The American Society of Mechanical Engineers

T urbulence an d E n erg y D issipation
T his paper incorporates a study o f th e origin and d issipation
o f turbulence energy w hich m akes possible a better
understanding o f th e m echanics o f energy losses th a t are
introduced by various flow -disturbing devices such as
expansions, bends, valves, etc. T he im p ortan t param eters
w hich characterize turbulence are th e root-m ean-square
values o f the fluctuatin g velocity com p on en ts, th e len gth
factor proportional to th e size o f th e sm all eddies responsible
for th e dissipation o f energy, and th e len gth
factor proportional to the average size o f th e eddies. The
effect o f variation o f th ese param eters on th e energy losses
occurring in turbulent flow are discussed, and also th e
change in these param eters in th e decaying turbulence
beyond turbulence-producing devices is in dicated. D ata
are presented show ing th e variation in th e k in etic energy
o f m ean flow and th e turbulence energy in a 15-deg conical
divergence. Visual studies o f th e start o f turbulence at
rounded entrances to sm ooth conduits seem to in dicate
that there is a regular vortex form ation a t th e boundary,
and th e dispersion o f th ese vortexes in to th e m ain fluid
stream gradually establishes norm al tu rb u len t flow.
T
H E flow of all real fluids involves energy loss due to the
frictional resistance of the fluid. The terms “energy loss”
or “energy dissipation” are used in this paper to describe
the eventual changing into heat of the energy producing flow.
Energy dissipation in a continuous fluid for either lam inar or
turbulent flow is due to the action of the viscosity of the fluid,
and such terms as “shock loss” or “im pact loss” do not describe
correctly how energy losses occur in continuous fluids.
Hydrodynamics indicates quite definitely how energy is
dissipated in a fluid in viscous flow. The rate a t which energy
is dissipated in viscous flow can be predicted from a knowledge
of the flow pattern and the characteristics of the fluid. However,
this is not the case for turbulent flow. I t is ordinarily
possible by the use of semiempirical formulas to calculate the
total over-all pressure drop and thus the total rate of energy loss,
however, we know very little about the mechanics of how this
energy loss takes place in the turbulent fluid itself. The purpose
of this paper is to analyze more thoroughly the mechanics of
energy dissipation in a turbulent fluid, and to relate such analyses
to various practical hydraulics problems. Liberal use will be
made of the fundamental ideas regarding the mechanism of
turbulence which have been put forth by L. Prandtl, G. I. Taylor,
Th. von K&rm&n, and their co-workers. The principles of the
statistical theory of turbulence will be extended to the practical
problems of energy dissipation in liquid flow.
F u n d a m e n t a l C o n c e p t i o n s
In order th at the meaning of various terms and expressions
used in this paper may be clearer they will be defined at this
stage.
By A. A. K A LIN SK E,1 IOWA CITY, IOWA
1 Assistant Professor of Hydraulics, College of Engineering, State
University of Iowa, and Research Engineer, Iowa Institute of Hydraulic
Research.
Contributed by the Hydraulic Division and presented at the
Semi-Annual Meeting, Milwaukee, Wis., June 17-20, 1940, of T h e
A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s .
N o t e : Statements and opinions advanced in papers are to be
understood as individual expressions of their authors, and not those
of the Society.
41
When a fluid flows in a straight conduit a t distances far enough
beyond bends or transition sections, so-called normal conditions
become established. In such a case, the am ount and nature
of turbulence present is unvarying and the shape of the meanvelocity
distribution remains constant. For this condition of
flow, the rate of creation of turbulence is in equilibrium with its
rate of dissipation. The kinetic energy of mean flow per pound
of fluid flowing is ordinarily designated as Um2/2g, where Um is
the mean velocity in the cross section. The total kinetic energy
of the mean flow is then equal to QyUm2/2g where Q is the discharge
and y is the specific weight. Because of the variation
in mean velocity, all particles of fluid passing any section do not
have the same kinetic energy, therefore the foregoing expression
can only be exactly correct for uniform velocity distribution.
For a circular cross section, the correct expression for the total
kinetic energy of mean flow is
where p =
V =
U =
unit density of fluid
distance from center
radius of cross section
mean velocity with respect to time a t point y
In true turbulent flow, the velocity a t a point varies irregularly
with time in direction and magnitude. However, in vortex motion,
as for example in the von Kdrm&n vortex street produced beyond
a body immersed in a flowing fluid, the variation in the velocity
vector with time is quite regular. Whenever the term “turbulence”
is used it will refer to a condition of flow where there is no
regularity in the variation of the direction and magnitude of the
velocity vector, except in the probability sense. I t is convenient
to designate the velocity vector a t any instant by the components
U, V, and W along the x, y, and z axes, respectively.
The x axis is in the direction of mean flow, the y axis is normal
to the mean flow, and the z axis is the other normal axis.
The kinetic energy per pound of fluid a t any instant is actually
equal to U2/2g. Separate the fluctuating component U into two
parts, such th a t U = U + u, where u is the fluctuating part.
Obviously, u = 0; (the bar indicating an arithm etic mean).
Thus the mean kinetic energy is in reality equal to (U + u)2/2g
or to ( t /2 + u2)/2g and the p art u2/2g is called the kinetic energy
of turbulence. Therefore, the total kinetic energy due to the
turbulence per pound of fluid will be
The total kinetic energy of turbulence for the fluid flowing in a
circular conduit will then be
E, = ttp U(u2 + v2 + w2)ydy [3]
Another im portant fundam ental idea is the concept of shear
in a moving fluid. For viscous flow the shear per unit area at
any point is equal to n dU/dy, or the product of the coefficient of
viscosity and the velocity gradient. The mean shear in a tu r­
bulent fluid is equal to
The term e is the so-called diffusion coefficient having the dimen-