Burgers’ model with a renormalized Wiener–Hermite representation

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Abstract
The use of the Wiener–Hermite expansion for the turbulence problem is reviewed. The expansion is known to give good results for lower Reynolds’ number flows, up to a fluctuation Reynolds’ number of 20 using more recent time‐dependent bases. The use and meaning of these bases is discussed. A new development in Wiener–Hermite expansion is used to calculate Burgers’ model: It is known that these expansions are not unique; by taking advantage of this arbitrariness, a renormalization is presented and used to improve the convergence of the expansion. The procedure involves calculation for turbulence using a time‐dependent base for a short time. The calculation is then stopped and the resulting functions are adjusted in such a way as to minimize the non‐Gaussian part of the energy, at the same time preserving the last value of the transfer function. Following this, the calculation is resumed using these new, adjusted values for the functions; then the process is repeated. For the first time it was possible to obtain the equilibrium form k−2, of the energy spectrum for Burgers’ model of turbulence. The calculation proceeds, holding the non‐Gaussian part of the energy to but a few percent. Good results are obtained up to fluctuation Reynolds’ numbers of 100.