Fluid Dynamics of Jet Amplifiers: Course held at the Department of Hydro- and Gasdynamics September 1970

110 Pages·1970·5.27 MB·English

by  Ario Romiti (auth.)

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R S E S A N D L E C T U R E S - No. 66
ARIO ROMITI
TECHNICAL UNIVERSITY OF TURIN
FLUID DYNAMICS OF JET AMPLIFIERS
COURSE HELD AT THE DEPARTMENT
OF HYDRO - AND GASDYNAMICS
SEPTEMBER 1970
Springer-Verlag Wien GmbH 1970
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81152-8 ISBN 978-3-7091-2880-0 (eBook)
DOI 10.1007/978-3-7091-2880-0
PREFACE
Jet amplifiers are the basic devices on which most of fluidic control
systems rely.
Signal amplification is obtained by controlling the deviation of a jet.
In order to have an insight of the phenomenon, one must investigate
the dynamic behaviour of the flow into the device.
The fluid dynamics of separated flow has to be considered, taking in
to account the eventual reattachment of the stream.
In the short course that is presented here, the fundamental laws de
scribing free jet spreading are first given ; then, the effect of nearby walls is con
sidered.
These preliminary concepts are used for examination of jet interactions,
stability, vibrations.
Finally, conclusions about jet amplifier design, either of bistable or
proportional type, are drawn.
Udine, September 1970
Chapter I
SUBMERGED JETS
The fluid flow issuing from a nozzle into a sur
rounding medium is called a jet.
A submerged jet spreads out in a medium made of
its same fluid.
The flow velocity at the nozzle output is approx
imately uniform. A potential core can be observed, that keeps
the same velocity at any point, but decreases in size until fi
nally it vanishes. A growing boundary layer appears, that is lim
ited at its inner side by the potential core, and at its outer
side it has free boundaries; it makes little difference if one
considers an infinite layer with asymptotic velocity profilas or
a finite thickness layer limited by a line joining the zero axial
velocity boundary points.(see Fig.l.l).
The boundary layer is due
to mixing and entra'inment ef
fects between the stream issued
from the nozzle and the surreund
ing fluid. Such effects are due
in their turn to the properties
of viscous fluid, that allow a
Fig. 1.1 continuous transfer of mass and
6 Chap. I. Submerged jets
momentum between the gas issued from the nozzle and the ambient
gas.
As any flow, jet flow may be laminar or turbulent,
depending upon the relative importance of viscous and inertia
stresses.
Pure viscous stresses obey to the Newton law:
'C=~g~
( 1.1)
where p 1. s the v1. scos1. t y coe ff.1 c1. ent , ecli nu , the velocity gra-
dient normal to the streamlines, and 1:' is the tangential stress.
Tangential stresses in laminar flow are given by
eq. ( l.l) .
Laminar flow is characterized by smooth, separate
streamline patterns.
The ratio between inertia and viscous stresses is
7V l. V
characterized by the Reynolds number' Re = where is
v
the flow speed, is the kinematic viscosity, and l, is a char-
acteristic dimension of the flow.
When the Reynolds number increases beyond a cer-
tain value, laminar flow becomes unstable; an eddy motion ap-
pears, that produces mixing among different fluid layers.
Transition from laminar to turbulent motion oc-
curs even for smaller Reynolds numbers, if some disturbances in-
teract with the stream.
The velocity in turbulent motion is given by a
Jets equations 7
mean velocity plus a fluctuation velocity, whose time-averaged
value is zero.
For example, in plane motion, the instantaneous
velocity components U. and 1Y are given as sum of the mean veloc
itios u. and '-\1' and of the fluctuation components u,I and '\1' I :
.
u, = u, + lA.' ' '\1' := ". + '\1' I (1,2)
where the time integrals of U: and v'along a sufficiently long
time interval are null.
The momentum equation relative to the X axis can
be written in the form:
+ ". i) u. 1 cJ p
( 1. 3)
i)':l ~ iJx.
In the case of laminar motion along the X axis, ~
is given by eq. (1.1), therefore it is
iJ
'txy
( 1.4)
~ iJy
In the case of turbulent motion, the pure viscous
stresses (1.1) if one considers only the mean values of the velo~
ities, must be increased by a factor due to the momentum trans-
fer from one to the other flow layer.
Consider now a motion one-dimensional on the aver
age.
A particle having a component velocity u, relative
to the X axis has a fluctuation velocity "\7' I relative to the y
8 Chap, I. Submerged jets
axis.
Being u. = lA. + U:, a momentum f? u; \T' is transferred
in each time unit through the unit surface of any layer.
By averaging, one obtains a pure turbulent stress:
( 1. 5)
One can note that U.I '\1 I may be different from ze
ro ( if a "correlation degree" exists _between the fluctuation
velocity components ~·and ~'), even if the averaged values of the
fluctuation velocities are always null according to their defi-
nition.
In turbulent motion, the stresses (1.1) and (1.5)
are both present, but usually the stresses given by (1.5) exceed
by far the pure viscous stresses (1.1), that can then be neglec~
ed.
An exact analysis of turbulent motion is not a-
vailable at the present time; several theories have been estab-
lished, that account for the global effect of turbulence on the
motion of sizeable quantities of fluid.
Prandtl's mixing lenght theory was issued in1925,
It states that a fluid particle preserves averagingly its initial
velocity until its transverse displacement ceases, after which
its velocity jumps to the value of the velocity of the new flow
layer.
cJU.
If cJy is the mean transverse velocity gradient,
Mixing lenght hypothesis 9
one has
(1.6)
Furthermore, the fluctuation velocity components
U: and v' are of the same order of magnitude. One obtains there
fore:
I
T == ntz. au . I ü ü, 0. 7)
r.y T Üy Üy
where t is the mixing lenght, a local characteristic quantity
of the fluid motion.
Prandtl's mixing lenght theory for free turbulence
advances the hypothesis of constant mixing lenght 1. in each
transverse section of the flow. This hypothesis is justified by
the absence of walls; near them the mixing lenght would decrease.
It remains to be taken into account the mixing
lenght variation in the stream direction.
The similarity of velocity profiles can be assumed
for "fully developed" jets, that is in a region sufficiently far
from the nozzle so that the boundary layer fills entirely the
cross section.
The mixing lenght, that varies proportionally to
a characteristic lenght,can be assumed proportional to the width
b of the jet.
If one compares formulas (1.7) and (1.1), one can
10 Chap, I. Submerged jets
write eq.(1.7) in the form:
( 1.8)
where E is an apparent kinematic viscosity, according to the
Boussinesq's definition.
The mixing length theory gives:
( 1. 9)
cJU.
Therefore & should be null when -iJy- = 0 , that is
not confirmed by experimental results.
Anyway, the gross results obtained from this hy-
pothesis are in good agreement with experiments.
In order to remove the inconsistencies of his for
rner hypothesis, Prandtl announced in 1942 a new free turbulence
hypothesis. Here the apparent kinematic viscosity, E , instead
of the mixing length t , is assumed to be constant in any cross
section,
According to this hypothesis, E is proportional
to the local rnixing region width, b , and to the difference bet-
ween minimum and maximum velocities in the cross section:
( 1.10)
where k is a constant.
If one considers a fully developed jet flowing
through fluid at rest, b is the jet width, and U.mi.n = 0 ; there