# The turbulence problem

### Abstract

The stochastic closure theory (SCT) of turbulence \cite{BB211} and \cite{BB312} has been extended to Homogeneous Turbulence \cite{KBSB18}, Boundary Turbulence \cite{BC16,KKB18} and Lagrangian turbulence. These extensions give excellent agreement with experimental data. We explain how the theory puts the Kolmogorov-Obukhov '62 scaling theory, with the She-Leveque intermittency corrections \cite{SL94} on a firm theoretical foundation and permits one to compute the scaling coefficients, in addition to the scaling exponents, and their dependence on the (Taylor) Reynolds (T-R) number.

For Homogeneous Turbulence, SCT is compared to data obtained in the Variable Density Turbulence Tunnel (VDTT) \cite{BBNSX14}, at the Max Planck Institute for Dynamics and Self-Organization in G"ottingen. Given a mean flow in homogeneous turbulence the comparison with the data reduces the number of parameters in SCT to just three, one characterizing the variance of the mean-field noise and another characterizing 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. This characterizes the smoothness of the turbulent velocity. For Boundary Turbulence, we compare the predictions of SCT to the generalized Townsend-Perry constants \cite{MM13} as estimated from measurements in the Flow Physic Facility (FPF) \cite{VK13} at the University of New Hampshire. Following \cite{KKB18} we show how the SCT can be used to compute the constants, explaining the sub-Gaussian relations of the constants. We then compare the SCT theory predictions with the data \cite{ZK14}, showing good agreement. Finally, we discuss the extension of SCT to Lagrangian Turbulence. Very surprisingly the comparison of SCT and the data also gives information about the smoothness of the turbulent velocity as T-R becomes very large, see \cite{KBSB18}.