Aousis 8 Pis Aousi hiion of n ulsoun sugil nsmi in h lin n nonlin gim of woing A. Posi B. Ivnčvić n D. Svil b Fuly of Elil Engining n Comuing ns 3 1 gb Coi b Bosi Insiu Avni Vćslv Holv bb 1 gb Coi nonio.osi@f.h 3579
Aousis 8 Pis Th mho fo msumn of iv ousi ow of n ulsoun sugil nif hs bn suggs in h f ousi fil oniions. Th ssu fil of h nsmi imms in h of qu wvlngh n vibing h funmnl fquny 5 H hs bn msu wih lib hyohon sysm iffn lil xiion lvls. In h lin gim h nsmi hs bn hoilly sib s n ousi iol. Th i ousi ow islmn n vloiy mgniu xiion fquny hs bn foun n goo gmn bwn hoil n ximnl suls is obin. Whn nsmi is xi wih high lil ow lvls h nonlin bhviou in loing mium s wih song viion iviy. In h vg ow sum of h o ousi ssu signl is vin h sn of hmonis n f subhmonis f/q ulhmonis n f/q of xiion fquny n lso h sn of viion nois wih oninouus fquny omonns vn u o wnih hmoni of funmnl fquny. Th sil ssu isibuion of h is fquny omonn in h f ousi fil hs bn msu n is onibuion o ol ousi ow hs bn lul. 1 Inouion In his wo msumn of iv ousi ow hs bn onsi using lib hyohon msumn sysm. Th nsmi wih sonoo i hs bn imms qu wvlngh λ/ on h woing fquny f 67 H in h nhoi ool Bosi Insiu Fig 1.. Th sil isibuion of ssu mgniu hs bn msu iffn fquny omonns in n y iion. In h lin gim of woing only xiion fquny omonn s in h ssu wvfom signl[1]. In h nonlin gim in fon of sonoo i ominn nonlin ff is viion. Subhmonis hmonis n ulhmonis in h o wvfom signl in fon of h sonoo i. In h lin gim of woing h non-il sou sonoo i hs bn moll s illy n nsvslly osilling sh[6]. Th ms of il sou sou vloiy n islmn mgniu foun using ousi ioiy hoy ssuming oin sou n ssu ls bouny w-i bouny [7]. In h nonlin gim h is n oninouus fquny omonns n i hs bn sn h vg mgniu of ssu h is fquny omonn of ins hs sil isibuion in h fom of C/. Lin moling In his of wo h mols of illy n nsvslly osilling shs hv bn nly in lin ousi oximion[8]. Volg n un x.1 Rilly osilling sh Mium i Mium 1 w Sonoo i y Digil osilloso Assuming h sonoo i is oin sou of ulsoun wih imnsions muh smll hn wvlngh λ 6m in h mium funmnl xiion fquny f 67 H. Mium x B&K mlifi B&K Fig. 1. lsoun oin sou in h f ousi fil - y Mium 1 Th signl ossing hniqus hv bn us o obin ssu mgnius in h ousi fil oins h fquny omonn of ins. In h lin gim of woing only xiion fquny is sn in h signl of ynmil ousi ssu. A high xiion lvls nonlin ffs in loing mium : ousi sming fini mliu ffs n viion[-].th ominn ff is viion u o nonlin osillions of i bubbls in h flui. Th is fquny omonns subhmonis hmonis n ulhmonis in h ow sum of ynmi ousi ssu[5]. Fig.. Rilly osilling sh n ssu ls bouny Th i ssu fil fom l sou wih ius n b win in h fom of q. 1. 358
[ ] [ ] 1 [m]- isn fom l sou o fil oin [m]- immsion h [m 3 /s]- volum sou sngh [m 3 /s]- nsiiy of loing mium w [m/s]- ulsoun vloiy in w [m -1 ]- wvnumb in w [m]- ius of hoil illy osilling sh Pssu fom img of h sou is givn in h fom of q.. [ ] Th ol ssu in h ousi fil oin is givn in h fom of q. 3. [ ] [ ] R 3 R- offiin of flion givn in h fom of q.. os os os os 1 1 i i R [g/m s]- hisi ousi imn of mium 1 [g/m s]- hisi ousi imn of mium 1 Th ssu mgniu is msu in h f fil in - iion n onnion vloiy onil funions of wo sous hv bn us o fin ssu fil on xis. Th vloiy onil ψ is givn in h fom of quion whn volum sngh sou mgniu is unnown. ψ sing onnion bwn ssu n vloiy onil in fom of q. 5 h ssu is foun. ψ ψ 5 Th ssu ssuming f fil oximion n only iffn in hs bwn l sou n is img is givn wih q. 6. C f P sin 6 f [H]- woing fquny P [P]- ssu mgniu C [P] - oximion m Whn immsion h λ/ hn vlu sin 1. Aoximing h ssu mgniu sil nny P using q. 6 h volum sngh of h sou n oh ms of h quivln sou n b sily foun iffn nsmi xiion lvls. Th suls of fiing shown on fig. 3. Fig. 3. Fiing msu ssu mgniu sil isibuion wih hoil uv Ali RMS lil ow in onsi s is P EL.6 W. Th oximion m msu in his s C 5. P m. Volum sngh mgniu lul fom C is.33 1-5 m 3 /s. Th iv ousi ow P is lul using q. 7 n i hs vlu P.63 W. Th loousi ffiiny fo η is lul s io of iv n li ousi ow n i is η %. 8 P 7 Knowing sou volum sngh mgniu h vloiy n islmn mgnius of illy osillng sh n b lul using quions 8 n 9. 1 u 8 In h lin gim of woing h onnion bwn vloiy nf islmn is givn using q. 9. ξ ξ ξ u 9 Mgniu of vloiy on h illy osilling suf.67 m/s n islmn mgniu on h sou is ξ 1.67 μm. Th ssu fil lul ssuming illy osilling sh n ssu ls bouny is shown on fig.. Fig.. Pssu fil mgniu of illy osilling sh n ssu ls bouny Th vloiy onil funion on -xis is shown on fig. 5. Aousis 8 Pis 3581
Aousis 8 Pis In h f fil oximion h xssion isn' so omli n i is givn wih xssion 11. os A sin [ os ] 11 Fig. 5. Vloiy onil of illy osilling sh n ssu ls bouny I n b sn on fig. 5 h gin of vloiy onil u ψ n ssu ls bouny is high. Th mns h vloiy u is ominn o h ssu. In h - iion n h xssion fo ssu mgniu n b fi wih C/ funion. Vloiy mgniu m/s n islmn mgniu ξ 13 μm. Ths vlus high hn in h s of illy osilling sh s onsqun of vibing boy only in - iion. Th hoil ssu mgniu sil isibuion in y iion of illy n nsvslly osilling sh is om wih ximnl on fig 7.. Tnsvslly osilling sh Th nx hoil mol onsi is nsvslly osilling sh. Th ssu mgniu sil isibuion oximion m C is sm s in illy osilling sh mol bu vloiy n islmn mgniu of sous n' sm u o iffn quions sibing mol. y u u Fig. 7. Psu mgniu sil isibuion on xis y m Th suls of omison isn 1 m fom sonoo i shown on fig. 8. u u os Fig 6. Th i of sonoo s nsvslly osilling sh Th ssu fil of isol nsvslly osilling sh n b win in h fom of q. 8. 1 A os 8 I n b ssn h ssu fil ns on ngl n sou m A. Th m A is givn in h fom of q. 9. 3 A 9 Th ol ssu fil ssuming suosiion of i n fl wvs fom ssu ls bouny is givn wih q. 1. os A R os A Fig. 8. Psu mgniu sil isibuion on xis y 11.5 m Fom suls on figus 7 n 8 i n b sn h b mol is illy osilling sh u o gomy of sonoo i hlf of h sh.. Th suls fo quivln hoil mols in h fom of illy osilling sh fo iffn xiion lvls shown in bl 1. P EL [W] [m/s] ξ [μm] P [W] η [%].65.6 1.67.6. 1.37.39.51.59 3.5 Tbl 1. Pms of illy osilling sh lul fo h S sou woing in h lin gim 358
Aousis 8 Pis I n b sn fom bl 1. h loousi ffiiny fo is %. I will b vy insing o s how his fo n b lul in h nonlin gim of nsmi woing whn sbl n nsin viion sn in h oing mium. 3 Nonlin gim of woing Th lin mol hs bn suggs in h nonlin gim of woing ssuming h ll nonlin sous osilling bubbls in h viiniy of sonoo i. Th ms of mium unhng in h f fil. Th ssu wvfoms hv bn o in h fils on -xis iffn lil xiion lvls. Th ssu wvfom is o 1m wo iffn lvls of xiion shown on fig. 9. 1 ϕ1 1 ϕ 3 ϕ3 3 P1 P P3 n P i i 1 φn i... 1 n- numb of fquny omonns in is sum snion Knowing ssu mgniu h fquny omonn of ins P i n ssuming h ms of mium n' hng in h f fil u o viion h volum sngh of h sous on h is fquny omonn of ins is givn in h fom of quion 13. P i i i 13 i Th iv ousi ow on h n b lul nowing volum sngh of sou in h fom of q. 1. i i Pi i i i 1 8 Th li ou fo luling iv ousi ow on h fquny omonn is shown on fig. 11. Figu 9. Pssu wvfoms o in h ousi fil wo iffn lvls of xiion lf-lin ighnonlin Th mon ossing hniqu wlh mho is us o lul ow signl nsiy. Th suls fo wo iffn lvls of xiion shown on fig. 1. Fig. 11. Div ousi ow on h fquny omonn of ins in h nonlin gim Th i ousi ow on xiion fquny is ominn n i is svl mgnius high hn ow on oh fquny omonns. Th suls fo lul iv ousi ow on som is fquny omonns shown in h bl. Fig. 1. Pow sum nsiis of o ssu wvfoms iffn xiion lvls I is vin iffn bwn ssu mgnius of fquny omonns in h signl. Th ow sum mgniu is nsfom o ssu mgniu h fquny omonn. Th sm lulion s suggs in lin mol is li fo h is fquny omonn of ins. Th ol ssu mgniu is suggs s sum of h iniviul fquny omonn in h fom of q. 1. Fig. 1. Pssu mgniu sil isibuion h fquny omonn of ins 3583
Aousis 8 Pis Figu 13. Pssu mgniu isibuion of subhmonis Fig. 16. Pssu mgniu sil isibuion n omison wih hoil suls f 1/ In h bl 3 h fiing suls shown fo f 1/. P EL [W] 1/ [m/s] ξ 1/ [μm] P 1/ [mw] 1.89.73.9 5. Tbl 3. Fiing ms on subhmoni fquny Th moiy of ousi ow is on funmnl fquny f.. Figu 1. Pssu mgniu sil isibuion fo hmonis Th fiing ssu mgniu sil isibuion wih hoil uv in h fom of C/ fo onsi li lil ow is shown on figu 15. Conlusion I n b sn h in nonlin gim of woing h loousi ffiiny fo is s fom % o 3 % n in nx ss h mo li lil ow lvls will b onsi. In fuu wo h sm ou will b fo iffn ys of sonoo is n mo li lil ows vlus Rfns Fig. 15. Pssu mgniu sil isibuion n omison wih hoil suls f I n b sn h sil isibuion of mgnius n b vy wll fi wih hoil lw n ms of quivln illy osilling sous on h is fquny omonns n b foun. Th suls of hoil fiing fo funmnl fquny shown in bl. P EL [W] [m/s] ξ [μm] P [W] η [%] 1.89.3.18.51 3.86 [1] Aousil hiion of ulsoni sugil vi.ieee lsoni symosium 1995. [] By RT. Nonlin Aousis. Amin Jounl of Physis 1973; 19:16-167. [3] Hmilon M.F. Blso D.T.: On h Coffiin of Nonliniy-B in Nonlin AousisJASA 1988; 831:7-77. [] Hmilon M.F. Blso D.T.: Nonlin ousis. Amin Insiu of Physis 1997. [5] Lubon W. Numil Invsigion of Nonlin Osillions of Gs-Bubbls in Liquis. JASA 1976; 59:83-93. [6] Pi A.D. Aousis An Inouion To Is Physil Pinils An Aliions Amin Insiu of Physis 199. [7] Kinsl L.E Fy A.F Cons A.B. Sns J.V. Funmnls of ousis. John Wily n Sons INC. [8] Ay A Hogson TH. Soun Riion fom An Aling Sh. Jounl of h Aousil Soiy of Ami 1976; 59:S88. Tbl. Fiing ms of hoil sou on f 358