Table 2 Single-time second-order correlations

From: Synthetic Lagrangian turbulence by generative diffusion models

 

DNS

DM-1c

DM-3c

E

3.0

3.0

2.9

A

1.7 × 10−3

1.8 × 10−3

1.6 × 10−3

ΣV

− 0.41

\({{\emptyset}}\)

− 0.39

ΣA

4.4 × 10−5

\({{\emptyset}}\)

2.4 × 10−5

  1. Quantities are related to both velocity and acceleration for DNS, DM-1c and DM-3c: \(E=1/3{\sum }_{i}\langle {V}_{i}^{2}\rangle ,A=1/3{\sum }_{i}\langle {a}_{i}^{2}\rangle ,{\Sigma }^{V}=1/3{\sum }_{i,j}\langle {V}_{i}^{2}{V}_{j}^{2}\rangle -\langle {V}_{i}^{2}\rangle \langle {V}_{j}^{2}\rangle ,{\Sigma }^{A}\) \(=1/3{\sum }_{i,j}\langle {a}_{i}^{2}{a}_{j}^{2}\rangle -\langle {a}_{i}^{2}\rangle \langle {a}_{j}^{2}\rangle\), where in the last two expressions, the summation is only for ij = xy, xz and yz.
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