Turbulence simulation in wavelet domain based on Log- Poisson model: univariate and multivariate wind processes

Transcription

1 he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, 0 urbuene simution in wveet domin bsed on Log- Poisson mode: univrite nd mutivrite wind proesses Cho in, eng Wu, Ahsn Kreem Nthz Modeing Lbortory, University of Notre Dme, Notre Dme, IN, USA ABSRAC: he wind in tmospheri boundry yer is hrterized s turbuent fow t high Reynods number. he ompexity in turbuene is miny from its noniner nd mutise struture. his pper simutes the wind proess utiizing the wveet domin bsed on the Log-Poisson mode where both muti-se struture nd the intermitteny oud be propery represented. Foowing simuting univrite proess, methodoogy for simuting mutivrite proesses is so estbished, where orretion tehnique bsed on the oherene of the wveet oeffiients t eh se is utiized. he resuting simution foows the hrteristi fetures of the smpe reord nd offers more reisti simution thn the spetr representtion bsed methods. KEWORDS: urbuene; non-gussin; wveet; Log-Poisson mode; mutivrite proesses INRODUCION he wind proess in tmospheri boundry yer is hrterized s turbuent fow t high Reynods number. A physi origin of the ompexity in turbuene is due to the noninerity ontroed by the Nvier-Stokes equtions. As resut, turbuent fows re ntury desribed bsed on the sttistis theory. ypiy, turbuent fututions re ssumed to be Gussin. Convention wind proess simution, suh s spetr representtion, is bsed on this ssumption. Another physi origin of the ompexity in turbuene is due to the muti-se property resuting from noniner intertions, where the turbuene re better desribed utiizing the sttistis of veoity differene t two otions ompred with tht of veoity fututions t singe otion. As strong disontinuous nture of turbuene, intern intermitteny seems ony hnging the veoity sttistis sighty ( brod bnded probbiity density distribution (PDF) ompred to Gussin distribution), whie it hs signifint effets on the sttistis of the veoity differene ( onsiderby rger ti PDF ompred with Gussin distribution). In order to tke into ount the effets of intermitteny, sever phenomenoogi modes hve been introdued in the iterture, suh s Log-norm nd other mutifrt modes. Obviousy, the onvention wind proess simution nnot tke intermitteny into ount. On the other hnd, these rndom sde modes re onvenient to be inorported in the wveet bsed simution [,]. Speifiy, the reenty deveoped Log-Poisson mode [3], whih is imed to be be to reve the physi mehnism of the intermitteny phenomenon, is utiized in this study. As resut, the intermitteny wi be preserved in the simuted turbuene utiizing wveet domin. he boundry ondition effets on the energy ontining (EC) rnge nd the non-o retionship between the ses in the mutipitive sde proess of the inerti subrnge re inuded in the simution. In ddition, the wveet domin so fiittes to inude nonsttionry feture in the simution often observed in tmospheri turbuene. Bsed on this deveoped frmework, univrite nd mutivrite wind proess simution shemes re proposed nd their effiy is demonstrted by exmpes. 93

2 ANALSIS OF WIND PROCESS Convention strtegies of simuting wind proesses re usuy restrited to the seond-order sttistis with impied Gussin ssumption. hese strtegies n be generized by Krhunen- Loeve (K-L) expnsion nd numeriy reized with the Gerkin Sheme [4]. he generized Gussin proess simution sheme is given s [5] N N N k V, t kkkt kk dm m t k k m () where is the origin rndom vribe; V, t time t; k re the eigenvues of K-L expnsion; k bution (i.i.d) stndrd Gussin sequene; t is the trget rndom proess with respet to is the identi independent distri- k re the eigenfuntions orresponding to k k; d m re the oeffiients orresponding to the presribed trunted orthogon bsis t m, m,, N. Suppose Fourier bsis is seeted, the simution is equivent to the spetr representtion. On the other hnd, the simution n be extended to the nonsttionry Gussin proess if wveet bsis is seeted [5]. A typi turbuent fow fied is not neessriy Gussin s demonstrted by some experiment dt [e.g.,6]. Speifiy, the PDFs of veoity differenes t the inerti se exhibit signifint onger ti thn Gussin distribution, whih is reted to the intermitteny in turbuene. Convention shemes bsed on the seond-order sttistis re insuffiient to desribe suh non-gussin proess. On the other hnd, sine typi dynmi informtion in turbuene is ontined in the sing quntities, bsed on whih the wveet oeffiients re generted, wveet expnsion utiizing higher-order sttistis is n pproprite too to nyze nd simute suh muti-se proess with non-gussin PDF.. Sing properties of turbuene here re two signifint sing quntities desribing the muti-se struture of turbuene: the energy dissiption rte nd the veoity differene v. In this study, the hrteristis of re utiized. For turbuene fow, within the inerti rnge, the p th order moment of v t se hs power-w dependene on [3], v p ~ p () where denotes the expettion of the rndom vribe; p represents the exponent t the p th order moment. Within the EC rnge, is reted to the boundry ondition nd my exhibit nonsttionry fetures s it refets the extern mehnism.. Sttisti property of the Log-Poisson rndom sde mode he univers sing w n be reized vi Log-Poisson proess, whih is rndom mutipitive sde proess []. Within the inerti rnge, for the se of p=, the veoity differene v t se nd the veoity differene v t se hs the foowing retion [,3] 3 n,, v W v v (3) where W, is the mutipitive ftor between v nd v ; nd re the sereted prmeters nd need to be estimted bsed on the re turbuene fow, however, refer- 933

3 he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, 0 ene [3] suggests tht /3, /3; n is the independent Poisson rndom vribe with men,, stisfying the foowing retion [,3], n (4) It shoud be noted tht the bove retion from direty to is identi to vi 3 to. 3 SIMULAION OF WIND PROCESS he wveet tehnique is utiized here to simute the muti-se struture of the turbuene fow. Suppose ony v in the EC rnge depends on the extern mehnism, it is resonbe to ttribute the nonsttionry prt in the turbuene fow to v within this rnge. herefore, it is neessry to determine the riti se, whih pinpoints the EC rnge nd the inerti rnge. Usuy, is ssumed orresponding to the frequeny f whih mkes fsvv f mximum, where SVV f is the power spetrum of the wind veoity V t with zero men vue. he frequeny f is funtion of men wind veoity n integr se s Krmn spetrum is utiized. Hr wveet funtion is hosen here sine it represents er physi mening of veoity differene. Correspondingy, the retion between the wveet se order nd the dominnt frequeny f of th Hr wveet funtion n be determined [Appendix A]. Prtiury, the retion between nd f is shown s f nint og 0.74 (5) where n is the highest wveet se order used in the simution. 3. Simution of univrite wind proess b time position, ssoi- m b is osey re- Use mv bk, to denote the wveet oeffiient t the se nd k ted with the normized form bk,. With Hr wveet trnsform, V k, ted to v t se [Appendix B]. 3.. Simution sheme he simution sheme of univrite wind proess in wveet domin is presented in Fig., whih is desribed s foows: () Determine using Eq. (5); n () Within the EC rnge, v (,, ;,, ) i, i re produed from i.i.d stndrd Gussin sequene nd trnsformed to,,, ;,, b k ; (3) Within the inerti rnge, k n v (,, ;,, ) i, n i re produed independenty from v or, i v bse on, i v W i,, v or v, i W i,, v, i (6) where W, nd then trnsform the generted n, n,, W, v to i,,, ;,, b n k, k ; (7) 934

4 (4) mv bk, is dusted from bk, ording to the wveet oeffiients orretion t the sme se, whih shres the foowing retion with onvention spetr density for sttionry [Appendix C] proess, m,,,, V bk bk mv bk bk b f SVV f df (8) where, b f is the Fourier spetrum of wveet funtion, t t se b SVV f is the onvention spetr density with men veoity. For non-sttionry proess, the bove eqution is ony pproximtey stisfied, whie the ext energy distribution of wveet oeffiients shoud refer to the tehnique s Eq.(). In essene, the eddies oud be treted s sedependent sttionry within the inerti rnge s they depend miny on the intern mehnism of the turbuene rther thn the extern ondition; m b,. (5) Sythesize the wind proess v from the generted V k ; Inputs: S f V Identify f Determine Inputs: m b,,, ; k,, V k es Within the EC rnge Generte mv b, k No Within the inerti se rnge Generte mv b, k Synthesize v Wind veoity (m/s) Wind veoity (m/s) Figure Simution of univrite wind proess. Mesurement ime (se) Simut ion ime (se) Figure he mesurement nd simution resut of univrite wind proess. 935

5 he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, Power spetrum (m /s) Mesurement Krmn Simut ion Frequeny (Hz) Figure 3 Power spetrum PDF 0 - Observtion Simution Gussin PDF 0 - Observtion Simution Gussin Normized wveet oeffiient Se Normized wveet oeffiient Se PDF 0 - Observtion Simution Gussin PDF 0 - Observtion Simution Gussin Normized wveet oeffiient Se Normized wveet oeffiient Se3 Figure 4 PDF of wveet oeffiients t some typi inerti ses. Wind veoity (m/s) Wind veoity (m/s) 0 Mesurement ime (se) Simut ion ime (se) Figure 5 he mesurement nd simution resut of univrite nonsttionry wind proess. 3.. Cse study Hurrine Ktrin is utiized in this study to demonstrte the effetiveness of the proposed simution frmework. he mesured dt nd the simution resut re shown in Fig.. u is estimted s 47m nd 7. he power spetr bsed on the mesured nd simuted dt re presented in Fig. 3 together with Krmn spetrum. PDFs of the wveet oeffiients t some typi inerti ses re given in Fig. 4. here is 4.59% error between the vrines of the simuted proess nd of the mesurement. A nonsttionry se is shown s Fig Disussion It is interesting to notie tht referene [] hs used simir pproh to simute univrite sttionry wind proess. his study offers resonbe strtegy for simuting nonsttionry 936

6 wind. More importnty, there is mor distintion in the tretment of the Log-Poisson mode between this study nd referene []: () Within the inerti rnge, it is known tht the univers sing w is obtined from the ergodiity ssumption whih impies sttionrity, however, the gorithm for generting the wveet oeffiients in referene [] distorts the ergodiity in three spets. First, the odd wveet oeffiients re generted by invoving n gebri verge of the wveet oeffiients t rger se, whie the even wveet oeffiients re not, whih re given here [], b,, m/3 /9 ki bk i bki, s /3 (9) m /3 b /9 ki, s /3 bki, (0) where s is binomi rndom vribe tking the vue + or -; m is equivent to n, in this study. he ergodiity ssumption indites tht bk, shoud hve identi property t eh se whie referene [] obtins the odd nd even terms using different shemes; seond, both b, k i nd b, ki devite from the tu rndom vribes bk, sine nother rndom vribe s is invoved in the expression. he se of b, ki is worse due to the verge opertion previousy disussed; third, there shoud be n bsi i.i.d Gussin rndom vribes to generte the wveet oeffiients t eh se insted of. () Referene [] simpy regrds the wveet oeffiient bk, equivent to the veoity differene v nd supposes they hve the sme sttisti property. Atuy, inferred by the b, shoud be determined bsed on the ontribution wveet deomposition gorithm, k from the v i s. he weighting of the ontribution from eh v, i depends on the seeted, wveet funtion. For exmpe, for Hr wveet bsis, whih is utiized in this study, the retion between the veoity differene nd the wveet oeffiients is presented in Appendix A. (3) In referene [], it is stted "when the rge fututions ourred t ertin se, the oeffiient fututions beome rge round the sme time t the other : the o sef-simirity ws reized with the gorithm in this study." Atuy, this o strong futution is fse in the iner-. In this regrd, the differene wi be pssed down whih uses simir orresponding dif- t smer ses. Atuy, the univers sing w hods true ony in the sttisti mening rther thn the smpe mening, therefore, the smpes of k, insted of reusing t eh time. pperne used by the mnipution in the gorithm: the smpes of bk, ti rnge re generted from the sme series of smpes bk, between the smpes of bk, ferene between bk, bk, shoud be generted independenty from the smpes of b, 3. Simution of mutivrite wind proesses he vibe informtion in simuting mutivrite wind proesses is the ross power spetrum SX f nd the trends represented by mx bk, nd m bk,,, ;,, k t the trget otions X nd. he min obet is to generte mx bk, nd m bk,,, ;, n k from the wveet oeffiient ross spetr density t the sme se [Appendix C] between X nd. he retion is shown s foows for sttionry proesses m,,, Xm mxm b X where Cm m, i i f () C f C e d f S f e is the wveet oeffiient ross spetr density (CPD) t the sme se. X Simir to the univrite se, for non-sttionry proesses, the bove equtions re pproximtey stisfied. 937

7 he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, Simution sheme he simution sheme of mutivrite wind proesses in wveet domin is presented in Fig. 6, whih is desribed s foows: 0 0 () Generte respetivey the initi wveet oeffiients mx bk, nd m bk, using the univrite simution strtegy desribed in setion 3.; () Within the EC rnge, produe the fin wveet oeffiients mx bk, nd m bk, using onvention mutivrite Gussin simution strtegy t eh se; (3) Within the inerti rnge, estimte the first four moments d Xi,, nd d i,, 0 0,, n; i,, 4 t eh se from the produed mx bk, nd m bk,. Cute D the trget Cm Xm f from S X f nd set the initi oherene vue mxm mxm, the itertion number it=, where C mxm f m Xm f () C f C f mxmx mm G G (4) A set of Gussin orreted proesses mx bk, nd k, re generted using stndrd mutivrite Gussin simution gorithm [8] ; (5) rnsform respetivey the obtined G G mx nd m through forwrd Modified Hermite rnsformtion nd spetr orretion [9,0] to the non-gussin m X nd m whih mthes their trget Cm Xm f, C X mm f nd trget moments d Xi,,, d i,, ; (6) Mesure the oherene mxm from the generted m X nd m nd ompre it with the trget oherene mxm to determine the error s err mxm mxm (3) D (7) If the error is beow the eptne eve, the itertion ends. If not, updte C mxm by D D D C it C it C it C C (4) mxm mxm mxm mxm mxm unti the error is epted; (8) Generte the wind proesses v X nd m bk,,,, m respetivey. v from X k, m b,, n m b Inputs: SX f,,, m nd X Generte m b, k X Generte m b, k it it it Generte m b, k X Generte m b, k Estimte dxi,, i Estimte di,, i D mxm mxm Gussin mutivrite simute G G m b,, m b,, it X k k Spetr orretion m b,, m b,, X k k mxm it it it D D D mxm mxm mxm mxm mxm No es mxm mxm? Figure 6 Simution of mutivrite wind proesses. 938

8 3.. Cse study wo mesurements drwn from different pssges of Hurrine Ktrin re empoyed to simute the mutivrite wind proesses. he generted time histories re show in Fig. 7. he ross orretion oeffiients of the mesurement nd simution exhibit good greement (Fig. 8). Mesurement (Lotion ) Mesurement (Lotion ) Wind veoity (m/s) ime (se) Simution (Lotion ) Wind veoity (m/s) 30 0 Wind veoity (m/s) ime (se) Simution (Lotion ) Wind veoity (m/s) ime (se) ime (se) Figure 7 he mesurements nd simution resuts of mutivrite wind proesses. Cross-orretion Coeffiient 5.9 x Mesurement Simut ion ime Interv (se) Figure 8 Coeffiients of ross-orretion funtion. 4 CONCLUSION A methodoogy of simuting univrite nd mutivrite turbuene fows in wveet domin is proposed. his method, utiizing the Log-Poisson rndom sde mode, emphsizes the intern intermitteny effet of the turbuene fututions in this simution. he simution resuts gree with both the power spetrum nd the sttisti informtion presented by the sing w, thus present better representtion of the ntur wind proess thn the onvention methods whih ony onsiders the seond-order moments. Furthermore, in the simution of the mutivrite proesses, the wveet oeffiients re itertivey updted to mth the wveet oeffiients oherene t the sme se, thereby gurnteeing the onsistene of the ross orretion funtion between the mesurement nd the simution resuts. 939

9 he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, 0 5 ACKNOWLEDGEMENS he support for this proet provided by the NSF Grnt # CMMI whih is grtefuy knowedged. We so thnk Dr. Kurtis R. Gurey for the ssistne in progrmming the mutivrite non-gussin simution nd providing the dt set. 6 APPENDIX A. When time interv is se, the Fourier spetrum of Hr wveet funtion is shown s 0.8 = = = f x 0-3 =0.74x -(n-) Frequeny f (Hz) Figure 9 Absoute Fourier spetrum of Hr wveet B. he Disrete Hr wveet reonstrution gorithm is shown s foows: n k mv bk, ~ v, k,, n (5) i, n i k In order to expin ery the retion between veoity differene v nd wveet oeffiient mv bk,, the bove gorithm is expressed in unnormized form. Speifiy, suppose wind proess vt inuding 8 disrete points, there re m b, ~ v v v, m b, ~ v v v,, m b, ~ v v v V 3 3, V 3 3, 3 4 V 4 3 3, V, i 3 4 V, i i i3 4, ~ V 4, i i When ppied in the reonstrution ike the simution in this study, v i, mbk, is nother rndom vribe different from v k,, ~,, ~ m b v v v v v m b v v v v v m b v v v v v v v v v represent i.i.d sequene. herefore,, furthermore, indited by Centr Limit heorem, mb pprohes Gussin distribution oser thn v. C. he retion between the wveet oeffiients ross-orretion t the sme se nd the onvention ross-orretion is shown s [] mxm *,,,,, C b b m b m b (-)/ (f) * * t b, t b, xty tdtdt * * t b, t b, xty t dtdt itf * itf it tf tbe t be RX t, te dtdt (6) (7) 9

10 If X nd re both sttionry, the bove eqution n be written further s mxm itf * itf,,, X C b b t b e t b e S f dfdtdt * f f S f df, b, b X hen it n derive the foowing expressions,, C f S f df (9) mxm b X,, C f f S f (0) mxm b X,, C f S f df () mm V V b VV,, C f f S f () mm V V b VV Obviousy, sttionry proess impies se-dependent sttionry proess. (8) 7 REFERENCES Gurey, K. nd Kreem, A., Appitions of Wveet rnsforms in Wind, Erthquke nd Oen Engineering, Engineering Strutures, () (999): Kitgw,. nd. Nomur, A wveet-bsed method to generte rtifii wind futution dt, Journ of Wind Engineering nd Industri Aerodynmis, 9(7)(003): She, Z.-S. nd E. C. Wymire, Quntized Energy Csde nd Log-Poisson Sttistis in Fuy Deveoped urbuene, Physi Review Letters, 74() (995): R.G. Ghnem, P.D. Spnos, Stohsti finite eements: spetr pproh, Springer-Verg, New ork, K.K. Phoon, H.W. Hung, et., Comprison between Krhunen-Loeve nd wveet expnsions for simution of Gussin proesses, Computers & Strutures, 8 (3-4) (004): C.W. Vn Att nd W.. Chen, Sttisti sef-simirity nd inerti subrnge turbuene, Sttisti modes nd turbuene, Leture Notes in Physis, edited by M. Rosenbtt nd C.W. Vn Att, Springer-Verg, Berin, (97), pp Z.-S. She nd S. A. Orszg, Physi mode of intermitteny in turbuene: Inerti-rnge non-gussin sttistis, Physi Review Letters., 3(99): Shinozuk, M. nd G. Deodtis, Simution of Stohsti Proesses by Spetr Representtion, Appied Mehnis Reviews, 44(4) (99): Gurey, K. R. nd A. Kreem, A ondition simution of non-norm veoity/pressure fieds, Journ of Wind Engineering nd Industri Aerodynmis, 7778 (998): Gurey K. R., Modeing nd simution of non-gussin proesses, PhD hesis, University of Notre Dme, (997). thesis, Rie University, Houston, (996). 94