Vorticity and Vortex Dynamics 2011 Part 2 doc
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
∂...
Vorticity and Vortex Dynamics 2011 Part 1 pps
... . . . . . 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...
Vorticity and Vortex Dynamics 2011 Part 3 pptx
... 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...
Vorticity and Vortex Dynamics 2011 Part 4 pdf
... 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...
Vorticity and Vortex Dynamics 2011 Part 6 pps
... 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...
Vorticity and Vortex Dynamics 2011 Part 8 pptx
... 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
...
Vorticity and Vortex Dynamics 2011 Part 12 ppt
... 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...
Vorticity and Vortex Dynamics 2011 Part 13 ppt
... 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...
Vorticity and Vortex Dynamics 2011 Part 14 pot
... ( 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/
√...
Vorticity and Vortex Dynamics 2011 Part 15 ppsx
... 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...