prediction of transient and steady turbulent free subsonic
TRANSCRIPT
Memoirs of the Faculty of Engineering,Okayama lIniversity,VoL25, No.2, pp.39-54, March 1991
Prediction of Transient and Steady Turbulent FreeSubsonic Air Jets
* **Felix Chintu NSUNGE , Eiji TOMITA**and Yoshisuke HAMAMOTO
(Received February 18 , 1991)
SYNOPSIS
Velocity distributions and related parameters of
transient and steady, turbulent air jets issuing under
atmospheric conditions at Mach 0.14, 0.33 and 0.5 have
been predicted using Navier-StokestN-S) equations for
compressible flow and incompressible flow independent
ly with the k-E model. The closeness and consistence
of the results predicted by the N-S equations for
compressible and incompressible flows as well as with
relevant measurement or similar prediction show that
the incompressible flow assumption for at least some
subsonic gas jets issuing at velocities higher than
Mach 0.3, the general limit for incompressible fluid
flow, can be reasonably accurate particularly in the
main fully developed flow region. This suggests that
the divergence term in source terms of the momentum,
turbulence energy and its dissipation rate equations
have negligible effects for some seemingly
compressible high speed, subsonic free gas jets. The
computation time is reduced by at least 18 % when
incompressible flow assumption is used.
1. INTRODUCTION
Experimental results from many sources(1) have shown that trends
of velocity and temperature distributions in incompressible and
compressible free gas jets are identical. This empirical fact has led
some researchers notably Abramovich[1 1, Pai[21, Kumar[31 to suggest
* Graduate School of Natural Science and Technology
** Dept. of Mechanical Engineering
39
40 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
that at least for engineering purposes, some high speed, subsonic tree
gas jets may be assumed to be incompressible not only in the inviscid
potential core where the observed velocities, pressure, temperature
and so density are constant, but also outside the potential core and
particularly in the fully developed main region where flow velocities
are low at less than about Mach 0.63 for say the extreme case of the
sonic gas jet. For instance, for the fastest air jet at Mach 0.5
investigated here, the maximum Mach number in the main region as
estimated from semi-empirical velocity decay laws is about 0.32 while
the maximum density variation is about 4 %. Generally in fluid flow,
low speed gas flow at velocities below Mach 0.3 is treated as
incompressible flow because density variation is negligible[2-4J.
other criteria that may be used for identifying incompressible flow is
the maximum density variation in the flow which some researchers set
at up to 10 %. Whichever criteria is used, the main impli~ation of
incompressible flow as applies to equations governing continuum fluid
flow particularly the continuity and Navier~Stokes equations as well
as the turbulence energy .dissipation rate equation when eddy viscosity
is estimated by the high version of the two-equation k-E turbulence
model is that the constant density causes the divergence in the
continuity equation to vanish and consequently the same divergence
term in the source terms of the momentum, turbulence energy and its
dissipation rate equations equally disappear from these equations'and
only the velocities, pressure and in addition temperature need to be
solved for non-isothermal flows. Since even in cold incompressible
fluid flows small variations in temperature do exist in order to
offset the pressure variations so as to maintain constant density,
temperature here may be solved or assumed to be constant. Here
particularly because of the need to approximate all initial variable
fields, this being inherent in numerical iteration methods which the
simple algorithm also uses due to non-linearity of the governing
equations, velocity, pressure and temperature fields have been
calculated.
The problems solved involve two kinds of single shot, turbulent
free air jets issuing at an initial velocity, Uo under atmospheric
conditions of the surrounding air. The transient jet is characterised
by a short air injection time period of 10 ms while the steady jet
is one in which injection continues for a much longer time to allow
the jet to develop to its steady state. Both types of jets are
injected from a small 1 mm diameter automobile injector orifice.
Prediction of Tmnsient and Steady Turbulent AirJets 41
Similar gas jets find applications in many fluid mixing systems like
diffusion combustion furnaces and internal combustion engines, thrust
force propulsion systems like rocket, gas turbine jet engines,
helicopter engines for vertical take off and landing as well as in
fluid mixers.
2. GOVERNING EQUATIONS
2.1 Compressible flow
The governing equations used in the compressible flow solution
approach are the complete N-S equations represented by the general
property transport equation used in prediction of methane free gas
jet[51 for all variables except the species mass fraction because in
the present study, the injected and surrounding gases are identical.
2.2 Incompressible flow
The equations used in the incompressible flow solution take the
following form after divergence terms, V.U , are removed from source
terms of momentum, turbulence energy and its dissipation rate
equations since the density is now assumed to be constant. Definitions
of variables and other related symbols remain the same as given in[51
P~t($) +P~x(U$) +P~~r(rv$) = ~x(r$~) + ~~r(rr$~) + S$ (1)
where t, p, r and x stand for time, density, axial and radial
coordinates, r$' S$ are corresponding effective diffusivity and
variable source term whose definitions are given in Table.1.
Table.1
$ r$1 0
u IJ e
v IJ e
hIJ e
°h
Definitions of coefficients and source terms in Eq.(1)
Source term, S$
S'
~x(IJe ~~) + ~~r(rlJe ~~) ~ ~~ - ~~x(pk) + S~
+ S'v
ap~x{IJe(1
1 a u 2~2 )} 1 a ( 1 1 a u 2 v 2
at + - - )-(- + + --{ rIJ - - )-(- +- )}0h ax 2 2 rar e 0h ar 2 2
a 1 ~ ) ak} 1 a 1 ~ ) ak}+ ax{IJ e (o - + --{ rIJ (- - + S'k 0h ax rar e ok 0h ar h
kIJ e G S'- - pE +ok kIJ e
~(C1G - C2PE) + s'E°E E
Here, G IJ {2«a u )2 +(a v )2 ) +( au + av ) 2 +2(~)2}e ax ar ar ax r
42 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
The k-£ turbulence model constants C~, C1 , C2 , ok' o£ used are 0.09,
1.44, 1.92, 1.0, 1.3 respectively while the effective Prandtl number,
0h is 0.7. The dynamic eddy viscosity, ~t is calculated from the
semi-empirical relation[6,71i
lJt = C~pk2 IE
2.3 Auxillary relations
In both compressible and incompressible flow solutions,
Sutherland's law[41 has been used to evaluate molecular viscosity
which varies with temperature. The ideal gas equation has been used to
calculate densities only in the case of compressible flow solution
approach because the density is non-uniform for this case. These
auxillary relations close the two systems of equations solved
independently for the compressible and incompressible solution
approaches. E.ach system of equat.ions is solved simultaneously in an
iteration scheme identical to that used in ref.5.
2.4 Boundary and initial conditions
Boundary conditions in both incompressible and compressible flow
solution approaches are essentially identical to those used for the
solution of the methane gas jet[51 with the species mass fraction
equation and its boundary conditions removed because the species mass
fraction is not solved for in the present air jet calculations. For
instance to simulate free jet conditions the temperature and density
of the surrounding room air were again set at 301.15 K and 1.18 kg/m 3
respectively'in both solution approaches. Initial velocities, Vo used
for the three independent cases studied were 45, 105 and 160 m/s
while the initial exit radial velocities Vo were set to zero. Initial
densities, Po and temperatures, To also set uniform across the nozzle
opening were estimated from the ideal gas and isentropic law[31 with
the effective injection to surrounding air pressure ratio obtained
through the energy balance formula are given in Table 2. Initial
conditions for turbulence energy and its
Table 2 Initial conditions at nozzle exit
v V T Po0 0 0
m/s m/s K kg/m 3
45 0 300.14 1 .9
105 0 295.7 1 .194
160 0 288.4 1 .228
Prediction of Transient and Steady Turbulent AirJets
Auid Air[,48 Boundaries
North Boundaryair--------=-..l--__+-:--,cO
~~____-'-w
X,u South BoundaryNozzle Diameter 1
Fig.1 Calculation domain
43
dissipation rate being functions of the initial velocity, Uo were set
as defined in the solution of the methane gas jet[51.
2.5 Numerical procedure
Using control volume formulation, Eq.(1) is discretized into a
general finite difference equation and the SIMPLE algorithm[81 is
applied on the calculation domain shown in Fig.1. As outlined in the
solution of the methane gas jet[51 again the pressure fields are first
guessed and tentative velocity u, v fields and subsequently other
variable fields are calculated. Their accuracy is checked in the
continuity equation before correction, if necessary, is made to the
pressure fields. This is repeated in an iteration procedure until the
pre-set convergence condition like the sum of mass efflux in all
control volumes or the velocity difference between adjacent iterations
and timesteps at suitable convergence monitoring points is practically
zero for transient and steady jet flow conditions respectively.
3. RESULTS
3.1 Numerical performance
Although coarser computational grids like 21x31, 41x31 for grid
nodes in the axial and radial directions as well as smaller and bigger
timesteps like 0.1, 1 ms were tried with reasonably good results, a
51x41 computational grid using timesteps 0.2 and 0.5 ms for transient
and quasi-steady jets respectively were finally found to be accurate
enough and used in all results here. Because the injector orifice is
small, finite space lengths were stretched at rates of 1.07 and 1.21
in the axial and radial directions respectively.
The iteration convergence was set to be attained when the net flow
on the whole grid was practically zero or the change in axial
44 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
velocity, u at a suitable grid node on the jet axis was less than
0.003m/s between adjacent iterations within the same timestep. The
total calculation time periods for transient and quasi~steady jets
were 20 and 247 ms respectively. With the number of iterations per
timestep varying from 30 to 110 to attain convergence for the
incompressible flow solutions, a total of about 7000 and 34580
iterations were executed in each complete run for transient and quasi
steady jets respectively on the ordinary NEC-ACOS2010 Computer.
However using the same 51x41 grid, the total computer CPU time, varied
with the jet initial velocity, Uo and varied from 200 to 400 sec for
transient jets and 300 to 631 sec for quasi-steady jets for the
incompressible flow solutions. These CPU times were at least 18 %
lower than those used in the compressible flow solutions for the jets
with the same initial velocity.
3.2 Calculated results
Some results of the two prediction approaches which involve
independent application of firstly N-S and k-E property transport
equations for compressible fluid flow and secondly those for
compressible fluid flow to solve the same air jet flow problems are
presented here mainly for the typical case of the fastest jet with
initial velocity, Uo =160 m/s. However, although all predicted results
for jets at initial velocity, Uo =45 and 105 m/s are not presented
here, the predicted global trends of their velocity distributions and
other related parameters closely for the jet at Uo ~160 m/s presented
here.
Figure 2 shows typical distributions of transient mean axial
velocity as predicted independently by the compressible and
incompressible flow solution approaches at several locations on the
jet axis for the air jet with initial velocity, Uo =160 m/s. At each
location near the injector, the velocity rises to a maximum steady
value with the rate of rise decreasing with the axial distance, x from
the nozzle while at locations far from the injector maximum steady
values are not attained by the time the air injection is stopped at 10
ms. After air injection is stopped, velocities decrease towards zero
with the rate of fall decreasing with the axial distance in a similar
way exhibited during their rise. Although there are some discrepancies
especially in the transition region of the jet, the compressible and
incompressible flow solutions seem essentially the same. Moreover,
these predicted trends of the velocity distributions compare
Prediction of Tmnsient and Steady Turbulent Air jets 45
reasonably well with experimental data obtained by other researchers
like Witze[9] for similar transient jets issuing from small round
orifices;
Uo= 160 m/s MACH = Q. SDo =Imm
X/Do;~:~
=3.8
=5.3
=10.3
=?1 .?
1(/ :: ",,0.3
~= 95.6
200
180
160
140III
E 120
E 100::>
>- 80>-
u 60a...JUJ 40>
...J20a:
xa: 0
-=3.8
If'. =5.3
\If'.. =10.3
= 21.2'(
= '0.3 l\.
/ I~
= 95.6
X/DD=! .2-2 5
Uo= J60 rrvs MACH = Q. 5DD =1 mm
200
180
160
1401II
e- 120
e 100::>
>- 80>-
u 600...JUJ 40>
...J20a:
xa: 0
I I I Io 5 10 15 20
TIME FROM JET INITIRTION. t ms
j I I Io 5 10 15 20
TIME FROM JET INITIRTION·t ms
(a) Compressible (b) Incompressible
Fig.2 Histories of mean axial velocities on jet axis
180 180 _en 160 Uo 160 mjs MACH = 0.5 .!!! 160 UO 160 m/s MACH = 0'5......E PRESENT THEORY: E
PRESENT THEORY:E 140 (') AT TIME=I ms E 140 (') AT TIME=I ms::> ::J
£ AT TIME=2 £ AT TIME=2>- 120 + AT TIME=3 >- 120 + AT TIME=3>- x AT TIME=4 >- x AT TIME=4u 100 ¢ AT TIME=5 u 100 ¢ AT TIME=5a .. AT TIME=6 a .. AT TIME=6...J " AT TIME=7 ...J
" AT TIME=7UJ 80 UJ 80> ;< AT TIME=8 > ;< AT TIME=8...J 60
y AT TIME=9...J 60 yAT TIME=9
a: "AT TIME=IO a: " AT TIME=IOx 40 STEADY JET: x 40 STEADY JET:a: a:___ TOLLMIEN'S - -- TDLLMIEN'S THEORYz THEORY za: 20 a: 20UJ UJ>:: >::
0 --- - 0
0 40 80 120 160 200 0 40 80 120 160 200RXIRL OISTRNCE. X mm RXIRL OISTRNCE. X mm
(a) Compressible (b) Incompressible
Fig.3 Histories of axial profiles of mean axial velocity on jet axis
46 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
In Fig.3 is shown typical histories of axial profiles of the mean
axial velocity U as predicted independently by the compressible andm
incompressible flow solution approaches at several locations on the
jet axis for the air jet with initial velocity, Uo =160 m/s. The
temporal development of unsteady and steady parts of the transient jet
is exhibited more clearly here in that the locations identifying the
end of the steady rear parts and the beginning of the unsteady frontal
parts of the transient jet are easily seen as the point where these
velocity profiles leave the curve profile of the steady jet.
Tollmien's steady incompressible jet solution at a divergence factor
of 11.8[1,2] is plotted here mainly for two reasons. Firstly, for
clarity of result presentation, it serves as a common reference
against which compressible and incompressible flow solutions are
compared and secondly for checking the accuracy of the present
predictions not only in the steady part of the transient jet but also
that of both the present transient and quasi-steady jet solutions
since it is noted that the steady state numerical solution is obtained
by calculation the unsteady flow problem over adequately long time
periods. In Tollmien's solution which is well supported by
experimental'data from many sources[1,2], the Prandtl mixing length
hypothesis was used to estimate eddy viscosity, Wt' Indeed global
trends of the present predictions by both compressible and
incompressible flow solution approaches compare reasonably with
Tollmien's solution particularly in the fully developed downstream
main region.
However, in the transition region the compressible flow solution
is lower than that of the incompressible flow solution by a maximum of
about 8 % at x =24 mm from the jet initiation plane(nozzle). Temporal
jet penetration may be estimated from velocity profiles shown in Fig.3
if the velocity of surrounding air pushed ahead of the jet tip is
taken into consideration because the pushed air velocity is also part
of the velocity profile in the unsteady front part of the jets and the
jet tip lies at some location on the jet axis where the axial velocity
decays to almost zero. It is seen from Fig.3 that both the
compressible and incompressible flow solution approaches predict
approximate temporal jet tip penetrations and axial lengths of the
steady rear as ~ell as unsteady front parts of the transient jet to
almost same degree of accuracy at times, t greater than 2 ms after jet
initiation. It is also seen that the axial length of the unsteady
front part decreases with time from a maximum of about 50 % of the
Predictitm of Transient and Steady Turbulent Air jels 47
approximate jet penetration at time t =1 ms to about 30 % at time 10
ms.
In Fig.4 is shown temporal radial distributions of the axial
velocity in the steady rear parts of the transient jet as predicted by
the compressible and incompressible flow solution methods at different
cross sections. Here, the velocity is normalised by the respective jet
INCOMPRESSIBLE
1.0
J 0.8'-::>
>- 0.6t-
u0-'w> 0.4-'a:
xa:
00.2
z
0.0
Uo
Time =3 ms160m/s MR'CH = O· 5
=1 mm
PRESENT THEORY:('lX/Oo=8.S'" X/Do = 10.3+X/Do=12.2X X/Do = 14.2<>X/Do=16.4
OTHER'S THEORY:3/2 PO~ER LR~
OTHER'S EXPT:'* TRUPEL (1)-GAUSSIAN
CURVE
1.0
EO.8::>'-::>
>- 0.6t-
u0-'w> 0.4-'a:xa:
00.2
z
0.0
Uo
Time =6 ms
160 m/s MRCH= 0'5Do =1mm
PRESENT THEORY:('lX/Do =8.5'" X/Do = 10·3+ X/Do = 12·2XX/Do=14·2'" X/Do = 16·4.. X/Do = IB.7l(X/Do=21·2Z X/Do = 23.9
OTHER'S THEORY:3/2 POWER LAW
OTHER'S fXPT'* TRUPEL( f J+"", -GAUSSIAN
."'Cl: CURVE
0.0 0.5 1.0 1.5 2.0 2.5
'NO RRDIRl DISTRNCE. r/Re
0.0 0.5 1.0 1.5 2.0 2.5
NO RRDIRL DISTRNCE. fiRe
COMPRESSIBLE
OTHER'S THEORY:3/2 PO~ER LA~
OTHER'S EXPT,*TRUPEL (I)
-GAUSSIANCURVE
Time =6 msUo 16orn/s MRCH=O'!i
Do =1 mmPRESENT THEORY:('lX/Oo=B.S'" X/Oo = 10·3+ X/Do = 12.2XXIDo=14·2'" X/Do = 16.4.. X/Do = 18.7l( X/Do = 21 .2ZX/Do=23·9
Time =3 msUo 160 rry's MRCH: 0.5
1.0 Do =1 mm 1.0PRESENT THEORY:
('l X/Do = 8. SA X/Do = 10.3
J 0.8 + X/Do = 12.2 EO.8XX/Do=14.2 ::>
"- "-::> <>X/Do=16.4 ::>
>- 0.6 OTHER'S THEORY: >- 0.6t- 3/2 PO~ER LR~ t-
uOTHER'S EXPT: u
0 0-' '* TRUPEL (1) -'w -GAUSSIAN w> 0.4 > 0.4CURVE-' -'a: a:x xa:
+\ a:0
0.2 0.20z z
0.0
0.0 0.5 1.0 1.5 2.0 2.5 0.0
NO RRDIRL DISTRNCE. fiRe
Fig.4 Radial distributions of
0.5 ].0 1.5 2.0 2.5
NO RRDIRL DISTRNCE. r/Reaxial velocity
48 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
axis velocity, U of the cross section obtained from predictions inm
Fig.3 while r is normalised by the radial distance Rc at which the
axial velocity of the cross section is one-half of Urn and the inner
turbulent core are located. Temporal values of R which represent the. cjet radial spread were evaluated from predicted axial velocity fields
by interpolation and found to increase linearly with the axial
distance at an approximate uniform rate, RC = 0.103x at all times,
2 ~ t ~ 10 ms in the fully developed region. Since predictions from
both methods very closely follow the Gaussian and 3/2 power law curves
which represent experimental measurement from many sources[1,10],
predictions of radial distributions of the axial velocity by the
compressible ~nd incompressible flow approaches are almost identical.
Both predictions indicate the similarity and self-preserving nature of
the jets in the steady rear parts of transient jets shown by the
collapsing of all velocity profiles at different jet cross sections
like those shown in Fig.9(a) for the predicted steady jet into a
single unique curve. In the steady part of the transient jet, this
same property is exhibited at different times after jet initiation.
The predicted axial velocity fields may be used for evaluating
entrainment by numerical integration.
Transient radial distributions of the radial velocity in steady
rear parts of the transient jet as predicted by the compressible and
incompressible flow solution approaches are shown in Fig.5. Here, A in
the non-dimensional radial velocity and radial distance definitions
has a value of 0.78 which is a quotient of a constant, a =0.071 and
slope of the half Urn velocity ray of about 0.091[1,2]. Predictions of
global trends by both approaches fit Tollmien's prediction fairly well
particularly in the inner turbulent. core, r ~ Rc although prediction
by the incompressible flow approach is not as good outside this core.
This discrepancy is possibly due to non-turbulent flow, complex
recirculatory and considerable entrainment flow occurring in the
transition and viscous superlayer as well as in the unsteady jet
boundary interface region. Like in the case of radial distribution of
axial velocity, the similarity and self-preserving properties of even
transient jets are exhibited in the steady rear parts of the transient
jets at all times by the collapsing of all dimensional velocity
profiles at different cross sections as typically shown in Fig.9(b)
for the predicted steady jet into a single unique curve with apparent
maxima and minima points.
Prediction of Tmnsient and Steady Turbulent AirJets 49
The minima point located at coordinates va1ues (3.57,-0.32) in
Fig.5 is the approximate location of the jet outer boundary where the
entrainment velocity, V occurs at all cross sections. At all crosse
sections in the steady rear part of the transient jets, present
predictions show that the normalized radial location of this minima
point does not vary with time which again indicates t·hat the
INCOMPRESSIBLE
'\.,~ \!)
, .."
OTHER'S THEORY:- TOllMJ ENfO
Time =6 msUo = 160 m/s MACH =0·5
Do =1PRESENT THEORY:
<!> X/Do = 9.37AX/Do 11.21
r + X/Do 13.18" x X/Do = 15.28~....\ <> X/Do = 17.53
" + X/Do = 19.94~ X/Do = 22.52z X/Oo = 25.28
~., OTHER'S THEORY:'\'. - TOLLMIEN(1l,.
\".,, "'.
\ .... ,z:.~
.... ~6 .z",>!!J x.e.
0.4
~ -0.2
">
>-t- 0.0U<:)
-.JW>
II
E::> 0.2
THEORY:9.3711.2113.1815.2817.53
PRESENT<!> X/Do =A X/Del =+ X/Do =x X/Do =<> X/Do =
Time =3 msUo 160 nys MACH = O' 5
Do =1 mm0.4-
E::> 0.2II
"->
>-t- 0.0U<:)
-.JW>-.J -0.2II
0IIa::0z
-0.4
o 1 2
NO RRDIRL DISTRNCE.o 1 234
NO RRDIRL DISTRNCE. r/(R Xc}
COMPReSSIBLE
THEORY:9.3711.2113.1815.2817.5319.9422.5225.28
<!> X/Do =A X/Do =+ X/Do =x X/Do =<> X/Do =+ X/Do =~ X/Do =Z X/Do =
PRESENT
Time =6 msUo = 160m/s MACH =0·5
Do =1 mm
0.4
:of,0.2
II
"->
>-t- 0.0U<:)
-.JW>-.J
-02 jII.-0IIa::0z
-0.4
THEORY:9.3711.2113.1815.2817.53
<!> X/Do =A X/Do =+ X/Oo =x X/Do =<> X/Do =r
'·_·-'/. ~'"
t. '1<'.,..''1<\.~.,
~,
.~ OTHER'S THEORY:'" - - TOLLMIEN [1)~
\....,.,'X .. _ + )(
~ O .. ---Cl
Time =3 msuo = 160 m/s MACH =0'5
Do =1 mmPRESENT
0.4
::>EII
0.2
"->
>-t- 0.0U<:)
-.JW>
-.J -0.2II
0IIa::
0z
-0.4
o 1 2
NO RRDIRL DISTRNCE.o 1 2
NO RRDIRL DISTRNCE.3 4
f/lR Xc}
Fig.5 Transient radial distributions of radial Velocity
50 Felix Chinlu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
axial velocity fields by numerical
the mass flow rates at the nozzle
radial spread of the steady part of the is uniform with time. With the
value of A = 0.78, the radial location, Rm of the jet outer boundary
may be estimated from the constancy of the normalized radial distance,
R /(AR ) = 3.57 as R = 2.78R while the constancy of V /(AU )=-0.32me me· emis used to derive the entrainment velocity, V near or at the minima
epoints which locate the approximate location of the jet outer
boundary as Ve = 0.249Um . Thus knowledge of these coordinate values
from present prediction results, R obtained by radial interpolationcof predicted axial velocity fields and jet axis velocities, Urn from
Fig.3 then gives all entrainment velocities and their radial locations
from which V -based entrainment of the surrounding air has beeneestimated and found to vary linearly with the axial distance according
to the relation, Q /Q = 0.3x for the steady jet. Th~s entrainmentx 0 .
variation is almost identical to the U-based entrainment distribution
determined from the predicted
integration. Here, Q and Q areo x
exit and any cross section at axial distance x respectively.
= 288 ,301 K PRANOT .MACH = 0.7, 0,5 = 288 .301 K PRANDT.MACH =0,1,0·5a - - - -- - - - --:c
PRESENT THEORY PRESENT THEORY
r-- -3 " AT TIME= I ms -3 " AT TIME=lms0
• RT T I ME= 2 • RT Tl ME= 2+ RT TI ME= 3 + RT Tl ME: 3
UJ x RT TI ME= 4 x RT Tl ME= 4
'";J -6 • RT TI ME: 5 -6 • RT T I ME= 5r--
• RT TI ME: 6 • AT Tl ME= 6cr
'" x RT TI ME = 7 x AT Tl ME= 7UJ z RT TI ME: B z AT T I ME= B£L>: y AT TIME=9 y RT Tl ME: 9UJ -9 • RT TIME= 10 -9 • RT TIME= 10r--
UJ>
r-- STEADY JET: STERDY JET:cr -12 SQUIRE'S EXPTI 2) -12 SQUIRE'S EXPT(2)-'UJ
'" DT = T - Ts DT=T-Ts-15 -15
I I I
0 40 80 120 160 200 0 40 80 120 160 200
RXIRL OISTRNCE. X mm RXIRL DISTRNCE,' x mm
(a) Compressible (b) Incompressible
Fig.6 Histories .of axial profiles of relative temperature on jet axis
Temporal axial profiles of relative temperature, DT =T-T ass
predicted by the two solution methods on the jet axis are shown in'
Fig.6 where Squire's semi-empirical law[2] for jet axis temperature
for the steady jet is also plotted as common reference for comparing
predictions by ,the two methods. Here, Ts is the surrounding air
Predutian of Transient and Steady Turbutent AirJets 51
temperature assumed to be same as room temperature and once again
solutions by the two,methods particularly the global trends of
variations in the main region are very nearly identical as they both
fit Squire's law closely. The trends of temporal and spatial variation
of temperature on jet axis very nearly conform to those of axial
velocity given in Fig.3 which may be expected since they are both
solved by the same general property transport equation (1) and the
present predictions indicate that approximate jet tip penetration
indicated by these temperature profiles would reasonably accurate as
well. It is also worthy noting the similarity of the predicted axial
velocity in the unsteady front parts of the transient jets. When these
are normalized as shown in Fig.7, they appear to collapse into one
single curve represented by the 3/2 power law;
( 2 )
where temporal velocity, UI and distances x, Xl, Xo are as shown on
the insert. The radial distribution of turbulence intensity in the
quasi-steady jet as predicted using the incompressible flow assumption
for Uo =160 mls is shown in Fig.8. The prediction is near to
measurement by Corrsin[2] in the inner turbulent core, but higher
further outward although the decay trend is similar to measurement by
RT TIME =245 ms
Uo = 16Dm/s MRCH=O·5 Do=lmmPRESENT THEORY:'" X/Do = 9.37• X/Do 13.18+ X/Do = 17.53x X/Do = 25.28¢ X/Do = 34.77+ XIOo 42.2S"X/Do 50.82z X/Do = 60.63
1.0 uo = 160 m/s tlACH = o· 5"-
~~PRESENT THEORY'
; • AT TIME= Ims0.8
,-ATTIME=2
"-
-'"::J • AT TIME= 3.~ • AT TIME= 4.
• AT TJME= 5>- .'..... : ' . .AT TIME= 6u 0.6 "- 'ATTIME=70 ,. z AT TIME= B--'UJ . 'l! vAT TIME= 9> Y\ , • AT TIME= 10--' 0.4 .' 312 POWER LAWa: f!).' +,x ,a:Cl
U .!,z 0.2 v ~.
U1 AX • ".,. '~- -J0.0 0 )Co
0.0 0.2 0.4 0.6 0.8 1.0
NO RXIRL OISTRNCE. I X-Xliii XD-XII
0.50
=>E 0 .40"-
=>
>- 0.30.....V)
zw..... 0.20z
wLJZW 0.10-'=>lD
'"=>.....0.00
f"x•
0.0 0.2 0.4 0.6
NO RROIRL OISTRNCE.
~z• x'. '".• x
•
1.0
Fig.7 Axial velocity on jet
axis in the unsteady front part
Fig.8 Radial profiles of
turbulence intensity
52 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
Corrsin. The turbulence velocity, u ' is determined from the predicted
turbulent kinetic energy using the local isotropy of turbulence
assumption[2]. It is maximum near the jet axis at about the same
location where the maximum energy dissipation and maximum shear
stress occur.
Typical radial distributions of mean axial velocity across several
cross sections for the quasi-steady jet at Uo =160 mis, time =245 ms
as predicted using the incompressible flow assumption are shown in
Fig.9(a). Again the predicted curves at almost all cross sections in
the steady rear part seem similar in shape. The radial distance at
which the axial velocity becomes zero roughly locates the jet
outermost boundary and so the approximate radial thickness
characterising the jet spread. Figure 9(b) shows typical radial
distributions of predicted radial velocity across some cross sections
of the quasi-steady jet at Uo =160 m/s. These profiles which are also
similar in the steady' rear parts have a more complicated shape because
at each cross section, there exists both positive and negative maxima
radial velocities whose magnitudes decrease with the axial distance, x
from the injector. Furthermore, radial location of the maximum
negative radial velocities increase with the axial distance.
>
2.~
162
144
III126
t 108:::>
90>-0-
72u0...J 54w>...J 36a:x 18a:
0
-18
AT TIME =245 msuo = 160 m/s MACH=0·5 00=1 mmX{DD
3~
12 >-t-
,SU
24 0-'
•• W
53 >
" -'\12IT
0ITa:::
4
o
-2
-4
AT TIME = 245mS
Uo = 160m/s MACH =0·5 . 0 0 =1 mmPRESENT THEORY
~ X/Do= 1.8.X/Do=6.1+ X/Do= 11 .2x X/Do= 17.5• X/Do= 25.3+ X/Do= 34.8• X/Do= 46.4z X/Do= 60.6
y X/Do= 78.1xX/Do=99.4
40o4010 20 30
RRDIRL DI5TRNCE. r mmD 10 20 30
RRDIRL DI5TRNCE. Y mm
(a) Axial velocity profiles (b) Radial v~locity profiles
Fig.9 Axial and radial velocities in the steady jet as predicted
using the incompressible flow assumption
Predicnun of Transient and Steady Turbulent AirJets 53
It is noted that negative and positive radial velocities represent
radial motion towards and away from the jet axis respectively. When
compared with the jet spread from radial profiles of predicted axial
velocity like those in Fig.9(a), it is worthy noting that the radial
location of these predicted maximum negative radial velocities is also
.the approximate radial location of the jet outermost boundary in the
steady rear part of the jet and therefore the maximum negative radial
velocities are the approximate entrainment velocities, Ve at the
minima points exhibited in Fig.5. The jet spread can therefore also be
characterized by the radial locations of the maximum negative radial
velocities.
INCOMPRESSIBLE INCOMPRESSIBLE
90 IBO 270 360 450NO "RXIRL DISTRNCE. X/Do
1.00
:::J 0.9-eO.B:;:)
>-0.7
t-
u 0.60-' 0.5w>
-' 0.4cr:x 0.3cr:0 0.2z
0.1
0.0
0
Do =1 mmPRESENT THEORY:
AT TIME=245 msUo =160 m/sUo =105Uo =45
OTHER'S THEORY:STEADY JET:.TOLLM)EN III
.... _--_.:::_::-_-----------
.:;:)00.2
.....O. I, E
;:)O. I
>-t-
(J)
Zw 0.1t-z
0.1wuzw-'::::llD0::::::lt-
o
Do =1 mm
PRESENT THEORY:
AT TIME=245 msUo =160 m/sUo =105Uo =45
OTHER'S EXPT.STEADY JET
x CORRSINI 2)
90 180 270 360 450NO RXIRL DISTRNCE. X/Do
Fig.l0 Axial velocity on jet axis
of the quasi-steady jet.
Fig.ll. Turbulence intensity on
the jet axis of quasi-Steady jet
In Fig.l0 is shown axial distribution of the mean axial velocity
on the jet axis for all quasi-steady jets as predicted using the
incompressible flow assumption. Again in non-dimensional form the
velocities are represented by a single curve except in the unsteady
front part. Finally, axial distribution of the relative turbulence
intensity for all quasi-st~ady jets as predicted assuming
incompressible flow is shown in Fig.ll. The turbulence iritensities
which are determined from the predicted turbulence kinetic energy
54 Felix Chintu NSUNGE, Eiji TOMITA and Yoshisuke HAMAMOTO
using local isotropy of turbulence[2] also collapse into a single
curve and vary in a similar way.that mean axial velocities upon which
they are superimposed vary as shown in Fig.l0.
4. CONCLUSIONS
(1) Some seemingly compressible subsonic, high speed free jets
discharging into same surrounding gas under atmospheric conditions may
be predicted with reasonable accuracy using the incompressible flow
assumption. The present calculation results seem to indicate that the
compressibility represented by divergence terms in source terms of
momentum and turbulence energy dissipation rate equations has
negligible effect in solution of some jet flows.
(2) The consistency with which the k-E has predicted similar turbulent
air jet flows at substantially different initial velocities, Uo over
short and long calculation times, in transient and steady jets using
different time steps and "different grid spacings with reasonable
accuracy indicates that the k-E model may be deemed quite reliable and
so the Prandtl mixing length hypothesis used by Tollmien.
(3) Present predictions show that the similarity and self-preserving
properties of radial distributions of axial and radial velocities as
well as temperature which are usually associated with steady jets are
also exhibited in the steady rear parts of the transient jets.
REFERENCES
l.Abramovich,G.N.,"The Theory of Turbulent Jets",(1963),MIT.
2.Pai,S-I., "Fluid Dynamics of Jets",(1954),Van Nostran.
3.Kumar,K.L.,"Eng.Fluid Mech.",(1980),Eurasia.
4.Anderson,A.A.et al.,"Comp.Fluid Mech.& Heat Transfer", (1985),
McGrawHill.
5.Nsunge,F.C. et al., Numerical Calculation of a Transient, Methane
Gas Jet Discharging into Quiscent Atmosphere at Mach One, Memoirs
of the Faculty of Eng., Okayama University,Vol.25,No.2(1991).
6.Launder,B.E.et al.,Comp.Methods Appl.Mech.Eng.,Vol.3,(1974),p.269.
7.Launder,B.E.et al.,"Math.Turb.Models",(1972),Academic.
8.Patankar,S.V.,"Num.Heat Transfer & Fluid F19W",(1980),McGrawHill.
9.Witze,P.O. ,AIAA J. ,Vo1.21 ,No.2(1983) ,p.308.
10.Rajaratnum,N.,"Turbulent Jets", (1976),Elsevier.