Vorticity and Vortex Dynamics 2011 Part 1 pps

... studies.The topics of Part IV, including Chaps. 11 and 12 , are somewhat morespecial. As an application of vorticity and vortex dynamics to external-ﬂowaerodynamics, Chap. 11 presents systematically ... scope of vorticity and vortex dynamics, and 1 This deﬁnition is a generalization of that given by Saﬀman and Baker (19 79) forinviscid ﬂow.2In Chinese, the words vorticity and vortex can ... 583 Part IV Special Topics 11 Vortical Aerodynamic Force and Moment 587 11 .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 11 .1. 1 The...

Vorticity and Vortex Dynamics 2011 Part 6 pps

... inner and outer radii R 1 and R2 and angularvelocity Ω 1 and Ω2, respectively, the constants can be determined asA = −Ω 1 1 −µη2 1 −η2,B= Ω 1 R2 1 (1 −µ) 1 −η2, (6. 21) where µ = Ω 1 /Ω2 and ... that (5 .10 4) and (5 .10 5) can be replaced bylimT →∞ 1 Tt0t0−Tω(s, t)dt =0, (5 .11 0a)limT →∞ 1 Tt0t0−Tωx(s, t)dt>0, (5 .11 0b)which are a direct generalization of (5 .10 6) to ... oppositesign are also called vortex dipoles. Lamb in 18 95 and 19 06 (the second and third editions of Lamb 19 32) and Chaplykin in 19 03 (see Meleshko and vanHeijst 19 94) independently found an...

Vorticity and Vortex Dynamics 2011 Part 15 ppsx

... (x 1 ,x2,x3)-frame read(∂if) ·ej=c 11 c 12 c 13 c 21 c22c23∂3f 1 ∂3f2∂3f3, (A.83a)wherec 11 = ∂ 1 f 1 − κ 1 f2− b 11 f3c 12 = ∂ 1 f2+ κ 1 f 1 − b 12 f3c 13 = ... Lines and Surfaces 715 (A.74) can be written in a geometrically clearer form:∂ 1 e 1 = κ 1 e2+ b 11 e3∂ 1 e2= −κ 1 e 1 + b 12 e3∂ 1 e3= −b 11 e 1 − b 12 e2∂2e 1 = −κ2e2+ b 12 e3∂2e2= ... b 11 e3∂ 1 e2=h 1, 2h 1 h2e 1 + b 12 e3∂ 1 e3= −b 11 e 1 − b 12 e2∂2e 1 =h2 ,1 h 1 h2e2+ b 12 e3∂2e2= −h2 ,1 h 1 h2e 1 + b22e3∂2e3= −b 12 e 1 − b22e2,...

Vorticity and Vortex Dynamics 2011 Part 2 doc

... τ-frame. Then(2 .14 4) reads{Dij} =S 11 S 12 ωr/2S 12 rs− S 11 0ωr/20un,n. (2 .14 6)We have seen in Sect. 2 .1. 2 that the ﬁrst and third eigenvalues λ 1, 3 and associated ... HWD basis vectors (2 .12 1)with k = 1and = +1, not normalizedu+(x)=|k| =1 u+(k)(e 1 +ie2)eik·x, |k| =1, where (e 1 , e2, k) form a Cartesian triad. By letting e 1 be along the plus ... ×u, u · (∇u)T= ∇u ·u.Then (2 .11 ) is cast to the vorticity form, very important in vorticity and vortex dynamics: DuDt=∂u∂t+ ω × u + ∇ 1 2q2,q≡|u|. (2 .16 2)Therefore, the advective...

Vorticity and Vortex Dynamics 2011 Part 3 pptx

... C 1 and C2.(a) α 12 =0,(b) α 12 = 1, (c) α 12 =2C 1 C2C=Fig. 3 .10 . Decomposition of a knotted vortex ﬁlamentcan go through C 1 an integer number of times (Fig. 3.9c), so that Γ 1 = α 12 κ2,where ... study of Casey and Naghdi 19 91) . Then by (2 .1) , (3 .10 7), and (3 .10 9), we have∂Ω∂τ= 0 orωρ=ω0ρ0· F, (3 .13 3b)known as the Cauchy vorticity formula. Thus, the Lagrangian vorticity isstationary ... Rossby (19 36, 19 40) and Ertel (19 42), governed by (the name will be explained in Sect. 3.6 .1) DDtωρ·∇φ= 1 ρ(∇×a) ·∇φ,DφDt=0. (3 .10 6)Because (3 .10 6) is the simplest form of (3 .10 5),...

Vorticity and Vortex Dynamics 2011 Part 4 pdf

... Vorticity Dynamics in Boundary Layers 16 12.00 1. 00 2.00 3.00 4.00 5.00 6.000.00 1. 75 1. 50 1. 25 1. 00Re 1 21st3rdF6pmUaFig. 4 .13 . Drag coeﬃcient of a sphere derived from (4.76) for m =1( dashed ... 18 10 .1 -0 .1 -0 .1 -0.20 .1 0 .1 0 .1 -0.20 .1 -0.50 .1 -0.8 -1. 4 -1. 5 1. 50 .1 0x00 .10 0.250.50 1. 002.004.00yFig. 4. 21. Early stage of the roll-up. The vortex- blob positions are plotted on theleft and an interpolation ... that u−= u− 1 e 1 but u+= u+ 1 e 1 + u+2e2.Thus,[[ u]] = ( u+ 1 − u− 1 )e 1 + u+2e2,¯u = 1 2[(u+ 1 + u− 1 )e 1 + u+2e2].(4 .14 0)++--++--e2e 1 SSS(c)(b)(a)Fig....

Vorticity and Vortex Dynamics 2011 Part 8 pptx

... ≡π2q2U2= O (1) . (7.78)Having obtained dynamic relations (7.76) and (7.78), Ahlborn et al. (19 98)note that Cpb and CDare related by 1 −Cpb= (1 γ) (1 + CD),γ=CD+ Cpb 1+ CD, (7.79) 12 Ahlborn ... al. (19 98, 2002) set A =√2.360 7 Separated Vortex Flows0.60.40.200.4 0.6 1. 50 1. 500.500.250.250.50 1. 00 1. 000.8y/sz/s 1. 0Fig. 7.20. Location of vortex sheet (solid curve) and line ... condition (4 .14 1), and the Biot–Savart formula (3. 31) inconvenient component form. The following algebra is based on the work ofClark (19 76) and J.H.B. Smith (19 78).As shown in Fig. 7 .19 a, let...

Vorticity and Vortex Dynamics 2011 Part 12 ppt

... 1. 0 1. 2 1. 4 1. 6 1. 8 0.20.0 0.4 0.6 0.8 1. 0 1. 2 1. 4 1. 6-0.02 -1. 0-0.5-0.50.00.00.50.5 1. 0 1. 0 1. 52.0-0. 01 0.000. 01 0.020.030.040.050.5 1. 0X (m) 1. 5 2.0 0.0-0.02-0. 01 0.000. 01 0.020.030.040.050.5 ... length scale of background588 11 Vortical Aerodynamic Force and Momentn-nSVfB∂BFig. 11 .1. Flow domain to be analyzed and notations 11 .1. 1 The Need for “Nonstandard” TheoriesThe general ... As mentioned inSect. 1. 2, these eﬀorts have naturally in turn enriched the content of vorticity and vortex dynamics (e.g., Melander and Hussain 19 93a and 19 94, Pradeep and Hussain 2000; Hussain...

Vorticity and Vortex Dynamics 2011 Part 13 ppt

... (3 .15 7) and Sect. 4 .1. 2, although the ﬂuid is treated as 11 .5 A DMT-Based Arbitrary-Domain Theory 6254422-2-400 6 8 10 1 214 164422-2-400 6 8 10 1 214 164422-2-400 6 8 10 1 214 16(c)(b)(a)Fig. ... the vorticity advection. 11 .5 A DMT-Based Arbitrary-Domain Theory 635 1. 5 1. 50.50.5-0.5-0.5 -1. 5 -1 -1 1 1 002xyS 1 S2S3Fig. 11 .24. A ﬂexible airfoil with centerline deﬁned by (11 .94). ... stress-related 11 .5 A DMT-Based Arbitrary-Domain Theory 6 21 The Eﬀect of CompressibilityBy an inspection of the structure of (11 .69) and (11 .70) as well as a comparisonof (11 .4) and (11 .13 ), we ﬁnd...

Vorticity and Vortex Dynamics 2011 Part 14 pot

... governedTrajectory -1. 5 -1. 3 -1. 1 -0.90.00.20.40.60.8 1. 0 1. 2-0.7 -0.5 -0.3 -0 .1 xyFig. 12 .19 . Universal vortex track (solid line) for the advanced stage of two-contour vortex, computed by (12 .11 0). ... initially. Then∇2ψ 1 + 1 2λ−2(ψ2− ψ 1 )=P 1 ,r<R (12 .89a)∇2ψ 1 + 1 2λ−2(ψ2− ψ 1 )=0,r>R (12 .89b)∇2ψ2+ 1 2λ−2(ψ 1 − ψ2)=0, (12 .89c)where λ = Λ/√2, P 1 = ω0= HP0− ... stage for 1 <t<δ 1 and 674 12 Vorticity and Vortices in Geophysical Flows(a) (b)(d)(c)Fig. 12 .15 . Vorticity evolution of a monopolar vortex with a perturbation of type (12 . 81) with n...

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