##### A quadratic Reynolds stress development for the turbulent Kolmogorov flow and comparison with the Channel flow: a DNS study

Abstract: We study the 3D turbulent Kolmogorov flow, i.e. the Navier-Stokes equations forced by a single-low-wave-number sinusoidal force in a periodic domain, by means of direct numerical simulations with microscale Reynolds numbers up to Rλ=200. This classical model system is a realization of anisotropic and non-homogeneous turbulence. As a simple realization of turbulence with a spatially dependent mean flow, it is a convenient test ground for turbulent transport models. Boussinesq’s eddy viscosity linear relation is checked, and it is shown that a quadratic nonlinear constitutive equation can be proposed: a linear term and two nonlinear terms in the form of traceless and symmetric tensors are involved and their coefficients are numerically estimated (see Figure). In large shear regions, Boussinesq’s hypothesis is approximately valid, and is in total failure for vanishing shears. The behavior of the coefficients in the tensorial development is considered and for vanishing shears it is found that the quadratic Reynolds stress development fails since some coefficients are diverging.

A periodic flow with non-sinusoidal forcing has also been considered, with the choice of a Gaussian shape. It was found that the shear stress term is proportional to the mean velocity derivative, indicating that for such forcing also the eddy-viscosity does not depend on the position. The shape of the normal stresses in this case is non-trivial and cannot be precisely fitted. A quadratic development of the constitutive equation can also be proposed for this flow.

Finally, the turbulent Kolmogorov flow (with sinusoidal as well as Gaussian forcing) is compared with the Channel flow using DNS databases. Some common properties and also differences are emphasized.