LIMITATIONS OF GRADIENT TRANSPORT
MODELS IN RANDOM WALKS
AND IN TURBULENCE
1. INTRODUCTION
Possibly starting with de St. Venant (1843). certainly with Boussinesq
(1877). turbulent transport (c.g., of momentum, heat, or contaminant) has
been modeled by more or less ad hoc analogy to the simplest molecular
transport, i.e., by linear mean gradient models. Virtually all turblent transport theories, such as the mixing length theories of Taylor (1915, 1932,
1935), Prandtl (1925). and von Karm&n (1930) the “homologous turbulence” theory of von Karman (1937), the dimensionally inspired free shear flow
theory of Prandtl (1942), the more continuous interaction model of Nevzgliadov ( 1945). and most of the more elaborate higher moment approaches of
rcccnt decades (see, e.g., the papers in the symposia proceedings edited by
Kline et d.. 1969; and by the staff of the NASA Langley Res. Cent., 1973)
assume a linear gradient transport model for one or more properties. Originally it was applied directly to mean velocity, temperature, and concentration. More recently it has been applied to transport of turbulent energy.
incan square temperature and concentration fluctuations, and even
(sccmingly inappropriately) to rcprescnt the energy transfer due to the work
donc by fluciuating static pressure gradient forces.
l’hc application of the sccond moment differential equations began with
Keynolds (1895) and turbulent energy balance, followed after a long interval
by von Kirmin (1937) and Taylor (1938a) with the differential equation for
thc balance of mean square vorticity fluctuation, and Chou (1945) with the
shear stress balance. Since that time, more moment equations have been
introduced, e.g., Rotta (1451a.b) with dissipation rate and integral scale
balances, Corrsin ( 1952, 1953) with temperature fluctuation, gradient fluctuation, and lieat flux balances.
Since any finite collection of statistical moment equations generated from
25
MODELS IN RANDOM WALKS
AND IN TURBULENCE
1. INTRODUCTION
Possibly starting with de St. Venant (1843). certainly with Boussinesq
(1877). turbulent transport (c.g., of momentum, heat, or contaminant) has
been modeled by more or less ad hoc analogy to the simplest molecular
transport, i.e., by linear mean gradient models. Virtually all turblent transport theories, such as the mixing length theories of Taylor (1915, 1932,
1935), Prandtl (1925). and von Karm&n (1930) the “homologous turbulence” theory of von Karman (1937), the dimensionally inspired free shear flow
theory of Prandtl (1942), the more continuous interaction model of Nevzgliadov ( 1945). and most of the more elaborate higher moment approaches of
rcccnt decades (see, e.g., the papers in the symposia proceedings edited by
Kline et d.. 1969; and by the staff of the NASA Langley Res. Cent., 1973)
assume a linear gradient transport model for one or more properties. Originally it was applied directly to mean velocity, temperature, and concentration. More recently it has been applied to transport of turbulent energy.
incan square temperature and concentration fluctuations, and even
(sccmingly inappropriately) to rcprescnt the energy transfer due to the work
donc by fluciuating static pressure gradient forces.
l’hc application of the sccond moment differential equations began with
Keynolds (1895) and turbulent energy balance, followed after a long interval
by von Kirmin (1937) and Taylor (1938a) with the differential equation for
thc balance of mean square vorticity fluctuation, and Chou (1945) with the
shear stress balance. Since that time, more moment equations have been
introduced, e.g., Rotta (1451a.b) with dissipation rate and integral scale
balances, Corrsin ( 1952, 1953) with temperature fluctuation, gradient fluctuation, and lieat flux balances.
Since any finite collection of statistical moment equations generated from
25