Page 251 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

222 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

6.1.1.1.3 Length and time scales in turbulent flows

For a correct setup of a DNS of a turbulent flow, it is crucial to derive estimates of the
smallest and the largest length and time scales. The smallest length scale of turbulence
(i.e., the dimension of the smallest turbulent structures), where mechanical energy is
dissipated by viscosity, was introduced by Kolmogorov for homogeneous turbulence
at high Reynolds numbers (see the textbook of Pope (2000) for background):

1=4
ν
3
η ¼ (6.1.1.5)
E
where ν represents the kinematic viscosity and E the dissipation rate of turbulent
kinetic energy per unit mass k. Eq. (6.1.1.5) is useful for a posteriori evaluations of
the turbulent dissipation. For an estimate, the expression:

η LRe 3=4 (6.1.1.6)

is used, where L represents the macroscopic scale used to evaluate the Reynolds num-
ber. It is generally accepted that for performing a DNS, grid spacings need to be of the
same order of magnitude as the Kolmogorov length scale (Coleman and Sandberg,
2010). While far from solid walls a model for the Kolmogorov scale (Eq. 6.1.1.6)
might be appropriate, the calculation of the local Kolmogorov length scale in wall-
bounded problems requires the value of the turbulent kinetic energy dissipation ε.
However, the calculation of such a quantity can be cumbersome, and, in addition,
it can only be accomplished a posteriori, as it needs well-gathered statistics. For
wall-bounded flows, the Kolmogorov scale can be modeled by
η + 1=4
κy Þ (6.1.1.7)
ð
δ v
+
where the von Ka ´rma ´n constant κ is around 0.41, y being the distance from the closest
wall in wall units, and δ v is the viscous length ν/u τ , see the book by Pope (2000,
p. 290). Eq. (6.1.1.7) can be considered valid for fully developed wall-bounded flows
and also as a good indication for developing boundary layers.
The smallest time scale that follows from Kolmogorov theory is:
η 2
τ η ¼ (6.1.1.8)
ν
in wall-bounded flows it can be rewritten as

δ 2
+ 1=2
τ η v ð κy Þ (6.1.1.9)
ν
It is clear that time steps in DNS must be shorter than the Kolmogorov time scale in
order to accurately capture the flow dynamics.