The correlation coefficients between u' and u', denoted as ρ(u', u',), are:
ρ(u'₁, u'₁) = 2.08
ρ(u'₂, u'₂) = 0.769
ρ(u'₃, u'₃) = 1.0
In homogeneous turbulent flow, the Reynolds stress normalized by the turbulent kinetic energy (k) can be represented as a matrix:
[1.08 -0.32 -0.32]
[0.0 0.4 0.0 ]
[0.0 0.0 0.52]
The angle brackets (< >) indicate averaging. To calculate the correlation coefficients between u' and u', we divide each element of the matrix by k to normalize it:
[0.0 0.0 ]
[0.0 0.0 1.0 ]
The correlation coefficients, denoted as ρ(u', u',), are obtained from the diagonal elements of the normalized matrix:
ρ(u'₁, u'₁) = 2.08
ρ(u'₂, u'₂) = 0.769
ρ(u'₃, u'₃) = 1.0
These values represent the correlation between the velocity fluctuations in the respective directions.