ࡱ> %{ABhIJKLMNh pC5@bjbj22 ,LXXOh::::FkFkFk8~kBl2bmXs(sssdy y ySUUUUUUR ~UyNx"pxyyU::sslV"""y:RssS"yS"""sVm DFkZ?l0 ::::(yy"yyyyyUUeFkFkTEHNI KO VELEU ILI`TE U ZAGREBU PROMETNI ODJEL DIPLOMSKI RAD Istra~ivanja turbulencije traga zrakoplova u funkciji sigurnosti Mentor: Student: Doc.dr.sc. Sanja Steiner Katica Maljkovi MB 2405 143 988 Split, 2003. Sadr~aj 1.Uvod--- ----------------------------------------------------------------------------------------------------2 2. Pojam i definiranje turbulencije traga  wake-vortex -----------------------------------------4 2.1.Svojstva vrtloga u tragu ----------------------------------------------------------------------------4 2.2. Etiologija nastanka aerodinami kog vrtlo~nog traga iza zrakoplova ------------------------5 2.3. Parametri turbulentnog aerodinami kog traga --------------------------------------------------9 2.4. Gibanje vrtloga------------------------------------------------------------------------------------ 13 2.5. Propadanje vrtloga --------------------------------------------------------------------------------14 2.6. Posljedice djelovanja na zrakoplov u turbulentnom tragu prethodnika -------------------18 2.7. Primjena minimalnog razdvajanja zrakoplova pri pojavi vrtlo~nog traga---------------- 21 3. Segmentacija kriti nih faza letne operacije ---------------------------------------------------- 23 3.1. Utjecaj turbulentnog aerodinami kog vrtloga prethodnog zrakoplova na sigurnost leta u fazi krstarenja-------------------------------------------------------------------23 3.1.1.Vjerojatnost ulijetanja u turbulentni trag prethodnog zrakoplova- ----------------- 24 3.1.2. Postupci izbjegavanja turbulentnog traga u fazi prilaza i slijetanja-----------------26 3.2. Utjecaj turbulentnog aerodinami kog traga na sigurnost polijetanja prateeg zrakoplova-------- ---------------------------------------------------------------------30 3.3. Utjecaj turbulentnog aerodinami kog traga na sigurnost zrakoplova u fazi krstarenja ------------------------------------------------------------------------------------------35 3.3.1.Vjerojatnost susreta vrtlo~nog traga-----------------------------------------------------35 3.3.2. Potencijalne opasnosti kod susreta s vrtlogom u fazi krstarenja--------------------39 4. Wake vortex istra~ivanja u funkciji preventiv-------------------------------------------------42 4.1. Smjernice buduih istra~ivanja wake vortex  a ---------------------------------------------42 4.2. Nove tehnologije nacionalnog ureda za aeronautiku NASA --------------------------------44 4.3. Utjecaj smanjenja vertikalne razdvojenosti zrakoplova na sigurnost zra nog prometa --------------------------------------------------------------------------------------------52 4.4. Primjena empirijskih metoda u procjenjivanju okomitog kretanja vrtloga i njegovog raspadanja ----------------------------------------------------------------------------53 5. Zaklju ak ----------------------------------------------------------------------------------------------56 6. Literatura ----------------------------------------------------------------------------------------------57 1. UVOD Veli ina pouzdanosti svakog segmenta u sustavu zra nog prometa izravno utje e na ukupnu razinu sigurnosti njegova funkcioniranja, a eventualni otkaz samo jednog elementa mo~e izazvati otkaz cjelokupnog sustava. Stoga je implementacija sigurnosne koncepcije u cjelokupnom sustavu zra nog prometa ujedno i uvjet njegova funkcioniranja. Sigurnosna koncepcija zaatite, regulacije i kontrole pretpostavlja sustavni pristup u definiranju svih elemenata sustava, normiranju mehanizama regulative, kvantifikaciji sigurnosnih minimuma rada, te pozicioniranju viaerazinskog procesa kontrole i prevencije. Posaban sapekt sigurnosne regulative, koji za sada joa nije dostatno aktualiziran, jest problem turbulentnog vrtlo~nog traga koji se stvara u letu iza zrakoplova  wake vortex turbulence. Razmjerno s poveanjem inteziteta zra nog prometa poveana je i u estalost letnih operacija zrakoplova, kao i trend normativnog smanjivanja minimuma separacija. S aspekta sigurnosti letenja, a poglavito u po etno  zavranim fazama, to postaje sve vei problem zbog utjecaja aerodinami ke vrtlo~ne struje zraka, koja se stvara iza zrakoplova u letu i zadr~ava u slobodnoj atmosferi odreeno vrijeme. Svaki zrakoplov generira turbulentni vrtlog u tragu u svim fazama leta kao sna~ne vrtloge zraka u smjeru nizvodno od krila. Vrtlozi mogu trajati nekoliko minuta i mogu uzrokovati zna ajne smetnje ukoliko kroz trag prolazi drugi zrakoplov. Postojanost i intenzitet vrtlo~nog traga ovisi o te~ini zrakoplova, obliku i razmahu noseih povraina, letnoj brzini te o lokalnim atmosferskim uvjetima. Najintenzivnije vrtlo~no strujanje generiraju teaki i isti (kota i i mehanizacija krila uvu eni) zrakoplovi pri malim letnim brzinama. Vrtlo~ni trag djeluje na zrakoplov koji u njega uleti destabilizirajue, a u odreenim uvjetim i krajnje opasno. Postoje odreena pravila kojima se osigurava dovoljna horizontalna, vertikalna ili vremenska udaljenost izmeu zrakoplova koji slijee ili polijee, tako da je vjerojatnost upada u vrtlog prethodnog zrakoplova prihvatljivo niska. Velike smetnje mogu se javiti kao rezultat susreta s vrtlogom za vrijeme polijetanja, slijetanja i krstarenja, a veina ih se dogaa kad zrakoplov uleti u turbulentni trag drugog zrakoplova koji se nedavno penjao ili spuatao kroz iste visine. Uporaba odreenih zra nih putova kod upravljanja civilnim zra nim prometom poveava mogunost nastajanja tih susreta jer zrakoplovi lete na istim ili recipro nim putanjama. Let zrakoplova u skupini ili zbijenom poretku uobi ajen je za vojne formacije, a vojni piloti su posebno uvje~bani za takvo letenje, no znanje o opasnostima zbog ulijetanja u vrtlo~ni trag sve viae je potrebno i civilnim pilotima i kontrolorima letenja radi odreivanja minimalnog razdvajanja izmeu zrakoplova, te odreivanja minimalnog sigurnosnog intervala izmeu uzastopnog polijetanja i slijetanja zrakoplova. U blizini aerodromske zone intenzivnost letenja je viaestruko porasla, te i civilni zrakoplovi lete u zbijenom poretku, a poseban problem vezan je za eksploataciju kategorije teakih zrakoplova (zrakoplovi te~ine 136 000 kg ili viae) koji stvaraju turbulentni vrtlog velikog intenziteta. Koliki je intenzitet prometa mogu, s obzirom na bit aerodinami kog vrtlo~nog traga i fenomene koji se u njemu pojavljuju, kako prepoznati i reagirati na pojavu trenuta ne destabilizacije zrakoplova odnosno kako razdvajati i osigurati minimalan razmak izmeu zrakoplova da se odr~i osnovna sigurnost letenja predmet je istra~ivanja fenomena wake- vortexa u svijetu. 2. POJAM I DEFINIRANJE TURBULENCIJE TRAGA  WAKE-VORTEX Zrakoplovi proizvode vrtloge u tragu u svim fazama leta kao sna~ne vrtloge zraka, u smjeru nizvodno od svakog krila. Vrtlozi mogu trajati nekoliko minuta i mogu uzrokovati zna ajne smetnje ukoliko kroz trag prije propadanja prolazi drugi zrakoplov. Mogunost susreta je mala osim ako zrakoplov nije prisiljen pratiti drugi zrakoplov na sli nom tragu i na sli noj visini. To se redovno dogaa kada zrakoplovi lete prema ili od aerodroma, a destabilizirajua djelovanja mogu biti osobito opasna u blizini tla. Stoga postoje pravila kojima se osigurava dovoljna odvojenost izmeu prateeg zrakoplova dok se spuata ili uzlijee tako da je vjerojatnost zna ajne smetnje od vrtloga prethodnog zrakoplova prihvatljivo niska.. Stvarne brzine zrakoplova su 3 do 4 puta vee u fazi krstarenja leta nego u fazi prilaza zrakoplova ili fazi polijetanja. Stoga vrtlozi tipi no traju na udaljenostima od 20 nm (40 km) kod penjanja, krstarenja ili poniranja u usporedbi s 5-6 nm (10 km) u fazi prilaza ili polijetanja u mirnim uvjetima. 2.1. Svojstva vrtloga u tragu Sva tijela koja proizvode uzgon kada preko njih pree fluid stvaraju cirkulaciju, odnosno veu brzinu preko gornjih dijelova povraine nego preko donje povraine radi stvaranja pozitivnog uzgona. Krila su dizajnirana da budu u inkoviti proizvoa i uzgona i imaju osobito jaku cirkulaciju. Kada doe do brze promjene u kru~noj distribuciji uzgona preko krila, tada se cirkulacija rasipa nizvodno. To se javlja na vrhovima krila, krajevima krilaca (kada se koriste) i na spojevima krila i otvora za gorivo. Ta podru ja cirkulacije obi no se mijeaaju unutar 10 krilnih du~ina nizvodno kako bi oblikovala par tzv. vrtlo~nih pletenica (vortices) koje se okreu u suprotnim smjerovima (slika 1.). To je sna~na i stabilna vrsta toka koja mo~e trajati nekoliko minuta.  Slika 1. Stvaranje vrtloga Izvor: Civil Aviation Authority of New Zeland, Good Aviation Practise-GAP Snaga vrtloga odreena je te~inom, brzinom i oblikom krila zrakoplova koji stvara vrtlog. Svojstva vrtloga bilo kojeg zrakoplova takoer se mogu promijeniti proairenjem krilaca ili drugih naprava za konfiguraciju krila. Meutim, osnovni imbenik je te~ina zrakoplova, pa se stoga snaga vrtloga poveava proporcionalno s operativnom te~inom zrakoplova. Najvea snaga vrtloga javlja se kod teakih zrakoplova iste konfiguracije (stajni trap i mehanizacija krila uvu eni) s malim letnim brzinama. 2.2. Etiologija nastanka aerodinami koga turbulentnog traga iza zrakoplova Nastajanje turbulentnog traga iza zrakoplova u letu vrlo je kompleksno, a na njegovo formiranje u najveoj mjeri utje u: - slobodni aerodinami ki vihori stvoreni cirkulacijom, koji nastaju kod opstrujavanja noseih povraina (krila zrakoplova) kada na njima postoji uzgon - ispuani mlaz zraka iza reaktivnog - turbulentno polje zraka koje se pojavljujc kao rezultat strujanja viskoznog zraka u grani nom sloju na povraini zrakoplova - sekundarna turbulcncija koja nastaje zbog lokalnoga neravnomjernog opstrujavanja elemenata zrakoplova (interfercncije na spojevima pojedinih elemenata zrakoplova - krilo i trup), pa dolazi do odvajanja strujnica (separacije) na pojedinim dijelovima konstrukcije. Prema teoriji potencijalnog strujanja kojom se tuma i i nastajanje induciranog otpora, bez cirkulacije () nema uzgona na aerodinami kom tijelu u realnom fluidu. Uz pretpostavku viskoznog fluida u grani nom sloju koji uspostavlja cirkulaciju fluida, Prandtl je po etkom dvadesetog stoljca ( 1905. godim) dokazao da je uzgon po jedinici razmaha izravno proporcionalan cirkulaciji i da se krilo mo~e zamijeniti hipotetskim sustavom vihora (vortex) oko krila i slohodnim vihorima na krajevima krila (slika 2).  EMBED MSPhotoEd.3  Slika 2 . Sustav vihora Izvor:Aircraft Wake Turbulence Advisory Circular, Oct.1991 Nispono strujanje (downwash) zraka odnosno sni~avanje, koje proizvode slobodni vihori na krajevima krila i vezani vihor uz krilo, omoguuje fleksibilan model, pomou kojeg je matemat ki mogue izra unati bilo koju ~eljenu distribuciju nisponog strujanja ili uzgon na krilu. Kretanjem zrakoplova velika se masa zraka kree opstrujavajui nosee povraine zrakoplova. Kru~no gibanje zraka je stabilno, ato u biti i odreuje trajanje vrtlo~ne struje iza zrakoplova u letu, koja s vremenom nestaje zbog viskoznosti zraka. Takvo je kru~no gibanje nerazdvojno povezano s vrtlo~enjem, pri em se kod svakog vrtloga stvara ravnomjerno kretanje zraka, ije se estice, ne rotirajui, premjeataju oko grani nog sloja vrtloga. Unutar tog podru ja nalazi se jezgra vrtloga, gdje se estice kreu po kru~nici i istovremeno rotiraju oko svoje osi. Zrak koji je proaao kroz grani ni sloj nakon silaska s opstrujavanih povraina zrakoplova stvara donji turbulentni trag. Po svojoj strukturi on podsjea na reaktivnu struju motora, s obratnim predznakom brzine. Zbog mnogo mannje kineti ke energije od reaktivne struje, donja turbulentna struja nestaje u neposrednoj blizini zrakoplova. Turbulencije i vrtlo~enja nastala interferencijom opstrujavanja odnosno na spojevima pojedinih elemenata zrakoplova (primjerice krila i trupa), koja se nastoje pri samoj konstrukciji eliminirati oblikovanjem aerodinami kih slivnika, predstavljaju manju masu zraka i manji intenzitet vrlo~enja, pa nisu presudan imbenik turbulentnog traga iza zrakoplova. Mo~e se zaklju iti da je glavni uzrok nastanka turbulentnoga vrtlo~noga iza zrakoplova formiranje aerodinami kih slobodnih vihora zbog cirkulacije noseih povraina. Slohodni vihori, nastali kao posljedica opstrujavanja krila, uzajamnim djelovanjem oblikuju tzv. vrtlo~ne pletenice. Iza krila se obi no stvara par, a vrtlo~ne pletenice svakog para imaju po veli ini jednaku, a po predznaku (smjeru) suprotnu cirkulaciju. Cirkulacija vrtloga mo~e se odrediti pomou izraza:  EMBED Equation.3 = EMBED Equation.3  gdje je:  EMBED Equation.3 - sila uzgona  EMBED Equation.3  - gustoa zraka  EMBED Equation.3  - brzina leta  EMBED Equation.3  - razmak izmeu osi vrtlo~nih pletenica Prandtl je ustvrdio da je popre na povraina mase zraka okruglog oblika koji je razmjeran razmahu krila  EMBED Equation.3 = EMBED Equation.3  dok je inducirani kut nisponog strujanja u funkciji uzgona  EMBED Equation.3 = EMBED Equation.3  gdje je:  EMBED Equation.3  - koeficijent uzgona  EMBED Equation.3  - povraina krila  EMBED Equation.3  - razmah krila Razmak izmeu osi vrtlo~nih pletenica odreen je rasporedom opstrujavanja uzdu~ razmaha, koje ovisi o planformnom obliku krila (tlocrtni oblik) i napadnom kutu. Kod veine krila koja nemaju prijelom u napadnom rubu, na srednjim napadnim kutovima relativni razmak iznosi 0.8. Relativni razmak osi vrtlo~nih pletenica je omjer stvarnog razmaka i razmaha krila.  EMBED Equation.3 = EMBED Equation.3  Poveavanje opstrujavanja oko sredianjeg dijela krila smanjuje razmak izmeu osi pletenica na 0.72  0.75. To je osobito izra~eno kod strelastih i delta-krila na velikim napadnim kutovima, kod krila s promjenjivom geometrijom na malim i srednjim kutovima zakoaenja, kao i kod izvla enja ureaja za promjenu opstrujavanja krila (zakrilca, spojleri) ako su smjeateni unutar razmaha i ne mijenjaju planformni oblik krila. Kod zrakoplova sa strelastim krilima vee vitkosti, trup je relativno rastereen, pa se obi no stvara joa jedan unutarnji vrtlo~ni par. Vrtlo~ne pletenice unutarnjeg (sekundarnog) para imaju relativno malu cirkulaciju koja je suprotnog smjera u odnosu na osnovni vanjski par. Podru je uz os vrtlo~ne pletenice gdje se brzina poveava od 0 do  EMBED Equation.3 naziva se jezgra pletenice. Polumjer te jezgrc r* iza ravnih krila u rasponu je od 0.01do 0.02 razmaha b i prakti no ne zavisi od napadnoga kuta. Iza zrakoplova sa strelastim krilima na malim napadnim kutovima, relativni polumjer  EMBED Equation.3  iznosi 0.015  0.025, a na veim napadnim kutovima se poveava na 0.04  0.06. Podru je neravnomjernih brzina unutar i blizu jezgre vrtlo~ne pletenice popraeno je promjenom tlaka. Poveanjem brzine w (ako se mjeri od vanjskog podru ja prema jezgri) tlak opada, ato odgovara Bernoulijevom zakonu. Unutar jezgre tlak i nadalje opada, da bi u osi vrtloga dosegao najmanju vrijednost. Relativni polumjer jezgre pletenice je omjer stvarnog polumjera i razmaha krila zrakoplova.  EMBED Equation.3 = EMBED Equation.3  2.3. Parametri turbulentnog aerodinami kog traga Osnovu turhulentnog aerodinami ko traga ine vrtlo~ne pletenice. Njihova struktura je sreena, a osi gotovo paralelne. Promjena parametara uzdu~ traga odreena je procesima dinamike plina. Brzina kru~nog toka oko svake vrllo~ne pletenice djeluje na slobodne pletenice, a brzina sni~avanja traga mo~e se izraziti u ohliku:  EMBED Equation.3  EMBED Equation.3 = EMBED Equation.3  Kada bi brzina bila stalna, sni~avanje traga u vremenskoj dimenziji izgledalo bi kao ravna crta. Medutim, praksa potvrduju da se brzina sni~avanja traga s vremenom smanjuje, a kasnije on lebdi. Lebdenje sc objaanjava slo~enoau traga koje je izazvano sljedeim fizi kim procesima: u polju sni~avanja vrtlo~nog traga (para) stvara se izolirano podru je u kojem je zrak odvojen od okolne sredine zatvorenom crtom toka. To se podru je naziva atmosfera aerodinami kog turbulentnog traga. U sni~avanju traga, u okolnoj se atmosferi poveava tlak, ato doprinosi zgusnutosti traga, a posljedica je poveanje temperature traga koje nadmaauje okolnu temperaturu. Razlika temperature pri istim tlakovima uzrokuje razlike u gustoi ( EMBED Equation.3 - EMBED Equation.3 < 0) te zbog toga nastaje centrifugalna sila koja usporava sni~avanje. Kako su tlakovi unutar traga i u okolnoj atmosferi jednaki, zbog razlike u njihovim gustoama zraka (naruaavanje barotropije) na granici traga nastaje sekundarno vrtlo~enje, koje je usmjereno na smanjenje cirkulacije vrtlo~nih pletenica i brzina u vanjskoj zoni toka. Sni~avanje traga zavisi od po etne brzine EMBED Equation.3 , vremena sni~avanja i zagrijavanja ispuanim plinovima reaktivnog motora. Zbog zagrijavanja poveava se razlika gustoe pri em se usporava sni~avanje traga, a poja ava sekundarno vrtlo~enje koje doprinosi smanjenju cirkulacije vrtlo~nih pletenica. Promjena paramtara atmosfere u odnosu na ISA (International Standard Atmosphere) -uvjete tj. temperaturnoga gradijenta (dT/dH), postojanje slu ajne vertikalne komponente vjetra uzdu~ traga te promjena cirkulacije izazvane kolebanjima napadnoga kuta zrakoplova u odnosu na neku srednju vrijednost, dovode do neravnomjernog sni~avanja traga. `to se trag manje sni~ava, to se dulje odr~ava cirkulacija pletenice. Veli ina sni~avanja traga  EMBED Equation.3  nezavisno od tipa zrakoplova, prakti no ne prelazi 250  300 m. Nadalje, turbulenlni trag se uvijek premjeata u suprotnu stranu od djelovanja sile uzgona. Zbog toga se trag pri manevru zrakoplova nakon izlaska iz vertikalne ravnine ne samo sni~ava nego i premjeata u horizontalnoj ravnini. Kod manevara s preoptereenjem, intenzivnost turbulentnog traga raste razmjerno proporcionalno koeficijentu optereenja. Postupno gaaenje vrtlo~nih pletenica zbog viskoziteta zraka uzrokuje i smanjenje najveih radijalnih brzina, te poveanje uvjetovanih polumjera jezgre vrtloga  EMBED Equation.3 . Na gaaenje turbulentnog traga iza zrakoplova pri manevriranju bitno utje e nestabilnost vrtlo~nog para. Ta se nestabilnost ogleda u poveanju po etnih mlalih deformacija osi vrtlo~nih pletenica, gdje se njihovi pojedini dijelovi meusobno spajaju te se pravilno formirana struja gasi. Po etne deformacije nastaju zbog turbulentne flukluacije brzine unutar pletenica, a u nemirnoj atmosferi joa i zbog turbulentnosti. Razlika u brzinama i poveanje deformacija dovode s vremenom do pojave valovitog oblika vrtlo~nih pletenica s promjenjivom duljinim vala. Pravilna simetri na struktura aerodinami ke struje po inje se naruaavati pri poveanju deformacija osi sve do njihova spajanja, a cijeli proces se zbiva u vremenu t.  EMBED Equation.3 = EMBED Equation.3  gdje je :  EMBED Equation.3  - trajanje simetri ne strukture traga  EMBED Equation.3 / EMBED Equation.3  - vrijeme spajanja Veli ina deformacije vrtlo~nih pletenica je jednaka  EMBED Equation.3 /2. Brzina deformacije vrtlo~nih pletenica razmjerna je cirkulacijii obratno proporcionalan razmaku  EMBED Equation.3 .Na taj je na in vrijeme zbli~avanja jednako odnosu puta i brzine, a proporcionalno je  EMBED Equation.3 /  EMBED Equation.3 . Koficijent proporcionalnosti uglavnom se odreuje veli inom deformacije vrtlo~nih pletenica. Poveanje relativnog polumjera jezgre vrtlo~ne pletenice poveava podru je turbulentnosti i samim time poveava po etne deformacije osi pletenica ato smanjuje relativno vrijeme trajanja simetri ne strukture traga  EMBED Equation.3 . Turbulentnost okoline atmosfere na malim visinama, zbog vjetra, takoer poveava po etne deformacije vrtlo~nih pletenica. Ako su ostali uvjeti isti, to poveanje zavisi od relativnog polumjera  EMBED Equation.3 . Kod zrakoplova s ravnim krilima vee vitkosti vjetar ja e utje e na vrijema naruaavanja simetri ne strukture traga negoli kod zrakoplova s krilima manje vitkosti.Drugim rje ima, kad je turbulentna pletenica "kompaktnija'', vjetar je lakae skree i zanosi.  Slika 3.`irenje vrtlo~nih pletenica iza zrakoplova Izvor: : Civil Aviation Authority of New Zeland, Good Aviation Practise-GAP Vrtlo~ne pletenice (vortices) se aire horizontalno iza zrakoplova, a na udaljenosti do 5 nm iza zrakoplova spuataju se 500  900 stopa. Tendencija spuatanja vrtlo~nih pletenica je otprilike 300  500 stopa u minuti u prvih 30 sekundi (slika 3.) Promjer srediata vrtloga Utjecaj jednog vrtloga na drugi zrakoplov ovisi, izmeu ostalog i o relativnoj veli ini vrtloga u usporedbi s rasponom krila i tetivom tog zrakoplova. Veli ina vrtloga obi no se opisuje polumjerom gdje je tangencijalna vrtlo~nost najvea, te gdje je srediate vrtloga unutar tog polumjera uz brzinu 0 u srediatu. Svojstvo srediata vrtloga ovisi o veli ini odnosnog zrakoplova i o domaaaju spajanja bilo kojeg vrtloga. U uvjetima krstarenja vrtlozi su obi no vrlo kompaktni i laboratorijski eksperimenti ukazuju kako polumjer srediata vrtloga tipi no nije vei od:  EMBED Equation.3  Za zrakoplov B747 koji krstari pri razini leta FL 360 radijus vrtloga otprilike 25 km (13 nm) nizvodno (otprilike 90 sekundi) iznosi oko 3m, ato se mo~e prigodno usporediti s podacima iz Bisgoodovih promatranja na zrakoplovu B747  100 od 2.4 m. Meutim, i Bisgoodova mjerenja i mnoga druga mjerenja na punoj ljestvici pri malim visinama ukazuju kako postoji mala promjena u promjeru srediata s prolaskom vremena. Stoga je bolja procjena polumjera izra~ena u jednad~bi:  EMBED Equation.3  EMBED Equation.3  Ukoliko vrtlog promijeni stanje rasprsnuem, tada e se radijus dramati no poveati. Distribucija tangencijalne brzine Razli iti matemati ki oblici predlo~eni su kako bi se opisala distribucija tangencijalne brzine s radijusom od srediata vrtloga u tragu. Istra~ivanja su pokazala kako je otprilike 40% sveukupne vrtlo~nosti u srediatu vrtloga s velikim Reynoldsovim brojevima koji su tipi ni za velike vrtloge u tragu zrakoplova. Veina formula ima znatno vei udio vrtlo~nosti unutar srediata, a to utje e na okretne momente koje proizvode vrtlozi na drugi zrakoplov. Mjerenje stvarnih vrtloga u tragu vrai se pomou matemati ke formula s 37% vrtlo~nosti unutar srediata. Ta formula koristi se za izra un okretnog momenta zrakoplova koje je uzrokovao vrtlog u tragu. Jednad~ba tangencijalne brzine:  EMBED Equation.3  gdje je:  EMBED Equation.3 = ukupna vrtlo~nost R = promjer maksimalne tangancijalne vrtlo~nosti r = polumjer vrtloga 2.4. Gibanje vrtloga Vrtlozi u tragu kreu se okomito s obzirom na zrakoplov te te~e gibanju prema dolje zbog tendencije gibanja zraka na dolje koje je stvorio drugi vrtlog. Po etni omjeri spuatanja ovise o vrtlo~nosti i odvajanju vrtlo~nih pletenica, te su obi no oko 3 m/s, ato odgovara po etnom kutu spuatanja srediata vrtloga od oko 1 u uvjetima krstarenja. Vrtlog propada eksponentno s protekom vremena i proporcionalno intenzitetu turbulencije. Poveanje tlaka komprimirat e vrtlog dok se spuata, ato e poveati gustou zraka i smanjiti vrtlo~nost. Vrtlozi se mogu pomicati prema gore i prema dolje pomou vjetra, okomitih kretanja zraka, strujanje zraka prema gore ili prema dolje zbog toplinskih u inaka, ili kretanja zraka preko planinskih lanaca. Vrtlozi koji nastaju iznad planina ili preko vrhova kumulus oblaka mogu se uzdii iznad razine leta odnosnog zrakoplova. Podru ja mlaznih struja, planinski valovi, vrhovi sna~nih temperaturnih inverzija i oblaci termalne aktivnosti se povezuju s zna ajnim turbulencijama koje ubrzano rasipaju vrtloge. Stoga postoji mala vjerojatnost da e doi do susreta s vrtlogom u tragu na ovim podru jima, te su vrlo male aanse za susret s vrtlogom u kombinaciji s drugom turbulencijom. Cirkulacija vrtloga je usmjerena prema vani, prema gore i uokolo vrhova krila, kada se promatra ispred ili iza zrakoplova. Ispitivanja kod velikih zrakoplova pokazala su kako vrtlozi ostaju udaljeni, ali razmaknuti za manje od iznosa raspona krila, kreui se s vjetrom, na visinama veim od raspona krila od tla. U svezi s tim, ako doe do susreta s upornom turbulencijom vrtloga, potrebna je mala promjena visine i lateralnog polo~aja (preferira se uz vjetar) ato e pru~iti putanju leta bez turbulencija. Ispitivanja letenja su ukazala kako vrtlozi veeg zrakoplova (transportna kategorija) tonu pri omjeru od nekoliko stopa u minuti, a s vremenom i zbog poveanja udaljenosti od prethodnog zrakoplova usporavaju svoje spuatanje, te im takoer nestaje snaga. Piloti moraju letjeti na putanji ili iznad putanje prethodnog zrakoplova, te moraju mijenjati kurs ako je to potrebno da bi izbjegli podru ja iza i ispod odnosnog zrakoplova. Meutim, okomito odvajanje od 1000 stopa smatra se dovoljno sigurnim. 2.5. Propadanje vrtloga Kada opisujemo propadanje vrtloga vrlo je va~na promjena svojstava diskretnih velikih vrtloga. Mali vrtlozi su tehni ki ipak joa uvijek vrtlozi, ali e imati zanemarivi u inak na drugi zrakoplov. Vrtlozi imaju energiju i snagu koja postepeno mo~e propadati na na in da se propadanjem stvaraju manji vrtlozi koji se spajaju s vrtlozima koji se okreu u suprotnom smjeru. Vrtlozi koje stvara jedan zrakoplov predstavljaju dva nominalno jednaka vrtloga koji se okreu u suprotnom smjeru s sveukupnim kutnim momentom u iznosu 0. Stvaranje manjih vrtloga i ukidanje putem spajanja smanjiti e vrtlo~nost diskretnih velikih vrtloga. Propadanje vrtloga predmet je velikog broja ispitivanja i mo~e se javiti na razli ite na ine, u razli itim omjerima, u ovisnosti od nekoliko razli itih parametara. Istra~ivanja su pokazala kako postoje zna ajne razlike izmeu propadanja u laboratorijskim uvjetima i u samom letu, osobito u blizini tla, te se veina ispitivanja usmjerila baa prema ovom pitanju. Posljedice ulijetanja zrakoplova u vrtlo~ni trag u blizini zra nih luka pokazuju kako je propadanje vrtloga osjetljivo na vrlo niske razine atmosferske turbulencije, te kako na propadanje utje e atmosferska termalna stabilnost (temperaturni gradijent). Tek nedavno, s napredovanjem metoda izra unavanja dinamike fluida uz primjenu neposredne numeri ke simulacije, teorijska prou avanja su dokazala zna ajan utjecaj tih parametara na vrtloge pune ljestvice. Na veim visinama mogu postojati podru ja s vrlo niskim razinama turbulencije i u takvim uvjetima vjerojatno je da e propadanje biti donekle sli no propadanju koje je promatrao Reynolds u visokim broj anim laboratorijskim testovima. (Reynolds broj predstavlja odnos brzine fluida i kinematske viskoznosti fluida ato zna i da se prijelaz iz laminarnog strujanja u turbulentno zbiva u ovisnosti od Re broja.) Zna ajno ispitivanje stalnosti vrtloga u tragu pri visinama krstarenja primijetio je Bisgood, koji je prou avao 37 kondenzacijskih tragova. Te tragove stvorila je vodena para u ispuanoj cijevi motora mla~njaka, koja je zatim unesena u vrtloge traga. Mogu se primijetiti jedino sa zemlje u jasnim uvjetima kada je turbulencija na visini krstarenja vrlo niska. Ukoliko postoji zna ajna turbulencija na visini krstarenja te~i e rasipanju vodene pare. Stoga je Bisgoodovo ispitivanje bilo povezano s uvjetima koji uzrokuju stalnost (izdr~ljivost) vrtloga. Najvea starost vidljivih vrtloga iznosila je oko 120 sekundi i do isteka tog vremena obi no su postojale vidljive promjene u obliku traga. Crow om e Parovi vrtloga u tragu mijenjaju se s prolaskom vremena na tri razli ita na ina. U nedostatku zna ajne turbulencije mogu razviti sinusni uzorak vala izmeu dva vrtloga ato na koncu dovodi do spajanja dva vrtloga u intervalima koji formiraju om e u obliku mnogokuta koje su poznate kao Crow om e (slika 4). Ime su dobile prema znanstveniku koji je objasnio njihovu znanstvenu osnovu. Razvijanje sinusne valovitosti je uobi ajeno kada vrtlog traje du~e od 30 sekundi, a vrtlozi se mogu spojiti kako bi oblikovali seriju zatvorenih om i izmeu 50 i 100 sekundi nakon ato su nastali. Meutim, vrtlozi su i dalje sna~ni kada se formiraju om e, ato je suprotno dojmu kako formiranje om i predstavlja kraj vrtloga kao potencijalne opasnosti. Ustrajanost snage vrlo~nih om i prikazuje se kao nedostatak rasipanja u kondenziranim tragovima, a jednan od primjera je zrakoplov TriStar (L1011) koji je osjetio seriju sna~nih udara ubrzanja dok je prolazio kroz niz om i. Om e propadaju prema jednom od dvije vrste starenja : rasprsnuem ili rasipanjem. Rasprsnue se smatra kao iznenadno poveanje radijusa vrtloga nakon ega slijedi brzo rasipanje. Rasipanje nastaje kao posljedica turbulencije, koja je prisutna u atmosferi ili unutar vrtlo~nog traga zrakoplova.  EMBED MSPhotoEd.3  Slika 4.: Formiranje Crow om i Izvor:  HYPERLINK "http://www.cerfas.fr/cfd/wake_vortex_26.php" www.cerfas.fr/cfd/wake_vortex_26.php Rasprsnue vrtlo~nog traga Rasprsnue predstavlja vidljiv dokaz propadanja snage vrtloga kao brzo formiranje odreenih oblika kojima se promjer poveava za 3-4 puta. Meutim, mogu postojati kratki bljeskovi naizgled monih vrtloga izmeu podru ja rasprsnua, te su poznati primjeri da se jedan vrtlog iz para rasipa kroz rasprsnue dok drugi ostaje netaknut. Uzrok rasprsnua nije poznat, iako na tu pojavu mogu utjecati varijacije brzine uzdu~ srediata vrtloga i u inci turbulencije na te brzine. Iako rasprsnue mo~e zna ajno skratiti snagu vrtloga u tragu, ini se kako danas joa nije mogue predvidjeti pojavu tog fenomena. Stoga sve procjene potencijalne opasnosti zanemaruju tu osobinu i koncentriraju se na najgori slu aj postepenog raspadanja. Postepeno rasipanje vrtloga Rije  je o sporijem procesu propadanja koji je vidljiv kao postepeni gubitak koherentne vrtlo~nosti u glavnom vrtlogu na manje nezna ajne vrtloge, te postoje indikacije da tu pojavu prati difuzija vrtlo~nosti prema vani od srediata vrtloga (slika 5.). Izvjeaa pojedinih istra~ivanja ukazuju na linearni odnos izmeu omjera propadanja i atmosferske turbulencije, ato openito podupiru razli iti dokazi o tome da ovaj proces ubrzava prisustvo ak i sasvim lagane atmosferske turbulencije. Meutim, nema dovoljno dokaza kojim mo~emo potvrditi kvantitativni odnos izmeu turbulencije i omjera propadanja vrtloga, te se za sada koristi linearni odnos. Zna ajne posljedice efekta turbulencija kao ato su podru ja mlaznih struja, planinski valovi, vrhovi sna~nih temperaturnih inverzija i oblaci termalne aktivnosti ubrzano rasipaju vrtloge. Stoga postoji mala vjerojatnost da e doi do susreta s vrtlogom u tragu na ovim podru jima, te su vrlo male aanse za susret s vrtlogom u kombinaciji s drugom turbulencijom.  EMBED MSPhotoEd.3  Slika 5. Raspadanje vrtlo~nih pletenica Izvor:  HYPERLINK "http://www.cerfas.fr/cfd/wake_vortex_26.php" www.cerfas.fr/cfd/wake_vortex_26.php 2.6. Posljedice djelovanja na zrakoplov u turbulentnom tragu prethodnika Ulijetanjem zrakoplova u turbulentni trag predhodnika mijenja se i slika njegova opstrujavanja. Mijenjaju se napadni kutovi nekih dijelova zrakoplova i kutovi klizanja. Sve to dovodi do preraspodjele tlakova na povrini zrakoplova i stvara iznenadne neuravnote~ene aerodinami ke sile i momente koji izazivaju njegovo nepravilno kretanje. Destabilizirajue djelovanje odreeno je intenzitetom i smjerom vrtlo~nih struja, vremenom njihovog djelovanja na zrakoplov i djelovanjem pilota putem upravlja ke palice. Intenzitet turbulentnog traga najvei je kod  teakih zrakoplova, kao ato su transkontinentalni zrakoplovi Boeing 747, Boeing 757, ili Airbus 320. Opasnost ulijetanja u turbulentne tragove tih zrakoplova dovodi do neo ekivanih i nekontroliranih evolucija zateenog zrakoplova, a ponajviae zbog toga jer je takva opasnost nepredvidljiva. Kod presijecanja tragova pod veim kutovima destabilizirajue djelovanje se o ituje u obliku nenadanog udarnog optereenja (gust load), koje prakti ki ne mijenja osnovne parametre leta. Najvei je rizik opasnosti kada zrakoplov presijeca trag pod malim kutovima , manjim od  EMBED Equation.3 . To se naj eae deaava u fazi slijetanja i polijetanja te pri letovima vojnih formacija u zbivenom poretku. Osim toga zrakoplov mo~e uletjeti u vlastitu vrtlo~nu struju pri izvoenju oatrih zaokreta. Inducirani okretni moment Rijetko se dogaa, ali susret s tragom mo~e uzrokovati strukturalna oateenja katastrofalnih razmjera. Meutim, uobi ajena opasnost povezuje se s induciranim okretnim momentom koji mo~e biti vei od sposobnosti zrakoplova (koji se susreo s vrtlogom) u kontroli okretanja (slika 6). Prilikom eksperimenata letenja, zrakoplovi su po etno namjerno usmjereni neposredno prema srediatu vrtloga velikog zrakoplova. Dokazano je kako sposobnost jednog zrakoplova da se suprotstavi induciranom okretnom momentu koje namee vrtlog u tragu primarno zavisi o rasponu krila i sposobnosti odgovora kontrolnog okreta zrakoplova koji dolazi u susret. Jezgra vrtloga mo~e prouzro iti veli inu okreta od 80 stupnjeva u sekundi, a kod lakaih zrakoplova ak znatna strukturalna oateenja.  EMBED MSPhotoEd.3  Slika 6. Inducirani okretni moment Izvor:  HYPERLINK "http://www.asy.faa.gov/safety_products/WakeAC.html" www.asy.faa.gov/safety_products/WakeAC.html Kontrola suprotstavljanja obi no je u inkovita, a nametnuto okretanje minimalno u slu ajevima gdje je raspon krila nadolazeeg zrakoplova proairen iza polja rotacijskog toka vrtloga. Te~e je zrakoplovu s kraim rasponom krila ( u odnosu na prethodni zrakoplov), on se ote~ano mo~e suprotstaviti nametnutom okretanju koje je uzrokovao vrtlo~ni tok. Piloti zrakoplova kratkog raspona krila, ak i kada je rije  o zrakoplovima izuzetno visokih performansi, moraju osobito paziti na susrete s vrtlozima. Trag veeg zrakoplova zahtijeva pozornost svih pilota. Ako drugi zrakoplov susretne vrtlog prije propadanja, tada on mo~e osjetiti velike smetnje okretanja ukoliko slijedi sli an put ili pak veliki puls ubrzanja pod pravim kutom i mjeaavinu tih u inaka na drugim putanjama. Veli ina smetnji ovisi o vrtlo~nosti traga, promjeru srediata vrtloga, lateralnoj odvojenosti izmeu vrtloga, propadanju vrtloga, te o snazi kontrolnog okreta odnosnog zrakoplova koji se susreo s vrtlogom. Vjerojatnost susretanja vrtloga zavisi i o relativnom polo~aju zrakoplova koji slijedi prema zrakoplovu koji je stvorio vrtlog, te o starosti vrtloga. Maksimalan omjer induciranog okretnog momenta uzrokvan turbulentnim vrtlogom prema dostupnom momentu kontrole okreta zove se Roll Ratio. Kontrolni okretni moment vrlo je sli an za sve civilne zrakoplove u fazi krstarenja. Brzine krstarenja biti e sli ne za veinu zrakoplova uz izuzetak nekoliko turbo-prop zrakoplova na velikim visinama koji su sporiji od mla~njaka. Funkcija omjera raspona krila i polumjera vrtloga i oatrice krila ne razlikuje se za veinu civilnih zrakoplova. Stoga RollRatio najviae varira kod bez-dimenzijske vrtlo~nosti n , te uz funkciju omjera raspona krila dva zrakoplova. Ne-dimenzijska vrtlo~nost je funkcija uzgona zrakoplova i te~ine, indicirane brzine i raspona krila zrakoplova koji je uzrokovao vrtlog, i to je glavni parametar u odreivanju sigurne udaljenosti izmeu zrakoplova. Veli ina vrtlo~nosti koja je relevantna u odnosu na veli inu smetnji koju predstavlja bez- dimenzionalnu vrtlo~nost:  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 = EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  Bez-dimenzijska vrtlo~nost u fazi krstarenja mo~e iznositi oko 50% vrtlo~nosti u fazi slijetanja zbog kombiniranih utjecaja vee te~ine i vee indikativne zra ne brzine u uvjetima krstarenja. Openito civilni zrakoplovi naj eae djeluju oko uvjeta minimalnog otpora. To se javlja pri skoro stalnoj vrijednosti indicirane brzine za odreeno optereenje i konfiguraciju zrakoplova, i neovisno je od visine. Indicirana brzina daje najbolje performanse penjanja i ekonomi nog krstarenja. Pri minimalnom otporu bez -dimenzijska vrtlo~nost odreene konfiguracije, kao ato je ista konfiguracija koja se koristi na putanjama, biti e neovisna od visine.Kod slijetanja i kod uzlijetanja naglasak je na uzgonu pri najni~oj prakti noj brzini. Zrakoplovi koriste krilca kod polijetanja kako bi poveali uzgon, a kod slijetanja kako bi poveali otpor. Indicirana brzina za minimalni otpor je ni~a zbog potrebe za smanjenjem brzine pri slijetanju i uzlijetanju. Stoga e bez-dimenzijska vrtlo~nost za vrijeme slijetanja i uzlijetanja biti znatno vea nego u istoj konfiguraciji za druge faze leta. Meutim, obi no je neophodno poveati maksimalni kontrolni okretni moment kako bi osigurali primjerenu sposobnost manevriranja pri manjim brzinama za vrijeme slijetanja i uzlijetanja, dok je kontrola okreta u krstarenju esto ograni ena zbog strukturalnih ograni enja zrakoplova. Destabilizirajuem djelovanju vrtlo~nog traga vrlo dobro parira autopilot. Autopilot u re~imu stabilizacije nekog kutnog polo~aja zrakoplova djelotvorno odr~ava stabilnost kod ulijetanja u vrtlog. To se objaanjava injenicom da je prolazak kroz trag s uklju enim autopilotom smanjeno vrijeme djelovanja destabilizirajuih momenata. Vea stabilnost leta je osigurana kada je manja inertnost autopilota i kada autopilot upravlja sa ato veim dijapazonom kutova. 2.7. Primjena minimalnog razdvajanja pri pojavi turbulentnog vrtlo~nog traga Minimalno razdvajanje pri pojavi vrtlo~nog traga primjenjuje se da bi se smanjio mogui rizik nesree. Ovo razdvajanje bit e primjereno kada kontrolor leta procijeni da postoji opasnost od vrtlo~nog traga. Ako zrakoplov uleti u vrtlog traga koji stvara vodei zrakoplov mo~e doi do rotacije zrakoplova, gubljenja visine, oateenja zrakoplova i u najgorem slu aju do pada zrakoplova. Minimalno razdvajanje u ovom slu aju definirano je odlukama meunarodne organizacije za civilni zra ni promet- International Civil Aviation Organisation (ICAO). Kako su zra ni vrtlozi nevidljivi, njihova pojava i lokacija ne mogu biti precizno odreene. U skladu s tim kontrolor letenja , jednako kao i pilot mora prepoznati situaciju kada postoji opasnost od ulaska u turbulenciju traga . Najveu opasnost za zrakoplov koji ulazi u vrtlog predstavlja ograni eno vrijeme reagiranja, a razlog tomu su velike brzine leta. Kategorizacija zrakoplova Standardi za razdvajanje zrakoplova u uvjetima turbulentnog traga zrakoplova bazirani su na tri kategorije maksimalne te~ine zrakoplova MTOW: 1. Teaki - HEAVY (H) - svi tipovi zrakoplova teaki 136 000 kg ili viae 2. Srednji - MEDIUM (M) - zrakoplovi lakai od 136 000 kg, a te~i od 7000 kg Laki - LIGHT (L) - zrakoplovi teaki 7000kg ili manje Na ini razdvajanja: Kod razdvajanja zrakoplova koristi se radarsko ili neradarsko razdvajanje. Radarsko razdvajanje Radarsko razdvajanje primjenjuje se kada zrakoplov leti izravno iza drugog zrakoplova na istoj visini ili manje od 300 metara ispod njega ili kada oba zrakoplova koriste istu stazu za slijetanje ili paralelne staze na razmaku manjem od 760 metara (tablica 1.). Tablica 1. Radarsko razdvajanje  Izvor: Civil Aviation Authority of New Zeland, Good Aviation Practise-GAP Neradarsko razdvajanje Neradarsko razdvajanje temeljeno je na vremenu i primjenjuje se za srednje i lake zrakoplove koji slijeu iza teakog zrakoplova (tablica 2.). Razmaci su sljedei: 1. MEDIUM- 2 minute 2. LIGHT- 3 minute Ovo razdvajanje primjenjuje se kada zrakoplovi koriste istu stazu za slijetanje i kada koriste odvojene staze, s razmakom manjim od 760 metara. Tablica 2. Ne- radarsko razdvajanje  Izvor: Civil Aviation Authority of New Zeland, Good Aviation Practise-GAP 3.SEGMENTACIJA KRITI NIH FAZA LETNE OPERACIJE 3.1 Utjecaj turbulentnog aerodinami kog vrtloga prethodnog zrakoplova na sigurnost leta u fazi prilaza i slijetanja Fizi ke zna ajke vrtlo~nog traga iza svakog zrakoplova stvaraju zna ajne nagibne momente i vibracije te dovode do veih optereenja zrakoplova. Iako su te zna ajke istra~ene i objaanjene, joa uvijek se dogaaju ulijetanja u aerodinami ki trag prethodnog zrakoplova. Mala rezerva visine i brzine leta te veliki napadni kutovi koji smanjuju djelotvornost komandi leta, dodatno optereuju izrazitim deficitom vremena za ispravak prora una za slijetanje. U tim uvjetima mijenja se profil sni~avanja, konfiguracija zrakoplova, re~im rada motora, poveava se intenzitet radio komunikacije i po pravilu se poveava psihofizi ko naprezanje posade. Najjednostavniji na in prevencije ulijetanja u trag prethodnika je poveanje intervala slijetanja, ato je u nekim zra nim lukama u suprotnosti sa zahtijevom slijetanja velikog broja zrakoplova u zadanom vremenu. Za procjenu vjerojatnosti ulijetanja u vrtlo~ni trag potrebno je dobro poznavanje zna ajki vrtloga odnosno najva~nijih elemenata trajanja i brzine sni~avanja traga. Najintenzivniji i najdugotrajniji turbulentni trag generira se iza zrakoplova Boeinga 757. Iza manjih i lakaih zrakoplova trag traje 30  60 s, a iza teakih zrakoplova 2  5 minuta. Vertikalna brzina sni~avanja traga mo~e biti u granicama 1  5 m/s, zavisno od tipa zrakoplova, njegove brzine u prila~enju i slijetanju ili preoptereenja. Treba poznavati i podru je moguih trajektorija u tijeku prila~enja na slijetanje, koja se konstruiraju za svaku zra nu luku na osnovu usvojenih normi, po visini i usmjerenju u nekoliko presjeka. Za uvjete sigurnog slijetanja potrebno je odrediti vjerojatnost ulijetanja u turbulentni trag prethodnika. 3.1.1. Vjerojatnost ulijetanja u turbulentni trag prethodnog zrakoplova Trajanje vrtlo~nog traga u zadanom podru ju trajektorija slu ajna je veli ina, koja se mo~e izra unati po formuli: t = EMBED Equation.3  gdje je: H EMBED Equation.3  - stvarna visina leta zrakoplova u zadanom kontrolnom presjeku H EMBED Equation.3  - visina donje granice promatranog presijeka u odnosu na zemlju u zadanom kontrolnom presijeku  EMBED Equation.3 - vertkalna brzina sila~enja traga U izrazu je veli ina H slu ajna veli ina, a brzina  EMBED Equation.3  je zadana za odreeni tip zrakoplova i konfiguraciju u slijetanju. Pri stro~em kriteriju to nosti , treba uzeti u obzir da se brzina  EMBED Equation.3  s vremenom smanjuje, pa se u tom slu aju koristi srednja vrijednost w EMBED Equation.3 . Ako su poznate vjerojatne karakteristi ne visine kontrolnog presijeka, mo~e se odrediti vrijeme zadr~avanja aerodinami kog vrtloga u zadanom kontrolnom presjeku. Matemati ko o ekivanje tog vremena izra~eno je formulom:  EMBED Equation.3  m EMBED Equation.3 =  EMBED Equation.3  gdje je: m EMBED Equation.3  - matemati ko o ekivanje vremena zadr~avanja vrtloga u zadanom kontrolnom presijeku m EMBED Equation.3  - matemati ko o akivanje trajektorije slijetanja u zadanom kontrolnom presjeku w EMBED Equation.3  - srednja vertikalna brzina sila~enja vrtlo~nog traga Zrakoplov se promatra kao to ka ( te~iate c.g.), pa je trajektorij zrakoplova putanja te~iata. Pogreake (razmimoila~enje) trajektorija zavranog prilaza pod injavaju se normalnom zakonu pogreaaka  asimetrija i eksces nisu zanemarivi. Pilot leti u zavranom prila~enju ne procjenjujui polo~aj traga prethodnika, ato je i pravilno. Vea su odstupanja po ravnini poniranja (glide path) jer se u prilazu mijenja konfiguracija zrakoplova ( izvla enje stajnog trapa, zakrilaca, itd.) i napadni kut, dok su pogreake po kursu manje. Na veim udaljenostima, dimenzije zrakoplova su manje u odnosu na dimenzije elipse o ekivanog razmimoila~enja, a njena poluos je na udaljenosti od 4 km vea od 120 m. Dimenzije elipse na udaljenosti manjoj od 1 km gotovo su identi ne dimenzijama zrakoplova koji ulijee u turbulentni trag. Kod zanoaenja traga uzima se u obzir utjecaj bo nog vjetra. Ako je poznata brzina vjetra v EMBED Equation.3  mo~e se odrediti i veli ina premjeataja vrtloga od osi o  z. L EMBED Equation.3  EMBED Equation.3 = V EMBED Equation.3 - t EMBED Equation.3  gdje je : L EMBED Equation.3  EMBED Equation.3  - linearni zanos vrtloga uzdu~ osi V EMBED Equation.3  - brzina vjetra t EMBED Equation.3  - vremenski interval slijetanja Prora un vrijednosti uz dane pretpostavke vrijedi i za vizualni VFR (Visual Flight Rules Rules) i za instrumentalni IFR (Instrumental Flight Rules) prilaz. Zrakoplovi opremljeni sofisticiranom opremom ILS (Instrumental Landing System) sustava za autoslijetanje, zrakoplovi novog tzv. fly-by-wire narataja s digitalnom tehnikom upravljanja i GPS (Global Positioning System)  navigacijskim sustavom, slijeu po to no odreenom trajektoriju, pa je podru je razmimoila~enja trajektorija i na veim udaljenostima vrlo malo, odnosno odgovara dimenzijama zrakoplova. Takvo podudaranje trajektorija smanjuje vjerojatnost ulijetanja u vrtlog prethodnog zrakoplova i doprinosi sigurnosti letenja jer nisponim premjeatanjem vrtlog izlazi iz podru ja elipse. 3.1.2. Postupci izbjegavanja turbulentnog traga u fazi prilaza i slijetanja Susret s vrtlo~nom tragom nije uvijek opasan, meutim mogu nastati udari razli itih ja ina, ato zavisi o smjeru susreta, te~ini prethodnog zrakoplova, veli ini zrakoplova koji slijedi, udaljenosti od prethodnog zrakoplova i o to ki susreta s vrtlogom. Vjerojatnost induciranog okreta poveava se kada je usmjerenje zrakoplova koji slijedi openito usklaeno ili paralelno s putanjom leta prethodnog zrakoplova. Preporu a se izbjegavanje podru ja ispod i iza prethodnog zrakoplova, osobito pri malim visinama gdje ak i trenuta ni susret s vrtlogom mo~e biti opasan. Piloti moraju biti osobito oprezni u mirnim uvjetima i situacijama gdje vrtlozi mogu: - ostati na podru ju dodirne to ke - udaljiti se od zrakoplova koji djeluje na susjednoj pisti - potonuti na putanju uzlijetanja ili slijetanja s popre ne piste - potonuti u putanje prometa drugih aerodromskih radnji. Piloti svih zrakoplova moraju vizualizirati lokaciju traga vrtloga iza velikog zrakoplova i uporabiti postupke izbjegavanja vrtloga. Jednako je va~no da pilr  & tTr*^|^ ` d @!"" 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650/;;`ғ̐ȑ̍ȘӏǛ'_\DD¿Ғʍȗҟڏǔ˞ҚΑÔțЅƕόĄș͒ȑʌȔК֐̉Ŏʓϑ͑͑͑͑͑͑͑͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͐͑Α͑ˑˑ͎΋ЊΑʡ“̓̒̒̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̏̑όʍ͊ȕՐ}ƏɔӂŽ͟ӯ*HHMnsܜјΝΔȘԗ׋͍͊ƎʌϏ҉ƊǑӑ֒׈ˑԖ׉ɓДГʜОήܥГ֚ӕїԒҎ̗֓ДБʔ͚ӌȔϒĿ~þ͉ؒŕϚև’щɊ̐ГӍʔБ˔ΎɔϘӎɍȒ͏ʑ̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑ΒϒΒ̒̒ΏόыϒɢÓ̓̒̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̏̌ȍːΔҍˈĔ̕˗̎Ş܏ϋțϩ=?Klq۝ҖʜѕɓЉɋ͊Ȗҗ҂đјՓЏьϋΐӌ͆Ǎ͑А͔ΙГɒęʚșʔˑˑ΍ʉɎ̐ІÌɕϔЏɛ،Ʉċˈʗח|՞يɑ΍͌̇ʊʑӉǖՓΖэȑ̔ϏʕКՐˑ̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̑̒ΒΓ̓ʓ̑ώьϓʢÔ̔̔̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̔̔̓̑̓̓̕ʗʔɒʏ̜̏̒Υ!GHFnpט̗̔͒ʒώʑΓ̔Δ̓ἈΓΑΑΑΑΑΑΑΓΓΓ̔̔̔̓̓̕̕̕̕Γ̑Α̑Γ̓Γ̓Γ̓Γ̑Α̑Α̑Γ̓Γ̓Γ̑Α̑Α̑ἈΓ̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓ΒΓ̕ʕʖ̓Αяϔ˥Ŕ̘̘̔̔̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̔̔̓̔̕̕̕̕ʒʏʐ˔˜Τ HJDpqזʕ̒ˏʑ͏ʓ̔̔̓̑̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̔̓̕̕̕Ε̖ʖȘ̖Βёϖ˥ŕ̗̗̗̘̘̔̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̔̔̕̕̕̕ʔˏʐ˔˜Τ!IKDpqؘʗ̓̏ʑ͐˓̔̔̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̓̔̔̕̕̕ϖ͗˗ə͗ϓғЗ̦Ɩ͖͖͕͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͕͖͕͖͙͙͖͘͘͘ʔːˑ̝̕ϥ!IKEqrؙ˘͔̐˓ΐ˕͖͖͕͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͔͕͔͖͗͘͘͘˗ə˗ϓғЗ̧Ɩ͖͖͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͖͖͕͖͙͙͗͘͘͘˕̒˓̝̕ϥ#ILGprٚ˘͕͓̖ϓ͖͖͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕͕̕͘͘͘Η̘ʘʚ̘ДєЙ˨ŗΗΗΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΗΚΙΗΖΗΙΜΜΘ͖̓̔̕͟Ч#ILHqsٛ̚Ζ͓̖ϔ̗ΙΚΚΚΙΗΖΖΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΖΗΗΗΗ̘ʘʚ̘ЕєЙ˨ř̙̙̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̙̙̗̙̙̜̜̘̚̚ʘ˔ʕ˘˟Χ&IKIqq؜ʜ̙̔ʗ͔ʗ̙̙̗̗̗̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̗̙̙̘͙͛̚̚̚˛˜͙ϖҖќ̫ƙ̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̜̙̗̙̝̝̚̚̚̚ʘ˕˕˘ˠΨ&IKJpq؞ʝ͙̖˙Ζ˙̜̜̜̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙̙͙̚̚˛ʛʜ˛ϘҖќ̫ƛ̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̝̝̞̞̚̚əʔɔɚ˞˵CCKrqנʞ̚˖ʙ͖ʛ̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̝̞̞̞̝͝ϝ̞̟͞͝ϚӚџέƛ̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̞̝̝̞̟̝̚˛̕ʕʚ˞˵ABMqqؠʟ̝̘ʛ͘ʝ̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̛̞̟̟̟̞̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̝̞͝͞ʟ̞ϚӚџήƛʛʛʝʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʞʞʝʛʚʛʝʟʟʠ̝̘ʗɝʠɶ@?Oqpףɢ˟˚ɝ̚ɝʞʟʟʟʞʝʝʝʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝʞʝ̝̞͝ʟʡ̞͚њϠ̮Ĝ˜˞˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˟˞˜˛˜˞ˠˠˢΟΙ˘ʞˠɶ 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