Flows Caused by a Rotating Disk

Statistical methods also are applied in turbulence theory. These methods use the averaging over the ensemble of possible realizations of the process and take into account the probability distribution density for each quantity. In this case, various hypotheses and experimental data for the probability distribution must be used to obtain specific results. The statistical approach is related to a fairly high level of complexity of describing turbulent flows [290,460, 492]. 

Another group of methods relies on straightforward numerical simulation of turbulent flows. Numerical analysis is based directly on the Navier-Stokes equations [228, 229, 288, 314-316, 376] or equivalent variational principles [160, 310]. The computations are carried out until statistically steady-state flow regimes characterized by steady values of average quantities are attained. This approach involves a lot of computation but does not require the use of physical hypotheses and empirical constants. Note that no rigorous mathematical estimates of the accuracy of the numerical method for the simulation of turbulent flows have been available so far. 

A variety of other methods for the investigation and mathematical modeling of turbulence are known, which are based on various arguments and hypotheses, e.g., see [36, 192, 426,435,451, 459]. 

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