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National Advisory Committee for Aeronautics, Technical Notes - Heat Transfer in Isotropic Turbulence During the Final Period of Decay

The problem of heat transfer in isotropic turbulence with a constant
mean temperature gradient is considered during the final period of decay.
The Reynolds and Péclet numbers are then very small, and all triple cor—
relation terms can be neglected in the equations for the double correla~
tions. On this basis, it is found that the temperature field ultimately
becomes independent of the initial conditions on the temperature and has
characteristics determined only by the mean temperature gradient, the
physical properties of the fluid, and_the characteristics of the turbur
lence. Detailed analytical and numerical results are obtained for the
asymptotic state.

The mean turbulent heat transfer is in the direction of the mean
temperature gradient, with a magnitude proportional to the magnitude of
the latter. Although it approaches zero when the Prandtl number approaches
zero, its dependence on the Prandtl number is not large for Prandtl num—
bers of order unity and larger. This type of Prandtl number dependence
is typical for many of the other results depending on both velocity and
temperature fluctuations. In fact, the rate of decrease with separation
distance of the two—point temperature-velocity correlation varies little
over the full range of Prandtl numbers and is always about the same as it
is for the double velocity correlation. In contrast, all results involving
only temperature fluctuations display a strong dependence on the Prandtl
number. For example, for small Prandtl numbers the double temperature
correlation falls off much more slowly with separation distance than the
velocity correlation does, while for large Prandtl numbers the opposite
is true.

The simplest case of turbulent heat transfer is the problem first
considered by Corrsin (ref. 1), in which the temperature of the fluid is
specified to have a constant mean gradient in some preferred direction,
while the velocity field, which is assumed independent of the temperature
field, is regarded as isotropic and known. Thus, though mean values
associated with the velocity field only'are isotropic, those associated
with the temperature field are axisymmetric. A study of this problem,
although it is highly idealized, is expected to give some idea of the
nature of turbulent heat transfer and, by analogy, also of turbulent mass
transfer.