t hc time isotropic) turbulence in an incompressible fluid; these conditions
arc approximately satisfied in a wind tunnel, but are very far from the real
situation in the atmosphere; near the ground the structure of the wind is
highly inhomogeneous and the vertical gradient of temperature (leading to
important variations of the density) plays a foremost role.
Thus one can oiily marvel at the number and importance of the results
that have been obtained during this pioneering period. The only theoretical
tools available then were the Reynolds’equations established in 1895 for the
turbulent motion of an incompressible fluid (Reynolds, 1895); the case of a
compressible fluid was. in a somewhat superficial ways touched upon for the
lirst time by Kellcr and Friedmann (1924): the complete equation taking
account of thc temperature fluctuations in il gas was given, for the first time
:is h r as I know, in 1937 in my paper (Kanipk de Feriet, 1937).
I t was J. Boussinesq (1877), who as carly as 1872 pointed out that the
flows which we call now ”turbulent,” are much too complicated to be completely described in all their details; using thc Eulerian point of view, every
quantityf(s. y, z, t ) , like pressure p , temperature T , density p , components u.
I-. M- of the velocity, has to be put in the form
fbeing the mean value andf’ the fluctuation of this quantity; only the mean
valueJcan be measured. The main problem for the foundation of a theory is
thus to devise a set of equations containing only the mean values of the
velocity, pressures, etc.
The situation is exactly the same as ignoring the individual motions of the
molecules, when we consider a gas or a liquid as a continuous medium; the
velocity u(.u. y, L. I ) . v(x. y. z, t ) , w(x, y, z, I ) of the continuous medium is
the nieilti value of the velocities (Urn, V , , W,) of the molecules present at time
I in thc neighborhood of (s. y, z ) ; the Navier-Stokes equations take account
of thc 1luctuiitit)ns of the velocity of the molecules (14 - CI”, c - V,, w - W,)
by thc viscosity terms.
The fundamental step was made by Reynolds (1895), who obtained a new
set or cquat ions by averaging the Navier-Stokes equations; the influence of
the fluctuations on the mean motion (r(, v, E) is expressed by the tensor of
the Reynolds stresses:
f(.u. J.9 2. t ) = f ( x , y, z, t ) + f ‘ ( x , y. 2, t )
-. .
T A X = -plc‘2,
Tx. = -pu‘c‘,
Txr = -pu’w’
-
t , = -pc’w’
-z
Tyy = - PC ,
_ _ -
T,,x = -/W’U’,
T,, = -()W‘U’,
T,,, = - p W ’ V ’ ,
T z z = - p W
_.
- -;5
The exact meaning of the process by which the mean valueJis obtained is
extremely important for correct use of the Reynolds equations; research on
this subject has developed along two different lines.
arc approximately satisfied in a wind tunnel, but are very far from the real
situation in the atmosphere; near the ground the structure of the wind is
highly inhomogeneous and the vertical gradient of temperature (leading to
important variations of the density) plays a foremost role.
Thus one can oiily marvel at the number and importance of the results
that have been obtained during this pioneering period. The only theoretical
tools available then were the Reynolds’equations established in 1895 for the
turbulent motion of an incompressible fluid (Reynolds, 1895); the case of a
compressible fluid was. in a somewhat superficial ways touched upon for the
lirst time by Kellcr and Friedmann (1924): the complete equation taking
account of thc temperature fluctuations in il gas was given, for the first time
:is h r as I know, in 1937 in my paper (Kanipk de Feriet, 1937).
I t was J. Boussinesq (1877), who as carly as 1872 pointed out that the
flows which we call now ”turbulent,” are much too complicated to be completely described in all their details; using thc Eulerian point of view, every
quantityf(s. y, z, t ) , like pressure p , temperature T , density p , components u.
I-. M- of the velocity, has to be put in the form
fbeing the mean value andf’ the fluctuation of this quantity; only the mean
valueJcan be measured. The main problem for the foundation of a theory is
thus to devise a set of equations containing only the mean values of the
velocity, pressures, etc.
The situation is exactly the same as ignoring the individual motions of the
molecules, when we consider a gas or a liquid as a continuous medium; the
velocity u(.u. y, L. I ) . v(x. y. z, t ) , w(x, y, z, I ) of the continuous medium is
the nieilti value of the velocities (Urn, V , , W,) of the molecules present at time
I in thc neighborhood of (s. y, z ) ; the Navier-Stokes equations take account
of thc 1luctuiitit)ns of the velocity of the molecules (14 - CI”, c - V,, w - W,)
by thc viscosity terms.
The fundamental step was made by Reynolds (1895), who obtained a new
set or cquat ions by averaging the Navier-Stokes equations; the influence of
the fluctuations on the mean motion (r(, v, E) is expressed by the tensor of
the Reynolds stresses:
f(.u. J.9 2. t ) = f ( x , y, z, t ) + f ‘ ( x , y. 2, t )
-. .
T A X = -plc‘2,
Tx. = -pu‘c‘,
Txr = -pu’w’
-
t , = -pc’w’
-z
Tyy = - PC ,
_ _ -
T,,x = -/W’U’,
T,, = -()W‘U’,
T,,, = - p W ’ V ’ ,
T z z = - p W
_.
- -;5
The exact meaning of the process by which the mean valueJis obtained is
extremely important for correct use of the Reynolds equations; research on
this subject has developed along two different lines.