Residual stress of approximate deconvolution models of turbulence

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.

Keywords :

• Large eddy simulation
• Approximate deconvolution model
• Turbulence

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 (3).

^†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 figure 2.