Reynolds Number Dependence of the Structure Functions in Homogeneous Turbulence

Abstract

We compare the predictions of stochastic closure theory (SCT) (Birnir in J Nonlinear Sci 23:657–688, 2013a. https://doi.org/10.1007/s00332-012-9164-z) with experimental measurements of homogeneous turbulence made in the variable density turbulence tunnel (Bodenschatz et al. in Rev Sci Instrum 85(9):093908, 2014) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen. While the general form of SCT contains infinitely many free parameters, the data permit us to reduce the number to seven, only three of which are active over the entire range of Taylor–Reynolds numbers. Of these three, one parameter characterizes the variance of the mean-field noise in SCT and another characterizes the rate in the large deviations of the mean. The third parameter is the decay exponent of the Fourier variables in the Fourier expansion of the noise, which characterizes the smoothness of the turbulent velocity. SCT compares favorably with velocity structure functions measured in the experiment. We considered even-order structure functions ranging in order from two to eight as well as the third-order structure functions at five Taylor–Reynolds numbers (\(R_\lambda \)) between 110 and 1450. The comparisons highlight several advantages of the SCT, which include explicit predictions for the structure functions at any scale and for any Reynolds number. We observed that finite-\(R_\lambda \) corrections, for instance, are important even at the highest Reynolds numbers produced in the experiments. SCT gives us the correct basis function to express all the moments of the velocity differences in turbulence in Fourier space. These turn out to be powers of the sine function indexed by the wavenumbers. Here, the power of the sine function is the same as the order of the moment of the velocity differences (structure functions). The SCT produces the coefficients of the series and so determines the statistical quantities that characterize the small scales in turbulence. It also characterizes the random force acting on the fluid in the stochastic Navier–Stokes equation, as described in this paper.