RENORMALIZATION FOR THE
WIENER HERMITE REPRESENTATION
OF STATISTICAL TURBULENCE
Onc of the methods of treatment of statistical turbulence in~olves the use
of the Wiener Hermite representation. The representation has been discussed in previous publications (Imainura cr ul.. 1965; Meecham and Jrng.
1968: Meecham. 1969). I t has been used with some success in the treatment
of diffusion problems by Soffman (1969). Early mathematical work on such
representations was done by Cameron and Martin (1957). The treatment is
biised upon the white noise process- -a process with a constant spectrum.
Linear combinations of such white noise processes are suffjcient to represent
a Gaussian process. Non-Gaussian processes may be represented through
the use of polynomial combinations of the white noise process. The favored
forms are patterned after Hermite polynomials (hence the name of the
rcprcscntntion). Some of the earliest work involving the use of white noise
processes was in connection with Brownian motion problems. For a review
0 1 this work, see Wiencr (19%).
'l'hc Wiencr I~icrmite (W H) rcprescntation has been applied in a number
ol' ciiffcrcnt problem arcas; we shill1 be interested here in Ruid mechanics
applications of coursc. The methods have been used in connection with the
full Navicr-Stokes equations (see Mcecham and Jeng, 1968). A number of
investigators have used the method in treating a model equation described
by Burgers (1950); for this work sec. for instance, Meecham and Siege1
(1964). The results of this previous work may be summarized as follows: the
method gives fairly accurate and useful results for initial fluctuation Reynolds numbers up to about 20 for computation times oforder 2 or 3 (in units
of scale size divided by rnis velocity fluctuation). For larger Reynolds numbers or longer times, the method beconies difficult to malliige in its simplest
form. This will be discussed more fully below. The reason is somewhat as
445
WIENER HERMITE REPRESENTATION
OF STATISTICAL TURBULENCE
Onc of the methods of treatment of statistical turbulence in~olves the use
of the Wiener Hermite representation. The representation has been discussed in previous publications (Imainura cr ul.. 1965; Meecham and Jrng.
1968: Meecham. 1969). I t has been used with some success in the treatment
of diffusion problems by Soffman (1969). Early mathematical work on such
representations was done by Cameron and Martin (1957). The treatment is
biised upon the white noise process- -a process with a constant spectrum.
Linear combinations of such white noise processes are suffjcient to represent
a Gaussian process. Non-Gaussian processes may be represented through
the use of polynomial combinations of the white noise process. The favored
forms are patterned after Hermite polynomials (hence the name of the
rcprcscntntion). Some of the earliest work involving the use of white noise
processes was in connection with Brownian motion problems. For a review
0 1 this work, see Wiencr (19%).
'l'hc Wiencr I~icrmite (W H) rcprescntation has been applied in a number
ol' ciiffcrcnt problem arcas; we shill1 be interested here in Ruid mechanics
applications of coursc. The methods have been used in connection with the
full Navicr-Stokes equations (see Mcecham and Jeng, 1968). A number of
investigators have used the method in treating a model equation described
by Burgers (1950); for this work sec. for instance, Meecham and Siege1
(1964). The results of this previous work may be summarized as follows: the
method gives fairly accurate and useful results for initial fluctuation Reynolds numbers up to about 20 for computation times oforder 2 or 3 (in units
of scale size divided by rnis velocity fluctuation). For larger Reynolds numbers or longer times, the method beconies difficult to malliige in its simplest
form. This will be discussed more fully below. The reason is somewhat as
445