Dymoe Use s Manual Two- and thee dimensional dynamic inflow models Contents 1 Two-dimensional finite-state genealized dynamic wake theoy 1 Thee-dimensional finite-state genealized dynamic wake theoy 1 Two-dimensional finite-state genealized dynamic wake theoy The unsteady aeodynamic foces acting on a two-dimensional aifoil wee computed by Theodosen, based on the potential flow appoximation and the Kutta condition 1,, 3. A finite state fomulation of the same poblem was developed by Petes et al. 4, and takes the following fom A µ+ V b µ = U 3 c, (1) whee µ is the inflow states aay, A and c ae a given constant matix and aay defined in eqs. (1) and (9), espectively, V is the magnitude of the components of the fa flow velocity, V, in the plane of the aifoil, i.e. V = (I ā 1 ā T 1 )V, b the semi-chod length, and U 3 the component of along unit vecto ā 3 of the elative velocity vecto at the aifoil thee-quate-chodpoint. The magnitude of the aveageinflow vecto, λ, is a linea combination of the inflow states λ = 1 bt µ, () whee b is a constant aay given by eq. (11). This aveage inflow acts along a unit vecto, ā λ, which is the coss poduct of the fa flow velocity V by unit vecto ā 1 Thus the aveage inflow vecto is ā λ = Ṽ ā 1 Ṽ ā 1, (3) λ = λ ā λ. (4) The govening equations of the poblem, eqs. (1), fom a set of coupled odinay diffeential equations; a cental diffeence scheme is used fo thei solution A µ f µ i t + V b µ f +µ i = U 3f U 3i t c, (5) whee the subscipts ( ) i and ( ) f denote quantities computed at the beginning and end of the time step of size t. The inflow states at the end of the time step ae then eadily computed as µ f = ( A+ τ ) 1 ( I (U 3f U 3i ) c+ A τ )µ I i, (6) whee I is identity matix and τ = V t/b is the non dimensional time step size. The coefficients of the theoy ae summaized hee fo an appoximation involving N inflow states. At fist, matix D is defined as n/ fo n = m+1, D mn = n/ fo n = m 1, (7) fo n m±1. 1
Next, the following two aay, d and c, ae defined d n = { 1/ fo n = 1, fo n 1, (8) c n = n. (9) With these definitions, matix A becomes A = D+db T +cd T + 1 cbt. (1) Finally, aay b is given as b n = ( 1) 1 (N n)! (n!), b N = ( 1) N+1. (11) n 1(N +n)! Thee-dimensional finite-state genealized dynamic wake theoy The fomulation fo the thee-dimensional genealized dynamic wake is based on quasi-steady, potential flow with small petubations 5. The fomulation begins with the continuity and the momentum equations q i,j =, (1) q i, V q i,ξ = Φ,i, (13) The q i ae the velocity petubations, V the fee steam velocity, Φ the pessue and ( ) ξ the non-dimensional time deivative along the fee-steam. Pessue has both a convective component, Φ V, and a petubation component in the diection of the flow, Φ A. Sepaating the two components gives Φ = Φ V +Φ A, (14) Φ A,i = q i,, (15) Φ V,i = V q i,ξ. (16) Diffeentiating eq. (15) and using continuity, Laplace s equation is obtained fo the pessue Φ,ii =. (17) This equation implies that both the convective and petubation components of the pessue each satisfy Laplace s equation. Each component can theefoe be seen as its own acceleation potential Φ A,ii =. (18) Φ V,ii =. (19) The bounday conditions fo the potential functions ae satisfied by matching the pessue distibution along the blades and setting pessue to zeo at infinity. Using sepaation of vaiables in ellipsoidal coodinates, Laplace s equation can be solved analytically. This paticula method of solution povides fo a pessue discontinuity acoss a cicula disk. The potential is then witten as a Fouie seies Φ(ν,η,ψ, t) = m=n=m+1,m+3 P m n Qm n (iη)cm n ( t)cosmψ +D m n ( t)sinmψ. () Hee Pn m and Q m n ae Legende functions of the fist and second kind and Cn m and Dn m unknown coefficients to be detemined. The vaiables ν, η and ψ ae the ellipsoidal coodinates. The discontinuity acoss the disk comes fom the fact ν is positive above the disk and negative below the disk. Integating the petubation along the fee-steam q i = 1 V Φ V,i ξ dξ (1) Φ A,i = q i,, ()
Now, only consideing the z-component of the induced velocity nomal to the inflow disk, eqs. (1) and (15) become w = 1 Φ V dξ V z (3) dw d t = ΦA z, η= (4) Viewing the ight hand side of eqs. (3) and (4) as linea opeatos on Φ A and Φ V the pessue can be witten as Φ A +Φ V = E 1 w + +L 1 w = Φ, (5) Since eq. (5) is assumed to be a linea opeation, it is also assumed to be invetible. The induced flow is then expessed in a Fouie seies in the azimuthal and adial diections w(,ψ, t) = = j=+1,m+3 Ψ j α j ( t)cosψ +β j( t)sinψ. (6) In ode fo the seies to popely expess the induced flow, w, the Ψ j linealy independent. The test functions ae chosen as must be a complete set of functions which ae whee ν = 1 and P n m(ν) = ( 1)m Pm n /ρ m n. The induced flow then becomes w(,ψ, t) = Ψ j = φ j ( ) = 1 ν P j (ν) (7) = j=+1,+3 The expansion functions φ j ( ) ae detemined by whee The wake state equations become φ j ( ) = (j +1)H j H j = j 1 φ j q=,+ α j ( t)cosψ +β j ( t)sinψ (8) q ( 1) (q )/ (j +q)!! (q )!!(q +)!!(j q 1)!!, (9) (j + 1)!!(j 1)!!. (3) (j +)!!(j )!! π Hm n α+ j +L 1 c Vn m α j = 1 τmc n (31) π Hm n β + j +L 1 s Vn m βj = 1 τms n (3) The velocities matix Vn m is diagonal and is constucted using the following V 1 = µ +(λ+ν) (33) Vn m = µ +(λ+ν)(λ+ν) V1 The aveage induced velocity is the souce of nonlineaity and is elated to the fist inflow state as (34) The blade loads fom the genealized foces. In non-dimensional fom, τ c n = 1 π τ mc n = 1 π ν = 3 α 1 (35) 1 ρω R 3φ n( )d (36) 1 ρω R 3φm n ( )d cosmψ q (37) 3
The wake influence coefficients τ ms n = 1 π 1 ρω R 3φm n ( )d sinmψ q (38) L mc L ms whee l = min(,m) and X = tan(χ/) and L mc X m Γ m jn (39) X m +( 1) l X m+ Γ m jn (4) X m ( 1) l X m+ Γ m jn (41) Γ m ( 1) (n+j )/ (n+1)(j +1) H j H m n (j +n)(j +n+)(j n) 1 (4) Γ m sign( m) (n+1)(j +1), (43) Γ m jn =. (44) The following table outlines the numbe of spatial modes coesponding to the numbe of adial shape function. Pevious wok has shown that simultaneously inceasing the numbe of shape functions with the modes povides the most suitable numbe and distibution of wake states. Highest Powe m Total Inflow of states 1 3 4 5 6 7 8 9 1 11 1 1 1 1 1 1 3 1 1 6 3 1 1 1 4 3 1 1 15 5 3 3 1 1 1 6 4 3 3 1 1 8 7 4 4 3 3 1 1 36 8 5 4 4 3 3 1 1 45 9 5 5 4 4 3 3 1 1 55 1 6 5 5 4 4 3 3 1 1 66 11 6 6 5 5 4 4 3 3 1 1 78 1 7 6 6 5 5 4 4 3 3 1 1 91 Table 1: Total numbe of states fo a specified numbe of spatial modes and polynomial ode. The wake influence coefficients matix takes on the following fom. Hee, the th, 1 st and nd modes ae selected each with 3 Legende functions. This gives a 9 9 matix of coefficients but distinctly patitioned with 3 ows and columns of sub-matices with each patition ow/column epesenting a single mode. L mc L 11 L 13 L 15 L 1 1 L 1 14 L 1 16 L 13 L 15 L 17 L 31 L 33 L 35 L 1 3 L 1 34 L 1 36 L 33 L 35 L 37 L 51 L 53 L 55 L 1 5 L 1 54 L 1 56 L 53 L 55 L 57 L 1 1 L 1 3 L 1 5 L 11 L 11 4 L 11 6 L 1 3 L 1 5 L 1 7 L 1 41 L 1 43 L 1 45 L 11 4 L 11 44 L 11 46 L 1 43 L 1 45 L 1 47 L 1 61 L 1 63 L 1 65 L 11 6 L 11 64 L 11 66 L 1 63 L 1 65 L 1 67 L 31 L 33 L 35 L 1 3 L 1 34 L 1 36 L 33 L 35 L 37 L 51 L 53 L 55 L 1 5 L 1 54 L 1 56 L 53 L 55 L 57 L 71 L 73 L 75 L 1 7 L 1 74 L 1 76 L 73 L 75 L 77. (45) 4
Refeences 1 R. Theodosen. Geneal theoy of aeodynamic instability and the mechanism of flutte. Technical Repot 496, NACA Repot, 1949. R.L. Bisplinghoff, H. Ashley, and R.L. Halfman. Aeoelasticity. Addison-Wesley Publishing Company, Reading, Massachusetts, second edition, 1955. 3 R.L. Bisplighoff and H. Ashley. Pinciples of Aeoelasticity. Dove Publications, Inc., New Yok, 196. 4 D.A. Petes, S. Kaunamoothy, and W.M. Cao. Finite state induced flow models. Pat I: Two-dimensional thin aifoil. Jounal of Aicaft, 3():313 3, 1995. 5 D.A. Petes and C.J. He. Finite state induced flow models. Pat II: Thee-dimensional oto disk. Jounal of Aicaft, 3():33 333, 1995. 5