Prediction of Transient and Steady Turbulent Free Subsonic

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


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


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


Fig.3 Histories of axial profiles of mean axial velocity on jet axis


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


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


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


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


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.

Documents

Prediction of Transient and Steady Turbulent Free Subsonic