The American Society of Mechanical Engineers

42 TRANSACTIONS OF T H E A.S.M.E. JANUARY, 1941
sions of a velocity times a length, and a good physical interpretation
of e is given by Bakhmeteff ( l).2 In fully developed
turbulent flow, except near a smooth boundary, the coefficient
pe is many times greater than thus the first term in Equation
[4] becomes quite small compared to the second term.
The rate a t which potential or pressure energy is used in producing
flow is equal to r(dU/dy) in foot-pounds per unit volume
of fluid. For viscous flow this becomes ti{dU/dy)2, and represents
the rate a t which the energy producing flow is dissipated
into heat a t any point in a fluid. In fully developed turbulent
flow, since the mean shear stress is not influenced by viscosity,
the quantity r(dU/dy) represents the rate at which the pressureenergy
producing flow is transformed into turbulence energy.
The turbulence energy is then dissipated into heat by the action
of viscosity on the small eddies. However, energy of turbulence
is not necessarily dissipated a t the point where it is created;
it can and is diffused by the mixing action of the turbulence.
For instance in a uniform conduit, the term r(dU/dy) is greatest
near the boundaries and is zero at the center. The energy of
turbulence produced a t the boundaries is in p art diffused to the
center of the conduit and dissipated there. Of course it should
be remembered that, in nonchanging uniform flow, the total rate
a t which energy of turbulence is created is equal to the rate of
dissipation of this energy by the action of viscosity.
Lamb (5) shows th a t the general equation for the rate of dissipation
of energy per unit volume in a viscous fluid is
In isotropic turbulence U, V, and IF can be replaced by u, v,
and w. Taylor simplifies Equation [5] by demonstrating the
interrelationship existing between the various mean squares
and mean products of the velocity gradients, and obtains
E n e r g y C o n s i d e r a t i o n s f o r I s o t r o p i c T u r b u l e n c e
Because of the complexity of the turbulence mechanism, any
theoretical considerations m ust start w ith the simplest conditions.
One of the types of turbulence th a t has received extensive
consideration by physicists interested in wind-tunnel turbulence
phenomena is th a t referred to as isotropic turbulence.
This type of turbulence is such th a t (a) the mean values of
squares and products of the fluctuating velocity components,
such as v2 and uv, and their derivatives, such
/d»v
W and
/ dv du\
are independent of the location of the point observed,
\cto by)
and (b) the same mean values are obtained if the axes of reference
are rotated or reflected. Fluids having isotropic turbulence
can have no mean shear or mean-pressure gradient, th a t is,
dU/dy = 0, dU/dx — 0, etc.
In practical hydraulics, isotropic turbulence is hardly ever
attained; however, it is sometimes approached. The turbulence
in the center of closed conduits is nearly isotropic; also, the
turbulence formed downstream from various turbulence-producing
devices such as screens, grids, expansions, and flow-control
apparatus tends to approach isotropy a t times. As G. I. Taylor
remarks, “there is a strong tendency to isotropy in turbulent
motion.” A study of the energy dissipation characteristics of
isotropic turbulence may throw considerable light on the more
complicated turbulence obtained in many practical hydraulics
problems.
The intensity of turbulence is usually designated by the ratios
"v/w2/ U, y/v2/U, and "v/w2/ U, and for isotropic turbulence
these are all equal. Investigations by Dryden (2) and others reveal
th at these ratios a t any point beyond any particular turbulence-producing
device tend to be independentof the mean velocity.
This relationship also appears to hold for nonisotropic turbulence
a t high Reynolds numbers. The decrease in the intensity of
the turbulence beyond screens and grids in wind tunnels, or
the decay of the turbulence, has been studied quite extensively
both theoretically and experimentally by Taylor (3), von K&rm&n
(4), and Dryden (2).
2 Numbers in parentheses refer to the Bibliography at the end of
the paper.
where X is a length proportional to the small eddies present since
they are primarily responsible for the dissipation of the turbulence
energy. Possible methods of experimentally determining
v2 and X were discussed by the author in a previous paper (6).
The mechanism by which the small eddies are produced from
the larger ones is a fundam ental problem of turbulence about
which little is known. I t is these small eddies, referred to sometimes
as the microturbulence, which are largely responsible for
the high rate of energy dissipation associated with turbulent
flow.
The scale of the turbulence, L, which is im portant in regard
to the diffusive action of the turbulence, is proportional to the
average size of the eddies. The product \ / v2L is proportional
to the transverse diffusion coefficient e, as used in Equation [4],
As the turbulence created by some obstruction in a fluid stream
is dissipated downstream, a change in the length factors X and
L takes place which is a characteristic phenomenon of decaying
turbulence. Qualitative visual observations of the turbulence
in w ater streams beyond grids, throttled valves, sudden expansions,
etc., seem to indicate th a t the average size of the eddies
tends to increase as the turbulence is dissipated.
The internal stresses in turbulent flow are proportional to the
product of the density and the mean square of the fluctuating
velocities such as pu1. Any such force will then dissipate energy
at a rate proportional to the product of the force and the associated
velocity, thus, p(tt')3, where u' = \ / w 2. The total area
on which these forces act will be proportional to the square of
the scale of the eddy system, or to L2, and the rate of dissipap
M 3
tion per unit volume will then be proportional to For
isotropic turbulence, this quantity should then be proportional
to the rate of energy dissipation given by Equation [6]. The
following pronortionalitv can then he writ,ten
The terms under the radical in Equation [8] have been referred