PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 4 potx

122 L.J.A. van Bokhoven et al.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8

t

S∂ ui i

(a)

Aref

A

Cref

C3

Dref

D

10-1

10-1 100

τ

S

ω3

(b)

∝t

0.75

A

C3

D

Fig. 3. (a) Time evolution of the velocity derivative skewness S∂iui during and

after the isotropic precalculation for different viscosities ν. Background rotation in

cases A, C3 and D is applied at tini = 5.0, 4.0 and 2.0, respectively. For reference, the

isotropic precalculations have been prolonged. (b) Log-log plot showing the vorticity

skewness Sω3 as a function of the scaled, shifted time τ for cases A, C3 and D.

Sω3 appears to depend inversely on time tini, i.e. shorter precalculations yield

higher final values of Sω3 . The behavior observed in Fig. 3(b) may partly be

ascribed to slight differences in S∂iui at time tini.

Refined vorticity statistics of decaying rotating 3D turbulence 123

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10

t

S∂ ui i

(a)

Bref

B1

B2

B3

B4

10-1

10-1 100

τ

S

ω3

(b)

∝t

0.75

B1

B2

B3

B4

Fig. 4. As Fig. 3, but for different durations of the isotropic precalculation. Back￾ground rotation in cases B1-B4 is applied at tini = 2.0, 4.0, 6.0 and 8.0, respectively.

Finally, Fig. 5 shows the time evolution of S∂iui and Sω3 for various back￾ground rotation rates, viz. f = 0.5, 2.5, 5.0 and 10.0 (cases C1-C4). Clearly,

a lower background rotation rate results in a larger final value of Sω3 . This

result expresses the fact that the asymmetry between cyclonic and anticyc￾lonic structures is more pronounced at low rotation rates than at high rotation

rates. It is remarked that similar results were extracted from lower resolution

(N = 144) calculations.

124 L.J.A. van Bokhoven et al.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10

t

S∂ ui i

(a)

Cref

C1

C2

C3

C4

10-1

10-1 100

τ

S

ω3

(b)

∝t

0.75

C1

C2

C3

C4

Fig. 5. As Fig. 3, but for different background rotation rates. Background rotation

in cases C1-C4 is applied at tini = 4.0.

Third Order Vorticity Correlations

Figure 6 shows the time evolution of all nontrivial VTCs for various back￾ground rotation rates. The following three observations are made: 1) ω3

1,

ω1ω2

3 and ω1ω2ω3 are much smaller than unity and fluctuate around zero;

2) ω2

1ω3, ω2

2ω3 and ω3

3 are clearly nonzero; and 3) the ratio ω2

1ω3/ω2

2ω3

(not shown) is found to fluctuate around unity. These results are consistent

with relationship (3).

Refined vorticity statistics of decaying rotating 3D turbulence 125

-0.1

0

0.1

0.2

0.3

0.4

〈ω3

1〉/〈ω2

3〉3/2

C1

C2

C3

C4

-0.1

0

0.1

0.2

0.3

0.4

〈ω1ω2

3〉/〈ω2

3〉3/2

C1

C2

C3

C4

-0.1

0

0.1

0.2

0.3

0.4

〈ω1ω2ω3〉/〈ω2

3〉3/2

C1

C2

C3

C4

-0.1

0

0.1

0.2

0.3

0.4

〈ω2

1ω3〉/〈ω2

3〉3/2

C1

C2

C3

C4

-0.1

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6

τ

〈ω2

2ω3〉/〈ω2

3〉3/2

C1

C2

C3

C4

-0.1

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6

τ

Sω3

C1

C2

C3

C4

Fig. 6. Time evolution of the minimal set of VTCs in axisymmetric turbulence for

various background rotation rates. All VTCs are normalized by
ω2

3

3/2.

4 Discussion

Our numerical results show that in most of the considered cases Sω3 initially

grows at a rate proportional to t0.75±0.1. The latter power-law exponent is

in good agreement with the 0.7 obtained from recent laboratory experiments

[14, 15]. However, the amplitude of maximum Sω3 and the (scaled) time at