ILL IN I
S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin Series No. 413
TRANSPORT OF MOMENTUM, MASS, AND HEAT
IN TURBULENT JETS
LLOYD G. ALEXANDER
formerly Research Assistant
Professor of Chemical
Engineering
THOMAS BARON
formerly Assistant Professor
of Chemical Engineering
EDWARD W. COMINGS
formerly Professor of
Chemical Engineering
Published by the University of Illinois, Urbana
3050--5-53-52523- H NV-E sr
:: PRESS ::
CONTENTS
I. INTRODUCTION 7
I. General Problem of Turbulent Transport 7
2. Method of Investigation 8
3. Experimental Program and Description of
Systems Studied 8
4. Acknowledgements II
5. Notation II
II. MATHEMATICAL BACKGROUND AND PREVIOUS CONTRIBUTIONS 16
6. Transport Equations 16
7. Phenomenological Theories 18
a. Prandtl's Model 18
b. Taylor's Model 24
8. Empirical Theories 29
a. Boussinesq's Hypothesis 29
b. Reichardt's Hypothesis 31
III. THEORETICAL ANALYSIS OF FLOW IN FREE JETS 33
9. Generalization of Reichardt's Hypothesis 33
10. Principle of Superposition 34
II. Distribution of Momentum Flux 35
12. Distribution of Flux of Sensible Heat 37
13. Distribution of Mass Flux 38
IV. EXPERIMENTAL INVESTIGATION 39
14. Apparatus 39
15. Calibration of Instruments 42
CONTENTS (Concluded)
16. Total-Head Impact Tube
a. Calibration
b. Effect of Angle of Attack
17. Sii
c. Effect of Turbulence
ngle Free Jets
a. Transport of Momentum Flux
b. Effect of Similarity Conditions
on Distribution of Momentum Flux
c. Effect of Induced Turbulence on
Spread of Jets
d. Transport of SensiblaeHeat Flux
e. Transport of Mass Flux
18. Application of the Principle of Superposition
to Jets
a. Single Free Jets
b. Two Parallel Jets
c. Single Jet Discharging Parallel to
a Plane
19. Applications of Transport in Jets
V. SUMMARY
APPENDIX: BIBLIOGRAPHY
F IGURES
1. Zones of Flow and Momentum Flux Profiles in Free Jets 9
2. Schematic Diagram of Experimental Apparatus 39
3. Air Compressor 40
4. Flow Control and Metering System 41
5. Cross-section of the 0.898-in. Semi-Streamlined Nozzle 42
6. Exterior View of the 0.898-in. Nozzle 43
7. Total-Head Impact Tube Coefficients 46
8. Cross-section of the 0.75-in. Nozzle with Screen and
0.25-in. Throat Extension 49
9. Comparison of Radial Profiles of Momentum Flux Density
Ratio in a Single Free Jet with the Probability
Distribution 51
10. Geometric Similarity: Variation of Negative Logarithm
of Axial Momentum Flux Density Ratio with Square of
Radial Distance Ratio 54
II. Dynamic Similarity: Variation of Axial Momentum Flux
Density Ratio with Velocity of Discharge at Nozzle 57
12. Dimensional Similarity: Variation of Axial Momentum
Flux Density Ratio with Nozzle Diameter 58
13. Radial Profiles of Momentum Flux Density Ratio in a Free
Jet Discharged Through a Nozzle with and without Screen 62
14. Heat and Mass Flux Density Ratio Profiles in a Free Jet,
from Hinze's Measurements 64
15. Radial Profiles of Momentum Flux Density Ratio Adjacent
to a Finite Nozzle 67
16. Axial Profile of Momentum Flux Density Ratio Adjacent to
a Finite Nozzle 73
17. Parallel Nozzles 75
18. Radial Profiles of Momentum Flux Density Ratio in Two
Parallel Jets 76
19. Radial Profiles of Momentum Flux Density Ratio in Single
Jet Discharaed Parallel to Plane Surface
TABLES
I. Radial Profiles of Momentum Flux Density Ratio from Single
Free Jet: Axia.l Distance Ratio Varying 52
2. Radial Profiles of Momentum Flux Density Ratio from Single
Free Jet: Discharge Velocity Varying 53
3. Geometric Similarity: Variation of Axial Momentum Flux
Density Ratio with Radial Distance Ratio 55
4. Dynamic Similarity: Variation of Axial Momentum Flux
Density Ratio with Velocity at Nozzle 56
5. Dimensional Similarity: Variation of Axial Momentum Flux
Density Ratio with Diameter of Nozzle 58
6. Radial Profiles of Axial Momentum Flux Density Ratio in a
Free Jet Discharged Through a Nozzle with and without
Screen 60
7. Radial Profiles of Momentum Flux Density Ratio Adjacent to
a Finite Nozzle 68
8. Values of Momentum Flux Density Ratio Calculated by
Graphical Integration of Eq. 92 (Cm = 0.075) 72
9. Axial Profile of Momentum Flux Density Ratio Adjacent to
a Finite Nozzle 74
10. Radial Profiles of Momentum Flux Density Ratio in Two
Parallel Jets 77
II. Radial Profiles of Momentum Flux Density Ratio in Single
Jet Discharged Parallel to Plane Surface
I. INTRODUCTION
I. General Problem of Turbulent Transport
In the transfer of momentum, mass, and heat from turbu-
lent fluid streams to solid boundaries, the major resistance
has been thought to reside in a laminar sublayer adjacent
to the surfaces. The large number of semi-empirical and
theoretical equations available for such cases involve the
concept of molecular transfer through the laminar sublayer,
with suitable corrections to account for the resistances in
the transition layer and in the turbulent core.
In another class of transport problems there are no
solid boundaries, and the entire resistance to transport of
momentum, mass, and heat is in the turbulent fluid. For
instance, in a gas burner a combustible gas issues from a
nozzle and entrains and mixes with air. The position of
the flame is determined by the design of the combustion
chamber, the turbulent transfer properties, and the rates
of the chemical reactions involved. Similar considerations
apply to the performance of combustion chambers using
atomized or powdered fuel. In general the problem of
transport in free turbulence is common to such operations
as atomizing, spray drying, combustion, and jet injection.
Combustion in a turbulent flame involves the turbulent
transport of heat, mass, and momentum, the generation of
heat, and the formation of the products of the combustion
reaction. The local rates of heat release and mass conver-
sion are functions of the temperature and of the composi-
tion, which in turn are governed in part by the transport
properties. Experiments in the complex situations just
mentioned would probably not yield an understanding of
the basic physical principles underlying the process.
Instead it seems best to study simple cases and to synthe-
size the general case from the results of experiments
on its elements.
Accordingly, the first case chosen for study was the
transport of momentum in an isothermal, turbulent, free
jet of air discharging at subsonic velocities into air
of the same density. The study of this case involves
several theoretical treatments and the presentation of
much experimental data.
ILLINOIS ENGINEERING EXPERIMENT STATION
The next steps would be to introduce temperature gradi-
ents, then density gradients, then both simultaneously,
and then finally to include a chemical reaction involving
heat release.
2. Method of Investigation
The purpose of the work reported in this bulletin was
to obtain and correlate information on mixing in fluid
streams. Particular emphasis was placed on the measurement
of momentum transport in turbulent flow, with the object of
correlating this information, together with other investi-
gators' measurements of mass and sensible heat transport,
in a form which will have general engineering utility.
Consideration was given first to turbulent mixing in
free jets. Of the available theories of turbulent trans-
port, that of Peichardt(581* was chosen as involving the
fewest objectionable assumptions and as offering most
promise of extension. It was discovered that the assump-
tion embodied in Reichardt's formula for turbulent shear
stress also linearizes one of the equations of motion when
applied to free jets. This important result makes possible
the mathematical solution of problems involving complicated
boundary conditions by the method of superposition of
particular solutions of the linear equation. The pre-
dictions resulting from the mathematical solutions are
compared with the results of experiments performed in
this laboratory.
Heichardt's hypothesis was also extended to cover the
cases of the transport of mass and heat in free jets. The
resulting equations are also linear, and their application
to the correlation of experimental data is considered.
The current theories of turbulent flow as applied to
free jets are reviewed. To compare the assumptions and
range of application of each of these and to clarify
certain statements found in the literature, it seemed
desirable to develop the theories on a more comparable
basis than was done by the original authors. In some cases
this procedure required slight deviations and extensions
from the treatment presented by the original authors. Some
unstated assumptions which seem implicit in their original
treatments are also mentioned.
3. Experimental Program and Description of Systems Studied
The experiments reported herein dealt with the transport
of momentum in isothermal, turbulent, free jets of air
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
discharging at subsonic velocities into stagnant air.
The nozzles used were designed to provide a flow of uni-
form velocity over the discharge opening and a low level
of turbulence.
In such a jet there are several well defined regions, as
shown in Fig. 1. The nonturbulent core is a cone-shaped
region having its base on the nozzle opening and its apex
-4-
4~)
~li~
Axial Distance Ratio, X/d
Fig. I. Zones of Flow and Momentum Flux Profiles in Free Jets
on the axis of flow about 4 nozzle diameters from the
nozzle. In this region the intensity of the turbulence is
low; the velocity is substantially uniform and equal to the
velocity at the discharge. Surrounding this core is a
wedge-shaped annular region with its apex at the lip of the
nozzle. Here the flow is highly turbulent and the velocity
decreases rapidly away from the nonturbulent core.
In the region of flow more than 10 nozzle diameters from
the nozzle the turbulence is said to be fully developed,
ILLINOIS ENGINEERING EXPERIMENT STATION
that is the profiles of the velocity versus the radius are
similar at successive sections taken perpendicular to the
axis of flow. The relative intensity of the turbulence may
become 20 percent at some points in this region; it varies
in a complicated manner. (20 Between the nonturbulent
core and the region of fully d.eveloped turbulence is a
transition region.
The many types of mathematical functions that have been
proposed for fitting the profiles of velocity in the region
of fully developed turbulence include power series,170)
trigonometric series, (6 and the probability function. (2
This last is the most popular and involves the fewest
arbitrary constants.(58)
The locus of points in the jet at which the momentum-
flux density is one-half the value of that on the axis of
flow defines the half-momentum-flux angle. This is an
angle of approximately 10 deg, whose vertex lies near but
not at the nozzle discharge.
Two types of single free jets have been considered--
the two-dimensional jet which issues from a long, narrow
slot, and the round jet which issues from a circular
nozzle. The latter type was used in the experiments
reported in this bulletin. The results of these experi-
ments indicate that all such free jets are essentially
similar when compared on a dimensionless basis involving
ratios of velocities and ratios of dimensions in the jet
field and the nozzle diameter. This finding conforms with
general jet theory.
Measurements were also made in the turbulent flow field
adjacent to a nozzle discharge, in the field formed by the
confluence of two parallel jets, and in the field formed by
the discharge of a free jet parallel to a plane surface.
Advantage was taken of the linearization of the equation of
motion by Heichardt's hypothesis to predict the distribu-
tion of momentum in these several cases of free turbulent
flow with complicated boundary conditions. The predictions
are compared with the experimental results in Section 19.
Although the experiments were restricted to the trans-
port of momentum, the extension of Peichardt's hypothesis
to heat and mass transfer is applied to correlating heat
and mass transfer data presented by other investigators.
The experimental data were obtained by measurements with
a total-head impact tube which was mounted in a special
traversing mechanism permitting precise measurement of the
position of the probe relative to the discharge nozzle.
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
4. Acknowledgments
The investigation formed a part of the research program
of the Engineering Experiment Station. Certain studies
included in the investigation were thesis projects for
advanced degrees in chemical engineering in the Graduate
College of the University of Illinois. The experiments on
effect of velocity changes and geometric similarity in a
free jet were performed by J. F. Taylor, (68) those concern-
ing nozzle diameter were performed by E. E. Polomik, 50)
and the two series of experiments concerning the flow field
of parallel jets and the jet discharging parallel to a
plane surface were performed by H. L. Grimmett. 32" The
research on the re'sponse of a total-head impact tube as
related to turbulence and the effect of induced turbulence
on the spread of a free jet was part of the doctoral thesis
of J. F. Taylor. (68 The assistance of S. A. Phillips,
mechanic in the chemical engineering division of the
chemistry department, is also gratefully acknowledged.
5. Notation
A0 Discharge area of flow nozzle
a Distance of equivalent point source from
nozzle
a0 Proportionality constant in Eq. 43
bV, bh, bm, b
Jet breadth parameter
b0 Proportionality constant in Eq. 46a
b1 Term in Eq. 46a to account for transport
of heat or mass in the x-direction by
the longitudinal fluctuations of velocity
bH Term in Eq. 44a to account for transport
of momentum in the x-direction by the
longitudinal fluctuations of velocity
C , Ch, C,, C,
Spreading coefficient defined by Eq. 58
Cf Impact tube flow coefficient defined by Eq. 76
C Specific heat of fluid
P
Tollmien's spreading coefficient defined by
Eq. 24
Eq. 24
ILLINOIS ENGINEERING EXPERIMENT STATION
D Distance between two parallel nozzles, or
twice the distance between a nozzle and
a plane parallel to its axis
DM Mass Diffusivity
D Generalized symbol for molecular diffusion
constant of heat, mass or momentum
d Diameter of flow nozzle
E Thermal expansion factor (see reference 3)
F , Fh Fraction of flux (mass or heat) transported
by mean-flow components
G Constant used in Eq. 66
hi Pressure drop across standard flow nozzle,
in. of water
i, j, k Unit vectors in the x, y, z directions, or
in the x, r, c directions respectively
K Flow coefficient in Eq. 73, includes nozzle
diameter
KR Strength of a free jet, Eq. 47
K Term to correct for the nonuniformity of
the velocity distribution at the nozzle
Ky Term to account for deviation from a uni-
form concentration or temperature dis-
tribution at the nozzle
k Thermal conductivity, heat units per unit
area per unit temperature difference per
unit length
L' , L, L' Vector mixing length for momentum, heat
and mass, and vorticity
I' , I' , ' Scalar components of L' in cylindrical
coordinates
U"' p c' w Magnitude of L' under conditions of iso-
tropic turbulence
lp , lp Prandtl's mixing length defined by Eqs. 20
1T Taylor's mixing length defined by Eq. 38
M Molecular weight
N Unit vector directed outward normal to
surface
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
P Pressure
PO Stagnation pressure
P1 Pressure upstream from standard flow nozzle,
psia
AP Pressure recorded by impact tube manometer
Q Generalized symbol for local rate of gen-
eration of heat, mass, or momentum
Qh Local rate of generation of sensible heat,
heat units per unit of volume per unit of
time
Qi Local rate of generation of i-th component
of fluid, mass units per unit volume per
unit time
QsCfs Discharge capacity of standard flow nozzle
in cu ft per sec measured at 60 deg F and
14.7 psia
R Universal gas constant. Also a vector dis-
tance
RPu, Rp9, RT Correlation coefficients defined by Eqs. 18
and 37
r, p Polar coordinates in any plane perpendicular
to axis of flow
S Unit vector in any given direction
s, y Polar coordinates in the plane of the nozzle
orifice
T Absolute temperature, deg R
T7 Temperature upstream from standard flow nozzle,
deg B
t Difference in temperature between that of a
point in a free jet and the ambient tem-
perature, t = (Txy,z-Ta)
u In both rectangular and cylindrical coor-
dinates: Instantaneous velocity paral-
lel to x-axis
V Velocity vector (components u, v, and w)
v In rectangular coordinates: Instantaneous
velocity parallel to the y-axis. In
cylindrical coordinates: Instantaneous
velocity in the radial direction
ILLINOIS ENGINEERING EXPERIMENT STATION
r, 2
Y, Z
ao Po ' I1
6p, 6T, 6B
A
p1
T
I
In rectangular coordinates: Instantaneous
velocity parallel to the z-axis. In
cylindrical coordinates: Instantaneous
angular velocity
Concentration of i-th component in fluid,
mass units per unit mass of fluid
Cylindrical coordinates
Bectangular coordinates
Any scalar or vector function of the space
and time coordinates
Constants in Eqs. 45 and 46
"Eddy diffusivity" of heat or mass defined by
equations following Eq. 27
"Eddy viscosity" defined by Eqs. 15, 35
and 43
Base of the natural logarithms
z or ¢ component of w, the vorticity vector
y or r component of ý.
Time
Proportionality function defined by Eq. 50
Viscosity of fluid, force times time per
unit length squared
x component of w
Density of fluid, mass per unit volume
Density of fluid upstream from standard flow
nozzle, lb per cu ft
Summation operator, sigma
Surface of an arbitrary volume, 7
Arbitrary volume in a flow field
Generalized symbol for molecular transport
potential. Also compressibility factor
in Eq. 73 (see reference 3)
Generalized symbol for concentration of
sensible heat, mass, or momentum
Vorticity vector (Components ý,7,ý)
Vector differential operator, del
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
Line above -symbol indicates a time average
2 1 20 2
2 = E; udO]2
primed symbol denotes a fluctuating component
The following subscript notation is also used:
a Ambient conditions
dA Conditions at an elemental point source jet
h Sensible heat
j j-th point source jet
m Momentum
s Streamline of flow
t Temperature
w Mass
x,r Conditions at points (x,r) in jet, any k
x,O Conditions at point x on axis of flow
0 or 0,0 Conditions at discharge of nozzle
Y2 Badius at which a flow variable has one-half
the value on axis of flow
II. MATHEMATICAL BACKGROUND AND PREVIOUS
CONTRIBUTIONS
6. Transport Equations
A generalized differential equation for the flux of mass,
heat, or momentum may be obtained by equating the rate
of accumulation in a volume 7 to the rate of generation in
the volume plus the rate of transport through the surface
of the volume a. The rate of transport may be considered
as that resulting from molecular motion plus that from
the bulk motion of the fluid. Let T denote the concentra-
tion of mass, heat, or momentum, as the case may be; then
the rate of accumulation within the volume is
Sfdr7 = f Qdr + f DV4-Ndo, - J TV'Nda (1)
where the first term on the right represents the rate of
generation within the volume r and the second and third
terms represent the net rate of transport across the bound-
aries of the volume by molecular processes and by motion of
the fluid, respectively. For the three cases the general-
ized variables, T, Q, D and $, are interpreted as shown in
the table below.
Interpretation of Generalized Variables
Q D
Mass flux pw D. pw i
Heat flux pCt Qh k t
Momentum flux pV-S -VP-S 1 V'S
Provided a volume exists which on the one hand is
sufficiently small that all the flow parameters differ from
their space-mean values only by second-order differences
throughout the volume and which on the other hand is
sufficiently large that the statistical properties of
velocity, pressure, etc., are defined, Eq. 1 may be trans-
formed by the divergence theorem, to
T= Q VDVP - V' (2)
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
For application to free turbulent jets it is now assumed
that the pressure is uniform throughout the flow field,
that there are no chemical transformations, no heat release
or absorption by a chemical or physical process, and that
the amount of sensible heat formed by conversion from
kinetic energy is negligible. Under these conditions the
generation term Q in Eq. 2 vanishes. Furthermore, under
most conditions of free turbulence, the molecular transport
term is negligible. Hence, for the quasi-steady state of
steady turbulent flow (i.e., no change with time in the
mean flow parameters)
V~ V = 0 (3)
The bar denotes the operation of time-averaging:
= lim 1 j"+O Zdo
0c7 eo0/
where Z is any fluctuating variable and where it is sup-
posed that the value obtained for Z is independent of the
arbitrary reference time e0. Hence
-a= 0
A special case of Eq. 3 is now developed. Substituting
pwi for T and summing over all components present,
V*(ýpww)V = 0
iw. = 1
whence
V'pV = 0 (4)
which is the familiar equation of continuity. Introducing
the concept of resolving the flow parameters into time-mean
and fluctuating components, 30) let
V = u + v + w = V + V'(vector addition)
S= u + u'
v = V + v
w= W + WV
p = p + p
ILLINOIS ENGINEERING EXPERIMENT STATION
where, by definition
V » 0
u' = 0, etc. (6)
For the special case of an incompressible, homogeneous
fluid in isothermal flow (flow with uniform density), T and
p do not fluctuate with time.
p' = 0
V'pv = 0 (7)
Expanding Eq. 4 for the instantaneous case (i.e., Eq. 4
with bar omitted)
V.pV, + V*pV' + V-p'V + V-p'V' = 0 (8)
Applying Eqs. 6 and 7 for the case of flow with uniform
density,
pV'V' = 0 (9)
For use in later arguments, consider now the product
ZV'pV, where Z is any fluctuating property of the fluid.
Let Z = Z + Z' and expand the above product by the aid of
this relation and Eq. 5. Now applying the same arguments
which led to Eq. 9, it follows that
ZV-pV = 0 (10)
for incompressible flow with uniform density.
7. Phenomenological Theories
a. Prandtl's Model(51)
In developing the mathematical analysis of isothermal
free jets, the following restrictive assumptions are
frequently made.
1. The mean pressure Pand the mean density p are
uniform throughout the flow field; hence their
derivatives are zero
2. The jet and ambient fluids are incompressible and
have the same density and temperature throughout;
hence p' is always zero
The second is an important assumption, for if p' assumes
finite values, Eqs. 7, 9, and 10 are only approximate. For
example if p' is not zero, then the ideal gas law is not
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
satisfied by the mean components of pressure, temperature,
and density. The law may be written
p .- T
P-7T
Substituting the mean and fluctuating components and
obtaining a time average of the resulting expression gives
P = R(pT + p'T')
M
Fluctuations in density and temperature due to pressure
fluctuations are always of the same sign; hence the product
p'T' never assumes negative values, and the mean value
therefore cannot be zero unless p' is always zero.
Consider a free turbulent jet issuing from a point
source into an ambient fluid having the same density and
temperature as the jet fluid, and choose a system of
cylindrical coordinates (x,r,o) with the jet discharging
along the x-axis. For the x-directed momentum flux, Eq. 3
becomes
V-puV = 0 (11)
Applying the two assumptions listed above, together with
Eq. 8, this becomes
V.Vu =- V.u'V' (12)
Ey introducing the concept of mixing length, Prandtl(511
deduced a relationship between the u'V' product and the
mean flow parameters and their derivatives. Though Prandtl
originally treated the case where the turbulent motion
as well as the mean motion is confined to two dimensions,
his method is readily extended to cover motion in three
dimensions, as follows: It is assumed that momentum
is conserved, so that a parcel of flui-d at the point
(x ,r ,# ) having the mean momentum associated with the
flow at that point is conceived to move without alteration
of its original momentum along a vector path L' to a mixing
point (x,r,04). Because of the turbulent motion, L' is
seldom parallel to the mean velocity vector at (x ,r ,4 ).
Further, where there are velocity gradients, the mean
velocities at the two points will, in general, be different.
The parcel of fluid, therefore, has an excess or deficiency
of momentum, and upon mixing with its surroundings at
(x,r,o) causes a fluctuation in the velocity at that point
ILLINOIS ENGINEERING EXPERIMENT STATION
proportional to the difference between the mean momentums
at the two points. That is, for x-directed momentum
pu' = pAu = P[lUx,r,- Ux ,r ,4
The model may be analyzed in the following way. The
increment, Au, may be expanded in terms of the derivatives
of u in a Taylor series, and if all terms of higher order
are neglected,
Au = u' (13)
where L' is the vector path along which mixing occurs to
produce u'. The vector mixing length L' is represented as
a fluctuation, since it has significance only in connection
with a fluctuating quantity such as u'. Substituting into
Eq. 12
V*7u = V-(L' *Vu)V' (14)
L' and V' may be expanded in terms of their scalar compo-
nents. To obtain Prandtl's result, it is assumed that the
turbulence intensity and mixing length are isotropic, in
which case the parallel products are all equal.
i' ' = i' v' = ' ' = EP (15)
xu 1 c w = (p
where ep is defined by this relation; it is commonly called
the eddy viscosity.
It is reasonable to assume that to a first degree of
approximation all cross products are zero.(30°
I xu = w = 1' ru = 1 w = 1 u = 1h ' I v' = 0
(16)
Substituting into Eq. 14,
V'Vu = V-e Vu (17)
P
A relation between Ep and the flow parameters is required
to solve Eq. 17. Let a correlation coefficient be defined
such that
1' v
ru
Rp = _ (18)
S (l' 2 v2) (
ru
In isotropic turbulence u'2 = v'2 = w'2. Combining this
and Eqs. 13, 15 and 18
p = Rp, [1'r 2(L'u .V )2 Y] (19)
U
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
Again, in isotropic turbulence, it is assumed that
1' 1' = l' l' = l' 1' = 0
xu ru xu =t ru ou
and
T' 2 = 'o 2 = l' 2 = l' 2
xu ru u u
where 1'u is defined by this last relation. Hence
ep = Rp 1'2 [(V )23]
u
or
E= l V (20)
U
where 1p is the Prandtl mixing length and contains (impli-
U
citly according to Prandtl's original derivation) the
correlation coefficient Rp Substituting into Eq. 17
U
V.Vu = V[lp 2 ) (21)
Imposing the condition of isotropy does not seriously
limit the application of Prandtl's result. For turbulent
shearing stresses to exist it is not necessary that the
mean-square fluctuating components differ from each other;
it is sufficient that the correlation coefficient in Eq. 18
be greater than zero. A reasonable first approximation is
that the difference between the mean fluctuating components
is of secondary importance compared to the degree of
correlation in determining the shearing stress.
Equation 21 is a generalization of Prandtl's early
mixing length theory. His result is equivalent to the
hypothesis that
Ep = 2r (22)
which applies strictly when the turbulent motion is confined
to two dimensions. The quantity |Vu| in Eqs. 20 and
21 reduces to 3u/3r if the terms u/f3x and (1/r)(Bu/3) can
be neglected. For axially symmetrical flow in free jets,
where 3u/fa is zero and Bu/3x can usually be neglected
compared to 3u/3r, Eq. 21 becomes
V-Vu =i--[rl 2L I (-)] (23)
r 3r u r 3r
ILLINOIS ENGINEERING EXPERIMENT STATION
which is the form given by Prandtl and applied by Tollmien
to flow in free jets.(70) The utility of Eq. 23 depends
on the existence of simple relations between lp and the
u
boundary conditions for a particular case. For flow near a
plane surface Prandtl supposed that Ip is proportional to
the distance from the surface, ( and for flow in jets or
wakes he assumed that lp was independent of r and was
proportional to the breadth of the turbulent zone. 53'
The spread of free jets is approximately linear with the
distance from the nozzle discharge, 170 and the radial
distribution of the momentum at successive sections through
the jet perpendicular to the x-axis are similar. Hence,
Tollmien t70) assumed
lp = CTx (24)
u
and
Ur= f-r) (25)
xu x
Corrsin '19) has shown that these two assumptions are not
independent, if it is assumed that the distributions of
both the mean and fluctuating motions are dynamically
similar at successive sections.
Tollmien(70) further assumed that the quantity 3u'2/3x
is negligible compared to (1/r)(3u'v'/3r) and that the
quantity Bu/3x is negligible compared to (1/r)(3u/1r).
These assumptions are equivalent to each other in view of
the relation between u' and Vu in Eq. 13.
The assumption of similarity enabled Tollmien to define
a stream function and then to separate the variables. He
obtained a series solution which he presented in the form
of tables of dimensionless parameters. From experiments
performed at Gottingen, he estimated that CT is 0.156
for axially symmetrical flow. When this value is used,
the agreement between predicted and experimental mean
velocities is fairly good.
The principal defect in this treatment is that Eq. 24
is incompatible with the fact that Rp must vanish on the
u
axis of a free jet because of the local isotropy of the
turbulence. Hence, for lp. to remain constant across a
U
section, 1' must become infinite at the axis -- an impos-
sibility in the model assumed. Furthermore, 1' cannot
even become very large as compared to the dimensions
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
of the jet section, or the assumptions embodied in Eq.13
therefore become invalid.
Profiles are similar only from a point source jet, and
the solution based on similarity would be expected to hold
only in regions where the distance from the discharge is
large compared to the diameter of the nozzle. Actually,
Tollmien's solution does not apply in the region within ten
nozzle diameters from the nozzle.
On the other hand, the application of similarity to the
problem of two-dimensional turbulent flow around a corner
leads to a solution that is valid in the immediate neigh-
borhood of the corner. Tollmien solved this case also, and
the solution was applied by Kuethe(40' to the annular
turbulent mixing zone adjacent to a circular nozzle. His
method gave good agreement with measurements of velocity in
this zone close to the nozzle but became impractical in the
region where the turbulent mixing zones from opposite sides
of the nozzle intermingle.
The foregoing derivation of Prandtl's theory may be
extended to the general case of heat and mass transfer by
retaining T in Eq. 3 so that instead of Eq. 12 there is
obtained
.V$V =- V*?'V' (26)
and corresponding to Eq. 13 there is
' =-- L'9*Vu
From relations corresponding to Eq. 15
= ' t '
where Sp is the eddy diffusivity of heat or mass as the
case may be and corresponding to Eq. 17 is
V'*V = V-8VT (27)
By defining a correlation coefficient similar to that in
Eq. 18 and proceeding as before, there results
Sp = R [' 21 2(V ) 2]
The simplest assumption is that Rp = Rp and lp = 1p =
Ip.130) Then Eq. 27 becomes
V*VT = V- lp2 V l
ILLINOIS ENGINEERING EXPERIMENT STATION
and by analogy to Eq. 23
7.V l- 1 2 [rl2 ( )] (28)
rr r r or
Howarth(3 states that since Eq. 28 is of the _same
form as Eq. 23 and since the boundary conditions of 7 are
the same as thcse of u, then the distribution: of Y must be
similar to that of u, namely
Jx.r = __.r
x,o "x,o (29)
where Ux,r/ux,o is a solution of Eq. 23.
Equation 29 is one solution of Eq. 28, since substitu-
tion in Eq. 28 reduces that equation to Eq. 23. However,
Hinze(33) measured the distribution of heat and mass in a
free jet and found that they are not the same as the
distribution of momentum; hence Howarth's assumption about
the equivalence of Rp and lp in the two cases is not valid.
Hinze also investigated the assumption that lp is dif-
ferent from lp , but was unable to obtain agreement between
U
experimental and predicted temperature distributions.
In general, both the theoretical and the experimental
objections to the assumptions required for a solution for
the free jet problem using the Prandtl mixing length are
serious. In addition, it is shown later that there is
reason to believe that the mean flow parameters (u, T,
etc.) can be correlated only approximately on the basis of
measurements performed with total-head impact tubes and
thermocouples in highly turbulent streams.
b. Taylor's Model
Prandtl's theory was based on the assumption that the
momentum is conserved during the mixing process. Taylor
investigated the corresponding assumption that the
vorticity is conserved. In order to apply this concept,
the vorticity is introduced explicitly into the equations
of motion. Equation 11 may be expanded to
pV-Vu + uV-pV = 0 (30)
However, the second term is zero by Eq. 10. Partially ex-
panding the first term into mean and fluctuating components,
and imposing the condition of incompressible flow,
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS 25
Su' 3u' w' Bu'
V*Vu =- u' - '- -- -
3x -r r 30 (31)
Let a vorticity vector be defined as
= fi + 7j + ýk (32)
where
1 3(rw) 1 dv
r ar r 30
1 -u Bw
r 30 3x
,v bu (33)
3x ar
and i, j, and k are unit vectors in the x, r, and 0 direc-
tions, respectively. From 7 and ( the following equali-
ties may be written
Bv' 6u'
' - - +--) = 0
ox -r
1 Bu' 3w'
w'(-' + --- - --) =0
Adding these two equations to Eq. 31 and combining terms
yields
V'u -- + v'' - w''
2 ?x (34)
At this point the assumption regarding the conservative
transport of the vorticity is introduced. Taylor's own re-
marks (66) on this subject are perhaps the most cogent:
Unfortunately, the assumption that the turbulent com-
ponents (of vorticity) are statistically isotropic does not
give rise to any simplification in the analysis. No fur-
ther progress can be made without introducing further
simplifying assumptions. One such assumption is that the
components of vorticity are transported unchanged by the
turbulence just as the momentum is conceived to be trans-
ported in the Momentum Transport Theory. A part of the
fluid is conceived to leave a certain position with the
vorticity comporents of the mean motion and to retain those
components till it mixes with its surroundings after
traversing the mixture length. This is clearly a very
drastic assumption, because it is certainly untrue in
detail, though it may be true when considered in relation
to the effects on the mean motion.
26 ILLINOIS ENGINEERING EXPERIMENT STATION
With the above assumption, equations analogous to Eq. 13
may be written
In isotropic turbulence, the cross products are presumed
to be zero
w'l' = w'l' = v'l' ' ' ' ' 'l' 0
while the parallel products are assumed equal
u'l'x = v'1 w l' = ET (35)
where ET as defined by Eq. 35 is an eddy viscosity and in
particular is the eddy viscosity associated with the vorti-
city transport theory. Substituting these in Eq. 34,
and neglecting the term dV'2/Bx as is done in Prandtl's
momentum transport theory,
V U1 =-
T dr r
The terms in parentheses may be expanded by Eqs. 32 and 33
and the resulting terms regrouped and combined to give
- - 2_ 3
-Vu = € [Vu (V*I)]
T ax
But by Eq. 7 V-V is zero for incompressible flow, hence
V.Vu = E2u (36)
which may be compared to Eq. 17.
To evaluate the eddy viscosity Ty, HowarthI361 simply
took it to be of the same form as the ep of the momen-
tum theory defined by Eq. 15. This result follows only
if important assumptions are made. Let a correlation
coefficient RT be defined as
v' l'
RT = (37)
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
If the turbulence is isotropic, u 2 = v 2. In order to
eliminate u'2 it is necessary to use the momentum transport
theory as embodied in Eq. 13. These substitutions in
Eq. 37 lead to
ET = RT[r 772 (L' u-)2]
For isotropic turbulence
' 2 = 1' 2 = 1 2 = r' 2
where l' is defined by this equation. Hence
E [( 2 1' 2)(V.)2]Y
or
CT = iT2IVu (38)
where 1T is the Taylor mixing length and contains RT. From
Eqs. 20 and 38 Ep and ET are of the same form, provided
that the momentum transfer theory may be used as well as
the vorticity transport theory. From Eqs. 36 and 38
V-Vu =-lT2 Vu| (Vu) (39)
Because of the rotational symmetry, derivatives with
respect to b are zero; and if, as in the momentum theory,
derivatives with respect to x may be neglected, then
Vu =- a 1 - (r-a) (39a)
T B-r r r 3r
This may be compared to Eq. 23. These procedures origina-
ted by Taylor and carried forward by Howarth, making use of
the vorticity transport theory, result in another equation
of motion (Eq. 36) which differs from that obtained from
the momentum transport theory (Eq. 17). The eddy viscosity
which appears in both equations is the same function of the
velocity gradient (Eqs. 20 and 38).
Taylor's theory suffers from the defect that it requires
one to make all the assumptions that were made in deriving
Prandtl's theory (e.g., conservation of momentum during
mixing) plus the assumptions regarding the vorticity. It
is uncertain whether some of these assumptions are not
mutually exclusive.
ILLINOIS ENGINEERING EXPERIMENT STATION
If Eq. 21 is expanded in rectangular coordinates, then
for the case of two-dimensional flow in the x-y plane,
partials with respect to z are zero and, to the same degree
of approximation as before, partials with respect to x
may be neglected. Further, let us suppose that lp is
independent of y. Then it follows that
VV -V p = 2_ a )2]
or, performing the differentiations indicated,
V-Vu =-21 p2 - .(40)
By By2
Proceeding similarly from Eq. 39,
- Bu B2
V*Vu =- T 2 -2
T y 3y2
which is identical with Eq. 40 provided that 1T = V2 lp.
Abramovitch"1' has taken this result to be a general rela-
tion between the two mixing lengths, but it holds only
incidentally for two-dimensional flow, probably because in
two-dimensional flow vorticity is conservedAo)>
No such simple relationship exists for the axially
symmetrical flow.
While the momentum and vorticity theories give the same
distribution for momentum flux in two-dimensional jets,
they do not predict the same distribution in axially sym-
metrical jets, as Howarth has shown.,36' He found that the
momentum theory provides the best correlation, and he
presented his solution in tabular form.
The application of Taylor's theory to the transport of
heat and mass is not straightforward. Equations analogous
to Eq. 32 cannot be written, because temperature and con-
centration are not vectors. Although an equation analogous
to Eq. 32 cannot be derived by these methods, one may
simply assume, as Howarth did,1(36 that
V*V =- 1 2 r (r -- (41)
SBr r 3r Br
where it was also tacitly assumed that the mixing lengths
for heat and mass are the same as for vorticity. A solu-
tion of Eq. 41, obtained from the solution of Eq. 39a,
provided a better correlation of data from axially symme-
trical jets than did the corresponding solution of Eq. 23.
However, in view of the complete lack of a phenomenological
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
basis for Eq. 41, this result can hardly be interpreted as
a confirmation of the vorticity transport theory.
The foregoing discussion of Prandtl's and of Taylor's
theories states clearly the comparable basis on which each
rests. It must be concluded that the usefulness of each is
limited to special cases. Neither theory offers promise of
extension for use in the general treatment of mass, heat
and momentum transfer in turbulent streams.
8. Empirical Theories
a. Boussinesq's Hypothesis
By analogy to viscous flow, Boussinesq(10) proposed
that the u'v' product in the expansion of Eq. 12 can be
expressed in terms of an eddy viscosity and the transverse
gradient of the mean velocity, thus
V-Vu =- V'(u'2i + C€ -rrJ + u'w'k) (42)
For flow in a pipe, the term EB was supposed to be a func-
tion only of the Peynolds Number, but was found to be a
complicated function of the position in the pipe. For this
reason Prandtl and Taylor abandoned the analogy to viscous
flow and developed their theories along different lines.
There is a formal resemblance between Eqs. 17 and 42. If
the terms ýu/ix and -u'2/3x are neglected, the equations
become identical. However, they are not based on the same
fundamental postulates.
Although Boussinesq's hypothesis did not prove satisfac-
tory for flow in a pipe, Hinze'331 felt that an investiga-
tion of its applicability to flow in free jets would be
worthwhile. He assumed eB to be a function of the velocity
along the axis of flow and the distance from an equivalent
point-source jet, and to be constant across a section of
the jet. Thus
E = a0ou ,o(x+a) (43)
where x is the distance from the nozzle and a is the
distance from the nozzle to the equivalent point source.
With these assumptions it is possible to solve Hinze's
equations to obtain
"x,r V'K /8 d 1
------ 2 (44a)
U0,0 (x+a)(ao+bH) [1+ r2 2
3 8ao(x+a)2
ILLINOIS ENGINEERING EXPERIMENT STATION
where bH accounts for the transport of momentum in the x-
direction by the longitudinal fluctuations, u'2. This
quantity was evaluated from Corrsin's data(18) and found
to be approximately constant with an average value of
0.00045. Hinze then determined the constants a and ao from
his experimental measurements, obtaining 0.6 d and 0.00196
respectively, where d is the nozzle diameter. K is a term
to correct for the nonuniformity of the velocity distribu-
tion across the outlet of the nozzle. By inserting the
numerical values of the constants and by making certain
other approximations and assumptions, Hinze obtained the
equivalent of
Ur 6.39d 1
-- =- } (44b)
Uo,0 (x+0.6d) [1+ r2 2
0.0157(x+0.6d)2
From his measurements, Hinze showed that the correlation
obtained on the basis of the assumptions leading to Eq. 42
is as good as that obtained by Tollmien and Howarth on the
basis of assumptions regarding the Prandtl mixing length.
The hypothesis of Boussinesq is readily extended to the
transport of heat and mass. When this is done Eq. 26
becomes
V*V1 =- V-(('u' i + B----j + T'w' k)
Hinze first supposed 8B to be proportional to eB, but found
experimentally that this is not so. The ratio 6B/eB
decreased with increasing distance from the axis of flow,
and was not even approximately constant in the central
portion of the jet.
He then proposed the relation
..B = -0 + '-1 u r (45)
B 0 a0 Ux,O
which leads to the solution
1 + _
x,r VK/8d '80 2 0ao,
o0,0 (x+a)(bo+ 1)2 -1 + 1+ 42
0 8aO(x+a)] (46a)
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
in which b1 accounts for the transport in the x-direction
due to the fluctuating motion and KT accounts for any
deviation from a uniform distribution at the nozzle outlet.
After evaluating certain of the constants and neglecting
certain others, he obtained the equivalent of
Tr 5.27 d 1.196 .82
2
o 0 "(x+0.8d)0.196+ [1+ r 12
o,o (x+0.8d) 0.157(x+0.8d)2 (46b)
Hinze computed 80/a0 to be 1.22 and 1i/a0 to be 0.24.
Equation 46 correlates very well Hinze's data for the
central part of the jet, but gives high values for the
concentration and temperature in the region where the
ratio of r/x is greater than 0.12.
The principal disadvantage in the use of Hinze's devel-
opment stems from the complexity of the expressions and the
large number of empirical constants introduced. Equation
43 is certainly not true, and the success of Hinze's corre-
lation, as well as those of Tollmien and Howarth, seems
largely due to the fact that any reasonable assumption
regarding the dependence of u'v' or 'V' on the mean flow
parameters results in a solution that fits the data reason-
ably well. As Corrsin has pointed out, (18) it does not
seem possible to differentiate among these theories on the
basis of mean-flow measurements made with a total-head
impact tube.
b. Reicharat's Hypothesis
An empirical approach to the problem of free turbulence
was proposed by H. Peichardt."(58 Noticing that experi-
mental measurements in a free turbulent jet can be well
correlated by probability functions of the type
- K -(r/b)2 (47)
b2
he investigated the conditions under which such functions
become solutions of the equation of motion. Equations such
as Eq. 47 have been used by several investigators. (2,61,68)
KR is a constant, depending upon the strength of the
jet; b is a parameter, depending only on x, and b is
independent of r.
Now for axially symmetrical flow and where density
fluctuations may be neglected, Eq. 11 becomes
ILLINOIS ENGINEERING EXPERIMENT STATION
-aU2 1 -a (48)
77 + r r uv) = 0(48)
Peichardt showed that Eq. 47 is a solution of Eq. 48
provided that
uv =- A-r (49)
where
b db
A =- ax (50)
and b is a function of x only.
No phenomenological basis for Eq. 49 was proposed, but
Beichardt remarked that the hypothesis must stand or fall
depending on whether it permits a mathematical analysis of
free turbulence and not on whether a mechanical interpre-
tation can be found. Actually such a physical basis can be
developed if one assumes that
1. The mean flow is essentially one dimensional (x-
direction)
2. The intensity of the turbulence is small
3. The radial velocity fluctuations are proportional to
the mean velocity
This is no more rigorous a theoretical basis than are the
mixing length theories and hence is not presented here.
In some of the preceding methods, the term Bu'2/ix was
neglected. This ommision may introduce considerable error.
Peichardt's procedure avoids the necessity of neglecting
these longitudinal velocity fluctuations, since these are
contained in u2. Corrsin(201 reports experimental values
of the ratio (u'2/u2)% at various points in a free jet.
Along the axis of flow it increased from a negligible value
near the nozzle to somewhat more than 0.20 at 12 nozzle
diameters from the nozzle, and remained substantially
constant thereafter. This indicates that it is probably
inadmissable to neglect u'2 and its derivatives.
The Beichardt hypothesis merits further investigation.
Equation 47 provides remarkably good correlation of ex-
perimental measurements and contains only one arbitrary
constant. Equation 49, because of its simplicity, pro-
mises to be useful for the analysis of more complicated
cases. An extension to cover these cases is described
in the following chapter.
III. THEORETICAL ANALYSIS OF FLOW IN FREE JFTS
9. Generalization of Reichardt's Hypothesis
The methods of Prandtl and Taylor neglect the contribu-
tion of the fluctuating motion to the transport of momentum
parallel to the axis of flow in a free jet. This imposes
an unreal conservative relation on the square of the mean
velocity u. Beichardt's treatment uses the momentum flux,
which is a truly conservative quantity and makes no assump-
tion regarding the distribution of this between the mean
and fluctuating components. The resulting equation contains
only one arbitrary constant, and several of its investiga-
tors12,58,69' have shown that it correlates the data well.
Beichardt's procedure can readily be generalized to in-
clude the transport of heat and mass as well as momentum in
free jets. This is especially useful because Reichardt's
hypothesis (Eq. 49) linearizes the equation of motion for
x-directed momentum in a free jet and offers promise of
making possible the mathematical solution of cases of free
turbulence involving complicated boundary conditions.
For axially symmetrical flow Eq. 3 takes the form in
cylindrical coordinates,
- + T- -(riv) = 0 (51)
Integration over a .section through the flow field of a free
jet in a plane perpendicular to the axis of flow leads to
(A u) oA = 0 u 2nTr dr (52)
This merely states the conservation of the flux of T and is
an obvious consequence of neglecting the generation terms
in Eq. 2.
By analogy to Eq. 49, it is assumed that
TV - (53)
ILLINOIS ENGINEERING EXPERIMENT STATION
where
bv dbvi
A= db (54)
= 2 ax
and b is a function of x only. Equation 51 becomes
3a u AT_ - -6 iu
3x .r ?r 3r
A solution of this equation is
u --= 26_(r/b (55)
It has been observed'70) that successive profiles of the
momentum flux density ratio are similar. Assuming this
relation to be general for the transport properties
(Tu r r
(ju)o0,o (56)
From Eq. 55
(u)x r - -(r/bT)2 (57)
(ku)0,
But Eqs. 56 and 57 are compatible only if
bp = CTx (58)
Substituting Eq. 55 into 52 and integrating leads to
K = (U)o,o0A0/7T
Therefore for a free jet issuing from a point source,
Eq. 55 takes the form
(q )o, oA -(r/C .)2 (59)
(u)x," rC x2
That is, Eq. 59 gives the distribution of the flux of any
entity which originates wholly in the nozzle discharge
based on the following assumptions: (1) the jet issues
from a point source; (2) successive distribution profiles
are similar; (3) Eq. 53 is valid.
10. Principle of Superposition
Upon substituting Eqs. 53, 54, and 58 into Eq. 51 there
is obtained
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
-L -_ - (r -) = 0 (60)
3x 2r Br dr
This equation is linear in Tu, and therefore a linear
combination of particular solutions is also a solution.
Thus, the distribution of Tu in a flow field formed by the
confluence of any number of point source jets issuing in
parallel can be described by superimposing the solutions
for the flow fields of the individual jets.
For point sources discharging parallel to one another
j (Tu),n A, e-(r /C xn)2
S= 1C c2X 2 (61)
n= 0 T X
In this summation it is not necessary that all the point
sources lie in the same plane perpendicular to the axis of
flow provided that no point source impedes the flow from a
source lying behind it. Furthermore, if a finite source may
be taken as an infinite number of elemental point sources,
then the summation may be replaced by the integration
A0 ( )dA -( rdA/CxdA A
Cr 0 Ca2X dA 2 (62)
II. Distribution of Momentum Flux
For the case of momentum flux, q is pu. Substituting
this in'Eq. 59 gives
(Pu2)x,r A0 -(r/Cmx)2
(pU2)0o, 7Cm2x2 (63)
The momentum flux density is measured approximately by a
total-head impact tube, and Eq. 63 can be compared with
such measurements.
In cases of mass and heat transfer it is frequently
necessary to employ the mean velocity u rather than the
momentum flux. Introducing the mean and the fluctuating
components into Eq. 63 yields
(p+p')(u+u')2 r A0
Z r _ _ (r/Cmx)2
(pu2)0,0 7Cm2X2
Expanding the left-hand member, neglecting third-order
moments, and recognizing that all terms containing only one
ILLINOIS ENGINEERING EXPERIMENT STATION
fluctuating component vanish, there results
(pu 2)x, 2p'u' u2 A0
(1+ --- +-- =- --2 (r/x (64)
(P O) 0,0 u u2 7TC 2x2
Following Corrsin, (20) the density fluctuations are ex-
pressed in terms of the temperature and concentration
fluctuations. For an ideal gas in turbulent flow p = MP/RT
to a first degree of approximation (see Section 7).
Expanding Ap in Taylor's series and neglecting terms
of higher order
MP P
Ap =- -M-AT +-- -AM
RT2 RT (65)
For a two-component mixture
M = 1
wM . l-w
I i
M + M
whence
AM - MAw
W. + MO/Ma
l-MO/Ma
Substituting for AM in Eq. 65 and setting Ap., AT, and
Aw. equal to p', t', w'i respectively
p = p(- +--" (66)
T w' .+G
where
G = 0o/a
1-MO/Ma
Substitution of Eq. 66 into Eq. 64 gives rise to the terms
t'u'//T u, w'i u'/(wi - G)u, and u'2/u2. The terms v 2/u2
and w' 2/u2 also appear in slightly different circumstances.
CorrsinC21) discusses methods of measuring these by means
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
of a hot-wire anemometer. The mean temperature, T, can be
measured by a thermocouple. The mean concentration of jet
fluid, w., can be obtained _by standard methods of gas
analysis. The mean density p can be calculated to a first
degree of approximation from the mean concentration and
temperature by the use of a suitable equation of state.
Thus, methods of evaluating all quantities in Eq. 64 are
available and u may be calculated from
2AP
2p'u' u'2
(1+ 2-
pu u
where AP is the total-head impact tube measurement.
Substituting 7d2/4 for A0 in Eq. 63 and setting r equal
to zero gives the axial distribution of momentum flux for a
circular nozzle as
(P x,__ = )2 (67)
(pu2)0,0 2Cmx
Substitution of this into Eq. 63 gives
(pu2)x,r ,= -(r/Cmx)2
(p 2)x,0 (68)
which expresses the similarity of the flux profiles when b
is linear in x.
12. Distribution of Flux of Sensible Heat
For the flux density of sensible heat under conditions
similar to those just discussed for momentum flux, T is
pC t. The generation terms in Eq. 2 were neglected, and
therefore changes in t due to chemical reaction are not
considered. Likewise, phase transformations, Joule effects
and the like are neglected.
Substituting for T in Eq. 59 leads to
(PC ptu)x,r A _-(r/ hx ) 2 (69)
(pCptu)0,0 C h2X2
ILLINOIS ENGINEERING EXPERIMENT STATION
Here again, pC tu cannot be measured directly but, as
before, one may write
t' u' u' ' t'p'
t u pu t p
[1+--+-+--]
t u pu t p
A0 (r/Chx)2
Ch2X2
where Cp is assumed constant. Equation 66 applies to this
case also.
13. Distribution of Mass Flux
In like manner for the mass flux density, T is pwi
and Eq. 59 becomes
(p iu)i x,r
(pwiu)o,0
A0 e-( r/Cwx)2
7TC 2x2
U,
(71)
Since there is no direct method of measuring pwiu, it is
again necessary to introduce the mean and fluctuating
components, giving the relation
(p w u) r w' u' p'u' w' ip' A0 (r/Cx)2
[1+- +---+- 1 =- (72)
(Pwiu)0,0 wiu p u wi p 7C 2x2
Equation 66 again applies.
(pC t u) x,r
(pcp tU)00
(70)
IV. EXPERIMENTAL INVESTIGATION
14. Apparatus
The apparatus comprised a source of compressed air, a
flow control and metering system, heat exchangers, flow
nozzles for generating free jets, impact tubes, and a
positioning mechanism for locating the impact tubes in the
jets. A schemetic flow diagram is shown in Fig. 2.
The compressed air was supplied by a diesel-powered two-
stage air compressor having a capacity of 500 standard cu
ft per min at a pressure of 100 lb per sq in. gage (Fig. 3).
The compressor was connected to an aftercooler through a
Fig. 2. Schematic Diagram of Experimental Apparatus
2-ft section of flexible hose which prevented vibrations
from the compressor from traveling down the connecting
lines to the experimental apparatus. The air passed
through an oil and moisture separator into two air receiv-
ers with a capacity of about 200 cu ft, and thence through
two air filters consisting of porous ceramic discs 10 in.
in diameter and 2 in. thick.
The metering system is shown in Fig. 2. It consisted of
nine interchangeable orifice plates covering a range of
flow from 0.5 to 350 standard cu ft per min. The plate in
use was inserted between two flanges fitted with three pins
located asymmetrically so that the plate could be oriented
only in one position. A bulb-type dial thermometer was
used to measure the upstream temperature +2 deg F. The
pressure on the upstream side of the orifice plate was
measured with a Bourdon test gage giving readings accurate
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 3. Air Compressor
to V percent of the full scale, while the pressure on the
downstream side was measured with a regular Bourdon gage.
The orifice was operated in such a manner that the ratio of
the downstream pressure to the upstream pressure was less
than the critical pressure ratio for the flow of air under
the conditions used. Hence the mass rate of discharge of
air through the orifice was independent of small changes in
downstream pressure, and the rate was determined entirely
by the upstream pressure and temperature.
For flow rates up to 50 standard cu ft per min the up-
stream pressure was regulated by a Moore Nullmatic control
valve; for rates of 50-350 standard cu ft per min, by a
Fisher Wizard control valve (Fig. 4). Both valves covered
the range of pressures from 0 to 100 lb per sq in. gage.
A part of the metered air from the orifice was passed
through a battery of heat exchangers. This air was mixed
with the main stream to adjust the temperature so that when
it expanded adiabatically through the flow nozzle, the jet
would be discharged at the ambient temperature of the room.
Four shell-and-tube exchangers were connected in series,
each having a transfer area of about 35 sq ft. These could
be supplied either with hot boiler water at about 215 deg F
or with cold water from the mains. Control of the main-
stream temperature was accomplished by varying the fraction
of the air sent through the exchangers.
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS 41
Three manometers were used for measuring the impact
pressures and the pressure immediately before the flow
nozzle--a 40-in. mercury manometer, a 40-in. water mano-
meter, and a Wahlen gage.( 74 This latter gage wasused for
measuring pressures from 0.001 to 1.000 in. of alcohol.
The pressure control system, the heat exchanger, the
flow control apparatus, and the instrument panel are
shown in Fig. 4.
Several flow nozzles having different throat diameters,
throat lengths, and exterior designs were used in the
experiments. All had an ASME long-radius, low-ratio
approach section.'ý3 Two typical 0.898-in. nozzles with
semi-streamlined exteriors are shown in Figs. 5 and 6.
A mechanism for traversing in three dimensions was used
to position the impact tubes in the flow field of the jets.
This mechanism traversed the region within 100 cm axially
from the nozzle, ±30 cm laterally, and ±25 cm vertically.
It consisted of two rails having longitudinal V-notches in
which a cross rail rested on ball bearing end-supports.
The cross rail was surface-ground on two adjacent sides and
provided a bearing surface for a cross carriage. Alignment
of the V-notched rails with the nozzle axis was held within
an angle of 0.2 deg.
Fig. 4. Flow Control and Metering System
42 ILLINOIS ENGINEERING EXPERIMENT STATION
Nozzle Attached Approach Section
to 2" Pipe with
Machined Threads
Fig. 5. Cross-section of the 0.898-in. Semi-Streamlined Nozzle
The impact tubes, discussed in detail in Section 16,
were mounted on a vertical rod, 7/8 in. in diameter, which
passed through a sleeve clamp. The elevation of the impact
tube was determined by a vertical cathetometer, while the
axial and lateral coordinates were indicated by pointers
bearing on centimeter scales attached to the V-notched
rails and the cross rail, respectively.
15. Calibration of Instruments
The standard for the velocity and momentum measurements
consisted of two orifice plates which were standardized in
the laboratories of the manufacturer. Water, measured
volumetrically to 1 part in 10,000 in carefully calibrated
tanks, was passed through the orifices at constant rates
for periods of time measured to 1 part in 3000. The
pressure difference across the orifices was determined as a
function of the flow rate in the same range of Peynolds
Number later used with air.
The orifices were then used to calibrate a stainless
steel ASME standard flow nozzle having a throat diameter of
0.952 in. The nozzle was calibrated by connecting it in
series with the standardized orifice and an air source.
The pressure difference across the orifice and across the
nozzle, along with the corresponding temperatures, were
observed at various flow rates; the discharge capacity of
the nozzle was expressed in terms of the pressure and
temperature just before the converging section by means of
the following equation 3
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
h P
Qscfs = K HE(--)n )
Pi 1- (73)
By a series of 42 calibration experiments in which the
pressure differences across the nozzle were less than the
critical difference, the following relation was found
hi 0.516 P1
Qscfs = 3.028$E(-1) (74)
Pf Ts(74)
which covered air flow rates from 0.06 to 5 standard cu ft
per sec. In another series of 12 experiments in which the
pressure differences were above the critical difference,
the relation
Qsfs = 3.028 E( ) (75)
was found covering the flow rates from 5 to 8 standard cu
ft per sec. The standard flow nozzle was used to calibrate
the orifice meter described in the previous section.
Calibration of the impact tubes is discussed in the next
section.
Fig. 6. Exterior View of the 0.898-in. Nozzle
ILLINOIS ENGINEERING EXPERIMENT STATION
16. Total-Head Impact Tube
a. Calibration
The total-head impact tubes were made from No. 16
hypodermic-needle tubing having an outside diameter of
0.065 in. and an inside diameter of 0.050 in. Before use
the ends were carefully squared and burrs were removed.
In addition the tubes were calibrated and their readings
interpreted according to the equation
1 pu2
AP =
Cf22 (76)
From reference (3) Cf may differ from unity and should be
determined experimentally. Goldstein°30' defines a cali-
bration coefficient for pitot tubes by the equation,
AP = Kpu2/2. Hence, K = 1/Cf 2 He reports that the value
of K at speeds varying from 20 to 60 ft per sec did not
vary from unity by more than ±0.1 percent, while the value
at speeds from 6 to 20 ft per sec did not differ from unity
by more than ±1 percent.
There are two explanations for values of Cf below unity.
Barker (4 points out that at low velocities impact tubes
of small radius are influenced by a viscosity effect, which
may be taken into account by the equation
pu2 3 I u
7 = -T 77
where u is the velocity, A the gas viscosity, and r the
internal radius of the impact tube. It follows that the
coefficient C at low velocities is given by
p2
C2 2
C 2 2
f pu2 3 k u
2 2 r
Using this equation with r = .050 and / for air at 70 deg F
and atmospheric pressure, the values for Cf were calcula-
ted as a function of u and are given below. Impact tube
coefficients of this order of magnitude were also reported
by Homann135) under similar conditions.
At high velocities the stagnation pressure, static
pressure, and kinetic head are related by the equation12
Po-P = 1 + -M + M + , +1..
q 4 40 1600
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
Values of Calibration Coefficient
at Low Velocities
ft/sec
4
10
20
50
Cf
0.971
0.988
0.994
0.998
where P0 is the stagnation pressure, P the static pressure,
q the dynamic pressure, and M the Mach number. The quan-
tity (PO - P) is the pressure difference AP indicated by an
impact tube, and the dynamic pressure q is given by pu /2.
Therefore
P 1 Po-P _ 121 1
u2- C -2 = 1+- +- +-M6+...
pu2 C 2 q 4 40 1600
Using this relation C can be calculated as a function of
u, and the values so obtained are given in the table below.
Values of Calibration Coefficient
for High Velocities
ft/sec
0.999
0.995
0.990
0.983
0.975
0.963
0.949
0.934
0.918
0.901
600
700
800
900
1000
These values of Cf for both low and high velocities are
shown in Fig. 7 as a function of the velocities.
b. Efiect of Angle of Attack
The "ideal" impact tube is defined as an instrument
which measures the pressure at a point of stagnation in a
flowing stream. Since the velocity is zero at such a point,
the ideal impact tube has characteristics whichare indepen-
dent of the direction it faces; that is, the pressure at a
ILLINOIS ENGINEERING EXPERIMENT STATION
point of stagnation is independent of the orientation of
the obstacle producing the stagnation.
Actual impact tubes, however, do have directional
characteristics. This fact was demonstrated in an experi-
ment in which the axis of the tube was rotated through
an angle of 90 deg with respect to the direction of flow
of a nonturbulent stream. Up to an angle of about 15 deg
Velocity, Feet per Second
Fig. 7. Total-Head Impact Tube Coefficients
the indicated pressure was substantially constant. Beyond
that angle the pressure decreased, becoming less than
atmospheric for angles greater than 60 deg.
Since in a turbulent stream the direction of flow at
the impact tube fluctuates, the question arises as to
whether the average angle of attack is large enough to have
an effect on the pressure readings. The ratio of the
root-mean-square of the transverse velocity fluctuations to
root-mean-square of the axial velocity, v'2/V2, may be
regarded as roughly proportional to the sine of the
root-mean-square of the average angle of attack on the
impact tube. Corrsin1201 found this ratio to be nearly 0.2
across the entire flow field of a free jet, corresponding
to an average angle of attack of about 12 deg.
Since this average angle is less than 15 deg, it was
concluded that impact tubes used in free jets are not
subject to significant errors which are due to fluctuations
in the angle of attack.
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
c. Effect of Turbulence
In actual use the momentum flux densities, pV2, were
measured throughout the flow field of turbulent jets by
impact tubes, and the calibration coefficient defined by
Eq. 76 was applied regardless of the level of turbulence
and hence regardless of the velocity fluctuations existing
at the point of measurement. Other investigators have
followed essentially the same procedure, because the basis
for taking the effect of the velocity fluctuations into
account is not well understood. In an attempt tc determine
the effect of the velocity fluctuations, the following
analysis and experiments were performed.
For steady-state flow, changes in static pressure in any
direction may be represented in vector notation by
dP = VP'dR (77)
where R is a vector distance. When the momentum flux
variables are substituted into Eq. 2 and the first two
terms for unsteady flow and viscous shear are neglected,
there results
-V-(pV-S)V-VP-S = 0
Successive expansion in the x, y, and z directions and
recombination lead to an expression for VP. Substituting
this expression for VP into Eq. 77 yields
dP =(V.puVi+V-pvVj+VpwVk)-dR
Finally application of the equation of continuity gives
dP = (pVVWui+pVVWvj+pVVW'wk)-dR
Consider now an instantaneous streamline through a
stagnation point and assume that the distance along the
streamline from a point where the velocity has the free-
stream value to the stagnation point is so short that this
streamline is substantially a straight line between these
points. If the x-axis is oriented parallel to this stag-
nation streamline, then the velocity components normal to
it, v and w., are zero and
du
dP= pua --- dr (80)
dx
where both the distance, xs, and the velocity, u., are
measured along the stagnation streamline. Since u, is
ILLINOIS ENGINEERING EXPERIMENT STATION
equal to the magnitude of the velocity vector, integration
of Eq. 80 along the streamline from the free-stream point
to the stagnation point, when averaged with respect to
time, gives
AP = 1pV2
2 (81)
This expression may be expanded in terms of the mean and
fluctuating components to give, for incompressible flow,
-P = P(U- +u 2+v '2 ) (82)
and solving for u yields
- 2AP ,2_ 2 2)(
u = ( - u-2 2-_W2 (83)
Hence strictly speaking, u may be obtained from impact tube
measurements in turbulent streams only if the magnitudes
of the fluctuating components are known.
In addition, the momentum flux density in the x direction
is given by
p.u2 = 2AP - pv"'2 - pw'2 (84)
and substitution of p from Eq. 81 into this equation
results in
- - rV2 W02
pu2 = 2P(1-- ----) (85)
V2 V2
Data concerning w'2/V2 are not available; however, if this
is approximately equal to u'2/V2, then the term in paren-
theses may be estimated from Corrsin's C20 work to be 0.92
along the axis of a free jet. From this analysis it would
appear, therefore, that in determining the x-directed
momentum flux, the correction to be applied to the pressure-
difference reading on the axis, in order to allow for the
velocity fluctuations in the y and z directions caused by
turbulence, amounts to about 8 percent of the reading.
As a more direct means of determining the effect of
turbulence on the impact tube measurements, several other
experiments were performed in which impact tube measure-
ments in nonturbulent and varying levels of turbulent
jets were compared.
The nozzle used in these experiments is shown in Fig. 8.
It had a throat diameter of 0.75 in., and the converging
section followed ASME specifications for long-radius low
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
Coanaarin Section
Converging Seciion
Fig. 8. Cross-section of the 0.75-in. Nozzle
with Screen and 0.25-in. Throat Extension
ratio flow nozzles. 3 The nozzle was equipped with four
removable throat-extension sections -- 0.25, 0.50, 0.75,
and 1.00 in. long. For generating a turbulent jet a
35-mesh copper screen having a wire diameter of 0.243 mm,
square openings of 0.441 ± 0.030 mm on a side, and a free-
flow area of 41.3 percent was inserted at the entrance to
the throat section. To determine the scale and the level
of turbulence at the discharge end of the throat section,
reference was made to empirical equations developed from
similar work by Dryden. (241 These values are tabulated
below as a function of the length of the throat section.
In the experiments themselves the impact tube was loca-
ted at the discharge of each throat section in turn while
the air flow rate through the nozzle was maintained at the
constant value of 180 ft per sec. To obtain measurements
in a nonturbulent jet for comparison, the experiment with
the 0.25-in. throat section was repeated with the screen
removed. Although the results are not presented here, they
were in fairly good agreement with the qualitative predic-
tions from Eq. 82. With the mean velocity held constant
Turbulence in Discharge from a Nozzle with Screen
Length of section Turbulence level Scale of turbulence
in. percent mm
0.25 7.8 0.12
0.50 4.2 0.13
0.75 3.0 0.15
1.00 2.3 0.16
ILLINOIS ENGINEERING EXPERIMENT STATION
while the turbulence level is raised, there is a definite
effect on the impact tube readings, but this may be partly
caused by minor changes in the velocity distribution
across the nozzle.
Consequently in accordance with the common practice of
other investigators, the impact tube measurements were not
corrected for the fluctuating components of velocity
because the basis for making proper corrections is not well
understood. Although these fluctuations in velocity may be
one of the chief sources of error, it is believed that this
error is not excessive in most regions of the jet. To
correct the impact tube measurements, therefore, the
calibration coefficient from Eq. 76 was used.
17. Single Free Jets
The experiments described in this section consisted of
obtaining impact tube traverses along the axis of single
jets and also along a radius of several cross sections of
the jet at various distances from the nozzle. Several
nozzles having different diameters were used including the
special nozzle described in the previous section. The
velocity at the nozzle was also varied in the experiments.
a. Transport of Momentum Flux
The results of several measurements of the momentum flux
density in a free jet of air from a 0.898-in. nozzle are
shown in Fig. 9. Two sets of data are presented in the
figure. One set (Table 1) was obtained by making radial
impact tube traverses at axial distances of 10, 15, 20, 25,
and 30 nozzle diameters while the discharge velocity was
held constant at 403.1 ft per sec. The other set (Table 2)
was obtained by making a radial traverse at 20 nozzle
diameters at each of four discharge velocities ranging from
177.9 to 803.6 ft per sec.
In Fig. 9 these data are compared with the probability
distribution of momentum flux which was used to arrive at
Reichardt's hypothesis. For this comparison Eq. 57 was
modified as follows. By defining ry as the radial distance
at which (Pu)x r is one-half the value on the axis,
(-u) x,0, it follows that
-(r/b)2 2
2
Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS 51
L. O
• s" 0.8 -- '-- ^ - - - - - - - - -
* - Equation 87
i 06 -- "' -- -
I 0.2---- - -- - ----- --
0.
0 0.4 0.8 1.2 1.6 2.0 24
Radius Ratio, (-r)
Fig. 9. Comparison of Radial Profiles of Momentum Flux Density
Ratio in a Single Free Jet with the Probability Distribution
from which
2..
2 r,2
In 2
Substituting for 6b,2 in Eq. 57 gives
(T xr 2-( )21n2 (86)
Thus Eq. 86 implies that the ration(-u)z,r/(-u)x, O is a
unique function of r/rv and that this function contains no
arbitrary constants regardless of the functional dependence
of b on x.
For the case of momentum flux T is pu, therefore Eq. 86
becomes
(p2)xr ,. -( ~ 21n2 (87)
(pu2) x, 0
ILLINOIS ENGINEERING EXPERIMENT STATION
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Bul. 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
t- CN 0 C> % CD
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t- 11
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ILLINOIS ENGINEERING EXPERIMENT STATION
Calculated values of Eq. 87 are represented by the solid
line in Fig. 9. This line, the probability curve, corre-
lates the data reasonably well. The points have a slight
tendency to lie below the curve in the range of r/ry from
0 to 1.0 and to lie above it elsewhere.
b. Effect of Similarity Conditions on Distribution of Momentum Flux
In Section 9 the similarity of radial profiles was
assumed and was used to deduce the relation between b and
x. There are three kinds of similarity involved in Eq. 63.
It requires that successive profiles of momentum flux must
be similar geometric similarity), and that the ratio
(pu2) xr/(pu2) x,0 must be independent of the discharge
velocity (dynamic similarity) and also of the nozzle
diameter (dimensional similarity).
The results of several impact tube measurements taken
through a free jet issuing from a nozzle at a velocity of
403.1 ft per sec are shown in Table 3 and Fig. 10. As
suggested by Eq. 59, the negative logarithm of the momentum
flux density ratio is plotted against (r/x)2 for traverses
taken at various distances from the nozzle.
~I- .~
~
~
0 0.005 0.01 0.05 0.02 0.025
Square of Radial Distance Ratio, (v)2
Fig. 10. Geometric Similarity: Variation of Negative Logarithm of
Axial Momentum Flux Density Ratio with Square of Radial Distance Ratio
Bul 413. MOMENTUM, MASS AND HEAT IN TURBULENT JETS
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