Dynamical Properties of Truncated Wiener-Hermite Expansions

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Abstract
The use of truncated Wiener‐Hermite functional expansions as a basis for the theory of turbulence is critically examined. An account is given of the application of such expansions to Burgers' model equation. The nature of Wiener‐Hermite expansions is such that certain important consistency requirements, such as realizability, follow immediately. In order to gain insight into other aspects of the predicted turbulence dynamics, a simpler model problem in which there are only three interacting modes is first studied. It is shown that Wiener‐Hermite closures do not faithfully represent the dynamics for this latter model. Then, the analysis is extended to Burgers' equation. It is shown that Wiener‐Hermite closures do not preserve fundamental properties of the exact dynamics in equilibrium. Inviscid equipartition solutions do not survive in the closures. In addition, it is shown that these closures do not treat the effects of large scales on small scales properly. Numerical calculations be extended to the corresponding theory of Navier‐Stokes turbulence. We conclude that truncated Wiener‐Hermite expansions are unsuitable for the theory of high Reynolds‐number turbulence.