4
J. KAMPE rx FERIET
11. Mtrth~wtr/rrd Poirrr 01 I’iew
Averaging the Nit\lier Stokes equations, Reynolds makes USC of m n c
properties of the mcan; it has been pointed out by Oseen in 1930 that neither
a time average over a finite interval of time nor a space average over a
bounded volume has the properties used by Reynolds. In 1935. 1
proved that the Reynolds equations are logical consequences of the NavierStokes equations if and only if the mean satisfies the following conditions:
(R2)
i$ = aT. 2 = const
(R,)
.& =T!j
(R, and R3 imply f = J thus 7 = 0)
(R4)
c’f/S~ = ~ J V X , I?~/JL. = aj./Sy,
. __
-
(ylaz = i w z , y/a = 2flat
This remark has lead since 1949 to a large number of purely mathematical
papers by Garrctt Birkhoff, Gian Carlo Rota, John Sopka, John Kelley, J. B.
Miller. and many others; except for trivial cases, it is extremely difficult
(Kampk de Feriet, 1946, 1949) to satisfy conditions (R,)-(R,); to obtain a
large class of solutions, one has to use a statistical interpretation, considering a function ,f(x, y, z, I ) as a sample of a random function f (x, y, z, r, a), a
Gibbsian ensemble of flows, o being chosen at rahdom in a given probability
space (n, .Up P ) ; with this interpretation the mean is considered as the statistical average (mathematical expectation):
.T(x. J’, z, t ) = j S(X, y, 2, t* a) dP
n
In Section 6 we will come back to this interpretation, which started at the
end of the period, slowly penetrated all theoretical research, and gave many
important results after 1945.
h. Experimenrul Point of View
Much more fruitful has been the discussion of the experimental data
collectcd about atmospheric turbulence: the clarification of the notion of the
scdo ofturhulencci, which emerged around 1930. is of paramount importance
for understanding turbulent atmospheric diffusion.
Every instrument is able to measure fluctuations in a fluid if they last more
than ; 1 time charactcriutic of this instrument; while in a given flow some
J. KAMPE rx FERIET
11. Mtrth~wtr/rrd Poirrr 01 I’iew
Averaging the Nit\lier Stokes equations, Reynolds makes USC of m n c
properties of the mcan; it has been pointed out by Oseen in 1930 that neither
a time average over a finite interval of time nor a space average over a
bounded volume has the properties used by Reynolds. In 1935. 1
proved that the Reynolds equations are logical consequences of the NavierStokes equations if and only if the mean satisfies the following conditions:
(R2)
i$ = aT. 2 = const
(R,)
.& =T!j
(R, and R3 imply f = J thus 7 = 0)
(R4)
c’f/S~ = ~ J V X , I?~/JL. = aj./Sy,
. __
-
(ylaz = i w z , y/a = 2flat
This remark has lead since 1949 to a large number of purely mathematical
papers by Garrctt Birkhoff, Gian Carlo Rota, John Sopka, John Kelley, J. B.
Miller. and many others; except for trivial cases, it is extremely difficult
(Kampk de Feriet, 1946, 1949) to satisfy conditions (R,)-(R,); to obtain a
large class of solutions, one has to use a statistical interpretation, considering a function ,f(x, y, z, I ) as a sample of a random function f (x, y, z, r, a), a
Gibbsian ensemble of flows, o being chosen at rahdom in a given probability
space (n, .Up P ) ; with this interpretation the mean is considered as the statistical average (mathematical expectation):
.T(x. J’, z, t ) = j S(X, y, 2, t* a) dP
n
In Section 6 we will come back to this interpretation, which started at the
end of the period, slowly penetrated all theoretical research, and gave many
important results after 1945.
h. Experimenrul Point of View
Much more fruitful has been the discussion of the experimental data
collectcd about atmospheric turbulence: the clarification of the notion of the
scdo ofturhulencci, which emerged around 1930. is of paramount importance
for understanding turbulent atmospheric diffusion.
Every instrument is able to measure fluctuations in a fluid if they last more
than ; 1 time charactcriutic of this instrument; while in a given flow some