MODELING OF TL'RRULENT TRANSPORT
19 1
preserving (is., is not proportional to yU, where U is the mean velocity
defect, an exact result of self-preservation) by about this percentage. The
direction of the error and trend (if it exists) in each component depends on
the initial values, and would not necessarily be the same in our computation
and in the real experiment.
Although the number of terms involved boggles the imagination, the cost
of such a calculation is not prohibitive. One run of the wake calculation, to
six doublings, programmed on the IBM 370/165, costs about $15.00. It must
be rcmernbered that this is an experimental calculation. and the program
was written so that the various constants could be adjusted at will; no
attempt was made to minimize cost. In the final form, the cost should be
substantially lower perhaps of the order of $10.00. If we envision a threedimensional calculation of a stratified flow with boundaries (which converges some five times as fast as a boundary-free flow) involving some lo4
points and 17 equations instead of six, we would have an estimated cost of
perhaps S3oOO. This is not a particularly accurate way to make such an
estimate, because the larger calculation cannot be done in core, and requites
reading out to disk storage. In addition, the pressure must be computed in a
threedimensional calculation. A detailed, direct estimate of the number of
computations involved, and storage requirements, suggests a cost of the
order of $1500. The difference between the estimates probably represents the
different proportion of I/O, CPU, and auxiliary storage access time, and
reflects the inefRciency of our present program. It is nevertheless clear that
such a calculation is within our grasp for a few thousand dollars.
ACKNOWLEDGMENTS
We have profited from many discussions with J. W. DeardorlT. R. Owen, W. C. Reynolds, H.
Tenneker and J. Wyngaard. A remark of Deardorff I d to the understanding or the dissipation
production: a remark of Owen suggested the importance of the second and third order terms in
the pressure gradient .velocity correlation; Reynolds carried out many calculations in an early
rtHge of the investigation, and contributed much analysis and comment which led to a better
understanding of the structure of the equations. and an appreciation of the importance of the
second order transport lerrns; Tcnnekes of course shares responsibility for the Reynolds
iiumhcr order-of-magnitude analysis, and has made extensive contributions by applying the
model lo simple situations. bringing flaws to light: Wyngaard has made independent calculations of the striitificd atmospheric boundary layer, and his findings influenced the development
of the model.
We are particularly grateful to our assembly language programmer, M. Hurvitz. who has
been ahsolutely indispensable.
Generally. the development of the theory ha8 been the responsibility of JLL and the execution of the calculations the responsibility of BK-N, although there has been intensive and
extensive interactions between the two, so that they Cannot be separated so simply. These
calculations will form part of the doctoral dissertation of BK-N.
We wish to acknowledge the assistance of the Applied Research Laboratory in the preparation of the final figures.
19 1
preserving (is., is not proportional to yU, where U is the mean velocity
defect, an exact result of self-preservation) by about this percentage. The
direction of the error and trend (if it exists) in each component depends on
the initial values, and would not necessarily be the same in our computation
and in the real experiment.
Although the number of terms involved boggles the imagination, the cost
of such a calculation is not prohibitive. One run of the wake calculation, to
six doublings, programmed on the IBM 370/165, costs about $15.00. It must
be rcmernbered that this is an experimental calculation. and the program
was written so that the various constants could be adjusted at will; no
attempt was made to minimize cost. In the final form, the cost should be
substantially lower perhaps of the order of $10.00. If we envision a threedimensional calculation of a stratified flow with boundaries (which converges some five times as fast as a boundary-free flow) involving some lo4
points and 17 equations instead of six, we would have an estimated cost of
perhaps S3oOO. This is not a particularly accurate way to make such an
estimate, because the larger calculation cannot be done in core, and requites
reading out to disk storage. In addition, the pressure must be computed in a
threedimensional calculation. A detailed, direct estimate of the number of
computations involved, and storage requirements, suggests a cost of the
order of $1500. The difference between the estimates probably represents the
different proportion of I/O, CPU, and auxiliary storage access time, and
reflects the inefRciency of our present program. It is nevertheless clear that
such a calculation is within our grasp for a few thousand dollars.
ACKNOWLEDGMENTS
We have profited from many discussions with J. W. DeardorlT. R. Owen, W. C. Reynolds, H.
Tenneker and J. Wyngaard. A remark of Deardorff I d to the understanding or the dissipation
production: a remark of Owen suggested the importance of the second and third order terms in
the pressure gradient .velocity correlation; Reynolds carried out many calculations in an early
rtHge of the investigation, and contributed much analysis and comment which led to a better
understanding of the structure of the equations. and an appreciation of the importance of the
second order transport lerrns; Tcnnekes of course shares responsibility for the Reynolds
iiumhcr order-of-magnitude analysis, and has made extensive contributions by applying the
model lo simple situations. bringing flaws to light: Wyngaard has made independent calculations of the striitificd atmospheric boundary layer, and his findings influenced the development
of the model.
We are particularly grateful to our assembly language programmer, M. Hurvitz. who has
been ahsolutely indispensable.
Generally. the development of the theory ha8 been the responsibility of JLL and the execution of the calculations the responsibility of BK-N, although there has been intensive and
extensive interactions between the two, so that they Cannot be separated so simply. These
calculations will form part of the doctoral dissertation of BK-N.
We wish to acknowledge the assistance of the Applied Research Laboratory in the preparation of the final figures.