Wiener-Hermite Expansion in Model Turbulence in the Late Decay Stage

Abstract

The Wiener-Hermite functional expansion, which is the expansion of a random function about a Gaussian function, is here substituted into the Burgers one-dimensional model equation of turbulence. The result is a hierarchy of equations which (along with initial conditions) determine the kernel functions which play the role of expansion coefficients in the series. Initial conditions are postulated, based on physical reasoning, criteria of simplicity, and the assumption that the series is to represent the late decay stage (in which the Gaussian correction is small and also decreasing with time). These are shown to justify an iterative solution to the equations. The first correction to the Gaussian approximation is calculated. This is then tested by evaluating the correction to the flatness factor, which for an exactly Gaussian function has the value 3, but which has been found by experiment (in real three-dimensional fluids, of course) to have a value which deviates from the Gaussian value increasingly rapidly with the order of the derivative. We utilize this effect as a test of the inherent ability of the Wiener-Hermite expansion to bring to realization the physical properties implicit in the Navier-Stokes or Burgers equations. The various contributions to the flatness-factor deviation, when computed, do show a potential capability of providing a theoretical basis for the effect.

Modeling uncertainty in flow simulations via generalized polynomial chaos

“…It is based on the original theory of Wiener (1938) on homogeneous chaos [9,10]. This approach was employed in turbulence in the 1960s [11,12,13]. However, it was realized that the chaos expansion converges slowly for turbulent field [14,15,16], so polynomial chaos did not receive much attention for a long time.…”

A study of the treatment of uncertainty in the Quantitative Fire Risk Assessment of offshore installations

“…This idea comes from the theory about polynomial chaos by Wiener (1938Wiener ( ), (1962, which was firstly applied in the study of turbulence in the 1960s (Meecham and Siegel 1964, Siegel et al 1965, Meecham and Jeng 1968. Nevertheless, its convergence rate was judged slow (Orszag and Bissonnette 1967, Crow and Canavan 1970, Chorin 1974: hence, it did not receive much attention until the work of Ghanem and Spanos (1991).…”

Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos

“…It is based on the original theory of Wiener (1938) on homogeneous chaos [9,10]. This approach was employed in turbulence in the 1960s [11,12,13]. However, it was realized that the chaos expansion converges slowly for turbulent field [14,15,16], so polynomial chaos did not receive much attention for a long time.…”