152
RlCIlhRU L. PESIIFi
distribution resulted in elimination of some of the fine scale details represented in Fig. 8.
It is evident that numerical simulation of Lagrangian statistics in twodimensional flow is feasible at least to the extent that single-particle statistics
can be studied. The use of physical space rather than k space computation
imposes severe limitations on the resolution of the Eulerian field. However,
the k space computation would impose serious problems of complexity of
computation far as determination of the Lagrangian field is concerned. I n
general, for this computation Taylor's formula was verified and the most
interesting result was the good correspondence obtained between the
Lagrangian autocorrelation as directly determined and that quantity calculated from the Corrsin hypothesis, It is apparent that the probability
displacement function for particles is not Gaussian except at long times after
release for the two-dimensional case.
2.2.3. Two-Particle Separation. As mentioned above. the filter function
arguments are not valid when considering two-particle relative separation.
Higher resolution of the Eulerian field is needed because this problem is
strongly characterized by initial separation lengths. Computations are
planned for this problem using k space rather than physical space representation for the Eulerian field. Lin (1972) estimated the two-point separation
law for two-dimensional or quasi-geostrophic turbulence based on the - 3
spectrum. His results indicate an exponential separation growth. Peskin
(1973) pointed out that the Langevin model theory would predict a t 3 law
and at present available data cannot resolve the issue inasmuch as the -3
spectrum may not be correcq. An important point is that the lack of energy
cascade in the - 3 region leaves open the question of physical mechanisms
for separation if the -3 for the Eulerian spectrum is correct. It is hoped that
the future acquisition of constant entropy level balloon data will resolve this
issue. The above-mentioned computer simulations were used to generate
computer movies of clusters of particles. Some qualitative information was
obtained from these films; in particular, the expected behavior of clusters,
namely, that they are persistent for fairly long periods of time and then
rapidly accelerate apart, was observed.
3. SIMULATION OF THREE-DIMENSIONAL LAGRANGIAN
TURBULENCE IN CHANNeL FLOW
While simulation of twodimensional Lagrangian turbulence is an interesting exercise and can be effected (for at least single particle statistics)
without necessity of ad hoc closure assumptions (other than wave number
truncation), the problem lacks certain appeal particularly because of the
omission of vortex stretching mechanisms. In addition, the practical importance of shear flow in diffusion dictated that a more complicated three-
RlCIlhRU L. PESIIFi
distribution resulted in elimination of some of the fine scale details represented in Fig. 8.
It is evident that numerical simulation of Lagrangian statistics in twodimensional flow is feasible at least to the extent that single-particle statistics
can be studied. The use of physical space rather than k space computation
imposes severe limitations on the resolution of the Eulerian field. However,
the k space computation would impose serious problems of complexity of
computation far as determination of the Lagrangian field is concerned. I n
general, for this computation Taylor's formula was verified and the most
interesting result was the good correspondence obtained between the
Lagrangian autocorrelation as directly determined and that quantity calculated from the Corrsin hypothesis, It is apparent that the probability
displacement function for particles is not Gaussian except at long times after
release for the two-dimensional case.
2.2.3. Two-Particle Separation. As mentioned above. the filter function
arguments are not valid when considering two-particle relative separation.
Higher resolution of the Eulerian field is needed because this problem is
strongly characterized by initial separation lengths. Computations are
planned for this problem using k space rather than physical space representation for the Eulerian field. Lin (1972) estimated the two-point separation
law for two-dimensional or quasi-geostrophic turbulence based on the - 3
spectrum. His results indicate an exponential separation growth. Peskin
(1973) pointed out that the Langevin model theory would predict a t 3 law
and at present available data cannot resolve the issue inasmuch as the -3
spectrum may not be correcq. An important point is that the lack of energy
cascade in the - 3 region leaves open the question of physical mechanisms
for separation if the -3 for the Eulerian spectrum is correct. It is hoped that
the future acquisition of constant entropy level balloon data will resolve this
issue. The above-mentioned computer simulations were used to generate
computer movies of clusters of particles. Some qualitative information was
obtained from these films; in particular, the expected behavior of clusters,
namely, that they are persistent for fairly long periods of time and then
rapidly accelerate apart, was observed.
3. SIMULATION OF THREE-DIMENSIONAL LAGRANGIAN
TURBULENCE IN CHANNeL FLOW
While simulation of twodimensional Lagrangian turbulence is an interesting exercise and can be effected (for at least single particle statistics)
without necessity of ad hoc closure assumptions (other than wave number
truncation), the problem lacks certain appeal particularly because of the
omission of vortex stretching mechanisms. In addition, the practical importance of shear flow in diffusion dictated that a more complicated three-