Introduction - Turbulence in the Atmosphere - page 10

Figure 1.3 shows the wall-flux ratios in the turbulent regime. The ratios for heat
and momentum differ by a constant factor of about 1.5-2 (a manifestation of the
Reynolds analogy between heat and momentum transfer) as they increase sharply
with Reynolds number.
If the flow over the earth's surface were laminar, not turbulent, the environmental
effects would be profound. In clear summer weather, for example, the earth's surface
temperature could routinely approach 100 ◦ C during the day and 0 ◦ C at night
(Chapter 9) .

1.4 How do we study turbulence?
Turbulence has long had a special attraction for physicists and mathematicians; it
has been called “the last great unsolved problem of classical physics.” † In practical
terms this means that we cannot analytically solve the equations of turbulent fluid
motion. The difficulty stems from their nonlinearity.
Leonardo da Vinci sketched turbulent water flows, and reportedly gave the sage
advice: “Remember when discoursing on the flow of water to adduce first experi-
ence and then reason.” ‡ Even today, some 500 years after da Vinci, much of our
understanding of turbulence is rooted in observations.
Since the 1960s turbulence has been studied numerically as well. One early
study had a revolutionary impact. Lorenz ( 1963 ) discovered the profound effects of
very small changes in initial conditions on the behavior of a very simplified, three-
equation, nonlinear model of turbulent convection. He found that two solutions with
slightly different initial conditions divergedwith time. This sensitive dependence on
initial conditions is now recognized as a fundamental property of turbulence. Gleick
( 1987 ) describes Lorenz' findings as the beginning of the field now called chaos .
The advances in digital computers and numerical techniques for solving differ-
ential equations after Lorenz' early work soon allowed the numerical simulation
of turbulence. There are two varieties. Direct numerical simulation (DNS) is the
numerical solution of the governing fluid equations. It is (within the numerical
approximations used) exact, but it is possible only in low Reynolds number flows
(Problem 1.9) . The Orszag-Patterson (1972) 32 32 (32 3 ) calculation of

isotropic turbulence is considered the first DNS. Large-eddy simulation (LES) is an
approximate technique that solves for the largest-scale structure of turbulence fields;
its underlying concepts were laid out by Lilly ( 1967 ). Deardorff's (1970a) study of
turbulent channel flow on a 24

× 32 ×

× 14 × 20 grid mesh (6720 grid points) is widely

† According to Holmes et al . ( 1996 ), precise references to such remarks are elusive. They have been attributed
to Sommerfeld, Einstein, and Feynman, and beginning in 1895 Horace Lamb expressed similar sentiments in
his Hydrodynamics .

‡

Rouse and Ince ( 1957 ) state that this quote appears in the Carusi and Favaro ( 1924 ) republication of da Vinci's
writings.