Direct numerical simulation of anisotropic turbulent flow for incompressible fluid

Equation for small-scale velocity or pulsation is a starting point to build almost all models of turbulence. Transport equation for Reynolds stresses, dissipation rate, kinetic energy are derived from this equation, adding some assumptions about structure of terms, including in these equations. Equation for small-scale velocity can be simplified, if we assume that large – scale velocity and its gradients are constants instead of linear profile for large-scale velocity. We implemented the direct numerical simulation of this equation under the simple shear, leading to anisotropy. The nonlinear helicity terms were computed in spectral space, using the three-dimensional Fast Fourier transformation, then, the inverse Fast Fourier transformation was used to return in physical space. Aliasing terms were not removed. Four – order Runge- Kutta method was used for integration in time. Evolution of Reynolds stresses in time were computed.


Introduction
A basis for direct numerical simulation of fully developed turbulence was laid down in [1,2]. Numerical solutions of Navier -Stokes equations for incompressible fluids with more powerful computers have been investigated in [3][4][5]. The pumping of energy in the isotropic flow was implemented by some random force with known statistical properties.
There is a more realistic picture, when energy from large-scale flow is transferred by an anisotropic shear term to small-scale eddies with a size less than the integral scale of turbulence. There are the results of the simplified models of anisotropic cascades [6] and the direct numerical simulation of anisotropic turbulence [7]. The prevailing assumption of the anisotopic turbulence simulation is a linear profile of large-scale or mean velocity [7]. A simplified form of the small-scale velocity transport equation, when large-scale velocity and its gradients were considered as constants, was proposed in [8].
The simplicity of this equation should lead to a better analytical analysis. Thus, the motivation of our simulation is to reveal unexplored properties of a new transport equation.
What is known about the simulation of small-scale velocity, when large-scale velocity has linear profile? The next system of differential equations was derived from Navier -Stokes equations [7]:  Unfortunately, results were given only for two -dimensional flow. In paper [9], the system of ordinary differential equation for the velocity Fourier -components was solved at the grid with  [10], a comparison with the dns results is not available. This system of differential equation was numerically solved also in [11], aperiodic oscillations of energy and Reynolds's stress were revealed. An anisotropy, caused by the gradients of mean velocity, was investigated in [12].
Velocity v was represented as a sun of small-scale velocity u and large-scale velocity U : We obtain the next equation from Navier-Stokes equation for incompressible fluid: Authors of the paper [12] considered shear   U  as anisotropic random forces:  are mutually independent, homogeneous distributed,  -correlated in time. We can transit to the coordinate system, moving with local velocity U , and nullify corresponding term in the transport equation. The inverse transformation in the initial coordinate system, being at rest, is given by Galileo transformation [13]: The results of simulation are in agreement with Kolmogorov's and Lumley's spectra.

Transport equation for polarization components of small-scale velocity
Using the approach [3], let us transform the small-scale velocity in Fourier series in spatial variables: -is a vector with the integer components, here − L is the integral scale of turbulence. We should consider n in numerical simulation as finite values: where 3 , 2 , 1 = i and N is a sufficient large number, being a positive integer power of 2 for using Fast Fourier Transform. These inequality can be rewrote as We seek the solution as: Let us define the collocation points as the following ones: ) , , ( Then, an approximate solution (3) of the small-scale transport equation in the collocation points shall be: The inverse Fourier transform with finite number of terms shall be: Due to the wave number cutoff, the nonlinear terms of the transport equation will have the wave numbers out of the range (2), it is so-called aliasing error. Since the small-scale velocity ) has the real components, there is an equality: kinetic energy of small-scale eddies is equal to: subgrid Reynolds tensor is defined as: Kinetic energy (9) and subgrid Reynolds tensor (10) have the similar expansion into Fourier series.
Combining the results of [3,8], transport equations for two polarization Fourier components of the small-scale velocity will be for anisotropic turbulence as: where summation over repeating indices is assumed ) directing along the unit polarization vectors 1 e and 2 e , these are orthogonal to each other and to vector ) , , ( Let us define the unit vector e , as: After that, we define the unit vectors 1 e and 2 e :  (16) and (17) are not zeros. We have a relation, following from (11), that ) , . Also, we obtain from formulas (14)-(17), that: The system of equations (12), (13) turns into the next system: Using well-known formula for double vector product: -arbitrary vectors.

Choice of equation's parameters
Parameter L is a spatial scale, dividing large and small fluid motion, The setting up of initial condition can be defined, following to [15] as: Following to [3], to compute

The computational results and discussion
Here, we give preliminary results of dns for anisotropic incompressible turbulence, using a new simplified transport equation for the small-scale velocity [8]. We used 8x8x8 grid of the wave numbers. Subgrid Reynolds number was . 10 Re = s The integration in time was stopped at .

= St
The averaging over the random initial conditions is absent. We give a typical evolution of the polarization Fourier velocity components. We see from figure 1 a tendency to isotropy. Also, we can describe the process as self-sustained one.