We consider the case of a homogeneous, isotropic, fully developed, turbulent flow. We show analytically by using the −5/3 Kolmogorov's law that the time-averaged consistency error of the Nth approximate deconvolution LES (large eddy simulation) model converges to zero following a law as the cube root of the averaging radius, independently of the Reynolds number. The consistency error is measured by the residual stress. The filter under consideration is a second-order differential filter, but the 1/3 law is still valid in the case of the Gaussian filter and large class of filters used in LES. We also show how the 1/3 error law can be derived by a dimensional analysis.
Acknowledgement
The first author is partially supported by NSF Grants DMS0207627 and 0508260.
Notes
1Here the‘filteredNSEsolution’ stands for the filtered solution to the Navier–Stokes equation deduced from the LES filter under consideration and ||·|| denotes any norm, defining the sense given to the notion of ‘accuracy’.
†Recall that the residual stress is defined by τ0 = (τ0
ij
)1 ≤ ij ≤ 3 = , the velocity u is the vector field, u = (u
1, u
2, u
3).
‡Recall that G
0
=
and one formally has G
∞
= u.
§Defined by its transfer function (Equation3).
†One must say here that we have tried to prove the mathematical convergence of the ADM models to the Navier–Stokes equations when N goes to infinity for a fixed δ. We failed with the mathematical fluid dynamics of the classical tools. This is mainly due to the lack of information on the fields in the high-frequency components.
‡The obstruction is of the same type when trying to use the functional analysis to solve this question of convergence: we are not able to keep a control on the high-frequency components, a difficulty well illustrated by the shapes of the transfer functions
N
given in .