Vorticity and Vortex Dynamics 2011 Part 2 doc

... Then, let S and A be the symmetric and antisymmetric parts of (∇ π b) π , from (2. 143) it follows that 2D =2nnu n,n + n(ω r × n)+(ω r × n)n +2S, (2. 144) 2 = n(ω ×n) −(ω × n)n +2A, (2. 145) where ... (∇u) T +(∇u) T =2 +(∇u) T , such that 2u · Ω = ω ×u, u · (∇u) T = ∇u ·u. Then (2. 11) is cast to the vorticity form, very important in vorticity and vortex dynamics: Du Dt = ∂u ∂...

Ngày tải lên: 11/08/2014, 17:22

... . . . . . 22 2. 2 Fundamental Equations of Newtonian Fluid Motion . . . . . . . . . . 25 2. 2.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. 2 .2 Balance ... Momentum and Angular Momentum . . . . . . . 26 2. 2.3 Energy Balance, Dissipation, and Entropy . . . . . . . . . . . . 28 2. 2.4 Boundary Conditions. Fluid-Dynamic Force andMoment...

Ngày tải lên: 11/08/2014, 17:22

... R m B B Fig. 4.1. Damped Alfv´en waves at high and low R m . 4 3 2 1 0 -1 -2 -3 -4 -4 -2 024 x -4 -2 024 x (a) (b) y 4 3 2 1 0 -1 -2 -3 -4 y Fig. 4 .2. A numerical Navier–Stokes solution of the ... by parts, the integrand of the volume integrals in (3. 92) and (3.93), k (1) V (x) ≡ 1 2 ω · ψ, (3.97a) k (2) V (x) ≡ (ω × u) · x =(x × ω) · u, (3.97b) 114 3 Vorticity Kinemat...

Ngày tải lên: 11/08/2014, 17:22

... the sphere: C D1 = 12 Re 4 − Re 2 − (2 + Re) 2 e −Re 1 − 1 2 Re 2 − (1 + Re)e −Re , (4.77) which for Re  1 is reduced to C D1 = 12 Re  1+ 3 8 Re − 19 320 Re 2  + O(Re 2 ), (4.78) the same ... u + 1 e 1 + u + 2 e 2 .Thus, [[ u]] = ( u + 1 − u − 1 )e 1 + u + 2 e 2 , ¯ u = 1 2 [(u + 1 + u − 1 )e 1 + u + 2 e 2 ]. (4.140) + + - - + + - - e 2 e 1 S S S (c)(b)(a...

Ngày tải lên: 11/08/2014, 17:22

... flow p = p ∞ + 1 2 [U 2 − (u 2 + v 2 )] −  1 2 k 2 ψ 2 for r ≤ a, 0forr>a. (6.111) Hence, at the extrema of vorticity, Chaplykin found that p min = p ∞ + 1 2 U 2  1 −4 J 2 1 (c) J 2 0 (ka)  = ... z)=− Γk 4πr 0  r 0 r  1 /2  K(k)+ r 0 2r k 2 1 −k 2 − 2 −k 2 2(1 −k 2 ) E(k)  , (6.63b) k =2  r 0 r z 2 +(r + r 0 ) 2 . (6.63c) In particular, at the...

Ngày tải lên: 11/08/2014, 17:22

... D and time average over T simply yield C D = 1 2 πβ 2 St q 2  U 2 , where q 2  scales with U 2 . Thus, we obtain the second dynamic relation (Ahlborn et al. 20 02) C D = β 2 KSt, K ≡ π 2 q 2  U 2 = ... p 0 ρ = 1 2 (w 2 0 − w 2 a ) − 1 2 C 2 a a 2 . Differentiating this equation with respect to z and neglecting the small amount dC/dz, we obtain d dz ...

Ngày tải lên: 11/08/2014, 17:22

... in Sect. 1 .2, these efforts have naturally in turn enriched the content of vorticity and vortex dynamics (e.g., Melander and Hussain 1993a and 1994, Pradeep and Hussain 20 00; Hussain 20 02) . We expect ... Saffman and Baker 1979; Leonard 1985; Hunt 1987; Ashurst and Meiburg 1988; Virk and Hussain 1993; Hunt and Vassilicos 20 00; Lesieur et al. 20 00; Schoppa and Hu...

Ngày tải lên: 11/08/2014, 18:20

... crucial 0. 125 -0. 125 0.075 -0.075 0. 025 -0. 025 x =0 .24 c 0.05 0.1 0.15 0 .2 0 .25 0.05 0.1 0.15 0 .2 0 .25 Sectional Cz distriution 5 4 3 2 1 -1 -2 -2. 5 0 0 Cz =1.6646 Czu =-1. 623 4 Czl =3 .28 80 y y (a) ... 625 4 4 2 2 -2 -4 0 0 6 8 10 121 416 4 4 2 2 -2 -4 0 0 6 8 10 121 416 4 4 2 2 -2 -4 0 0 6 8 10 121 416 (c) (b) (a) Fig. 11.17. Distribution of integrand of...

Ngày tải lên: 11/08/2014, 18:20

... ( 12. 51) we find the dispersive relation ω = − βλ 2 k 1+λ 2 k 2 , ( 12. 52) so the phase velocity c = ω/k and group velocity c g = ∂ω/∂k are c = − βλ 2 1+λ 2 k 2 ,c g = −βλ 2 1 − λ 2 k 2 (1 + λ 2 k 2 ) 2 . ... ψ 1 )=P 1 ,r<R ( 12. 89a) ∇ 2 ψ 1 + 1 2 λ 2 (ψ 2 − ψ 1 )=0,r>R ( 12. 89b) ∇ 2 ψ 2 + 1 2 λ 2 (ψ 1 − ψ 2 )=0, ( 12. 89c) where λ = Λ/ √...

Ngày tải lên: 11/08/2014, 18:20

... b 11 f 1 + b 12 f 2 c 21 = ∂ 2 f 1 + κ 2 f 2 − b 12 f 3 c 22 = ∂ 2 f 2 − κ 2 f 1 − b 22 f 3 c 23 = ∂ 2 f 3 + b 12 f 1 + b 22 f 2                        . (A.83b) Thus, b αβ and κ α appear ... b 11 e 3 ∂ 1 e 2 = h 1 ,2 h 1 h 2 e 1 + b 12 e 3 ∂ 1 e 3 = −b 11 e 1 − b 12 e 2 ∂ 2 e 1 = h 2, 1 h 1 h 2 e 2 + b 12 e 3 ∂ 2...

Ngày tải lên: 11/08/2014, 18:20