# INDICIAL RESPONSE OF THIN WINGS IN A COMPRESSIBLE SUBSONIC FLOW

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4 he same isa (=0), he railig vorex of he mos dosream ro of paels are emied i he vorex shee ad are dragged by he flo. I he ex sep of ime (=), occurs he same process: a e doubles superficial disribuio is geeraed; his disribuio is replaced by a disribuio of vorex i he boudaries of he paels, i he isa =, ad he free vorex are emied io he vorex shee. The process coiues i he subseque isas uil he seady flo is reached. The discreizaio of he ime (d) used is such ha he disace raveled by he free vorex, hich are dragged ih he freesream velociy (U), durig a ime sep is equal o he legh of oe pael i he chord direcio. For a discreizaio of he chord i m paels ad a chord legh c, i implies ha d = c / (U m). This relaio beee ime sep ad chord discreizaio is suiable o sho ell he ime evoluio of he lif over he ig. Figure 4, preses he process for a sep ig, i a op vie. Figure 4. Numerical process: appearig of he vorex ad emissios i he vorex shee i hree differe isas. The iesiy of he poeial jumps i each pael ad isa of ime is compued hrough he applicaio of he boudary codiio (Eq. ), a he corol pois. This applicaio is doe i he half of he ime sep. So, if ij k is he velociy iduced i he corol poi i, i he isa d/, by a uiary poeial jump creaed i pael j, i he isa k, i follos, from he Eq., ha: a a a x φ φ φ = U U U j= k = 0 j= k= 0 j= k= 0 The oly ukos of he Eq. 4 are he compoes of he vecor of he poeial jumps. pplyig Eq. 4 i each isa of ime, i is possible o compue he variaio of he poeial jumps ih ime. The marix ha muliplies he vecor of he poeial jumps is called he ifluece coefficies marix. I eeds o be compued jus oce. The vecor of he righ side of he Eq. 4 eeds o be compued i each sep of ime, because he iduced velociies vary ih he ime. Iside he ifluece coefficies marix, here are he erms a ii, hich are he impulsive velociies i each pael, creaed ih he impulsive geeraio of poeial. These erms appears oly a he pricipal diagoal of he marix, because is value i a give pael depeds oly o he iesiy of he poeial jump over he pael iself. They ca be compued, usig he liear piso heory (isplighoff, shley ad Halfma, 955) ad applyig he boudary codiio i he isa =0. I his isa, all he erms ij k are ull (i a compressible flo, he perurbaio creaed i he flo by he appearig of a poeial jump is o isaaeous). Thus, a isa =0, Eq. 4 reduces o: a φ U (5) ii i = k j k j k j (4)

5 roceedigs of COEM 005 Copyrigh 005 by CM 8h Ieraioal Cogress of Mechaical Egieerig November 6-, 005, Ouro reo, MG The liear piso heory saes ha, i he iiial isa, he pressure differeial over he ig is cosa ad equal o: U 4 = - ρ a C = ; C = (6) M ρ U I he above equaio, a is he free sream soud velociy ad ρ, he free sream desiy. From Eq. 3 ad 6, as φ x is ull, because he poeial disribuio is cosa over he ig i he iiial isa: U U = - ρ a φ = - d (7) M Thus, from Eq. 5 ad 7: a ii = (8) a d s sho i he Eq. 7, he values of he poeial jumps i he isa =0 are give direcly from he piso heory. Thus, he Eq. 4 is applied oly from he isa =. The erms ij k, of he iduced velociies by he vorex disribuios i each isa of ime are compued hrough he expressio for he iduced velociy by a vorex filame i a compressible flo. I he icompressible flo, here he perurbaio creaed by he vorex is fel isaaeously i all he pois of he flo, he expressio for he iduced velociy by a sraigh vorex filame, of iesiy Γ ad exremiies, (X,Y ), ad, (X,Y ), i a poi, (X,Y ), of he flo is give direcly by he iegraio of he io-savar La alog he filame (see Fig. 5). Tha is: W Γ = 4π ( x x ) ( x ( y x ) ( y ( x ( y x ) ( y (9) Figure 5. Illusraio of he io-savar La (lef side) ad vorex pael ih exremiies,, C, D. Figure 6. Ifluece regio, a isa, of a perurbaio (X V, Y V ) appearig i he flo a he isa 0.

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