) XS-XS-DXT PPP«5.9000*PPP -«PE«PRMS*PMG*PPP PE«2.0*PE MODIFICATION OF PRESSURE AMPLITUDE TO ADJUST RMS PRESSURE LEVEL DO LOOP FOR STEPPING THROUGH MODEL STORAGE LOCATIONS DXXE-DXE DO 1<» NX«NXI,NXM OXE-DXXE MODEL STATION X-LOCATION XS«STRT(NX) ARRIVAL TIME OF PRESSURE FLUCTUATION DXS-XS-X ADX>ABS(DXS) DXS»(DXS+ADX)/2. TGO«DXS/UC*T FLUCTUATION DEPARTURE TIME DXO«STP(NX)-XO TSTP-DXO/UC+T DOFS TSTP EXCEED TMAX IF(TSTP-TMAXS)<?,<,7 7 IF(TG0-TMAXS)8»8,1 8 NSTOP«NDIM GO TO 10 9 NSTOP«TSTP/DTT 10 NGO-TGO/DTT»JMI t^aü Of* - * i A *; J> ji 65 v--:i

80 -. ; ~^.-." T "! -,-. 1.".." V "? V ~> V". -: : - J 1 - -» v «'«- «-«-» I I i Vti't. ' '. i«r» t~.»~. v.'i IF(NG0-NREL)99,99,98 98 NTINUE AGX»DX0**0.6 ARG»-4.20*AGX DH»0.5+0.fe5*EXP(ARG) DXEE«OH*OXE DR-DXE-DXEE RE»DR/<DTT*UC) RE»RE*0.5 NP»RE NSTOP«NSTOP-NP DXE-DXEE IF(DXE.LE )GO TO 2 DO LOOP FOR SUCCESSIVE TIME NTRIBUTIONS TO THE SAME X LOCATION DO 12 NT«NGO,NSTOP TC»NT*DTT DELT-TC-T XTF-X + UCMDELT) XTB-XTF-DXE THET1»(XTF-STP(NX))*TPI/DXE TH1A«ABS(THET1) THET1«(THETl+THlA)/2. THET2»(STRT(NX)-XTB)/DXE TH2A«ABS(THET2) THET2«TPI-(THET2+TH2A)*PI XOT«UC*DELT IF(XOT-O.OOOOOOl)19,19,18 18 NTINUE FREQUENCY DEPENDENT VARIABLE DECAY RATE ADJUSTMENT AR-XOT/DXXE ARGX»-2.8**(AR**0.0) IF(ARGX.LE.-0.)G0 TO 7 DECA«EXP(ARGX) GO TO 2 7 DECA-O.O GO TO 2 19 NTINUE DECA-1. 2 NTINUE DTH1-2.*THET1 DTH2«2.*THET2 CFl-(C0S(DTH2)-C0S(DTHl))/4. CF2-C0S(THET1)-C0S(THET2) 0P-PE*(CF1*CF2) DP«DP*DECA*DXE/(TPI*DWT) B0BBA«B0BFAC*DELT**2*IF<B0BBA.GT.00.JBOBA-O0.0 DP"DP*(1.-EXP(-B0BBA)) NTIME-NT-NREL P(NX,NTIME)«P(NX,NTIME)*0P ^"^r^ 66

81 -. *"." - ITj. I"..". 4"....».-.«.. ' -._- V- - ^ NTINUE NTINUE NTINUE NTINUE F0RMAT(5X,F1 GO TO 2 NTINUE NWRT«NTM/8 CNT-NTM DO 11 J*1>NX PA(J)-0.0 PR(J)-0.0 DO 21 J»1,NX DO 20 I«1*NT PFI«P(J,I) PA<J)«PA(J) PR(J)»PR(J)+ NTINUE PA(J)«PA(J)/ WRITE(NW>192 PSJ-PA(J)*PA PRJ«PR(J)/CN T PR(J)«SORT(P RJ-PSJ) PRMSR«PR (J)/ WRITE(NW,19 NTINUE DO 22 J-1,NX DO 22 I«1,NT P(J,I)«P(J.I NYM-8000 ANN»NYM NKL»NXM-1 DO 24 KK»1,N KL DO 25 K«l,50 0 Kl-K-1 SUM.O. SM-O.SSKKM»0 SUM1-0. DO 40 M»1,NY SUM«SUM+P(KK SUM1»SUM1*P( SKKM-SKKM+P SKKMK«SKKM SM»S!i+P(l» SKKOMK-SKK NTINUE SKKM«SORT(SK SKKMK«SORT( S*«SOPT(SIV 0.6,6E14.6) M M M PFI PFI*PFI CNT )J>PA(J) (J) PRMS )J>PR(J)>PRMSR n H )-PA(J).$SKKOMK»0.$SKKMK«0.»M)*P(KK*M+KD 1#M)*P(KK+1,M+K1) (KK,H)**2 K+P(KK*M+K1)**2 M)**2 0MK+P(KK+1,M+K1)**2 KM/ANN) SKKMK/ANN) ANN) < ' -1 m. -. ' L^t^_Ll_VJ^J ^_^J^ ^^_ -' V v v - -.' _

82 " «' Q SKKOMK»SORT(SKK0MK/ANN) SUMX(K)»SUM/ANN/SKKM/SKKMK SUMY(K)«SU*11/ANN/SKK0»1K/SM NTINUE WRITE(NW,191) WRITE(NW>200)(SUMXCK),K*1,500) WRITE(NW,191) WRITE(NW,200)(SU"Y(«),«1,500) NTINUE FORMAT(lHl) F0P.MAT(//,5X,"UNC0RRECTED PAVG( ", 1, " ) IS",E20.6) FORMATtlOX, "RRECTED PRMS ( ", 1, ") IS",E20.6," RATIO", F10. ) F0RMAT(5F20.6) JXY« J»1,NXM WRITE (1 HP(J,KJ),KJ»1, 16500) NTINUE STOP END ',* - : J^L^ ^

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