The fluctuating field of a jet excited by transient mass injection is simulated numerically. The model is developed by expanding the state vector as a mean state plus a fluctuating state. Nonlinear terms are not neglected and the effect of nonlinearity is studied. The results show a significant spectral broadening in the flow field due to the nonlinearity. In addition, large scale structures are broken down into smaller scales

Bayliss, A.

Maestrello, L.

Turkel, E.

English

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NASA Technical Memorandum 84506 NASA-TM-84506 19820026315
• 1
STIMULATIONOFTHEFLUCTUATINGFIELD
OFA FORCEDJET
Alvin Bayliss, Lucio Maestrello, and Eli Turkel
AUGUST1982
I
N/ A
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23665
https://ntrs.nasa.gov/search.jsp?R=19820026315 2020-03-21T06:13:26+00:00Z

SIMULATION OF THE FLUCTUATING FIELD OF A FORCED JET
Alvin Bayllss
Courant Institute of Mathematical Sciences
Luclo Maestrello
NASA Langley Research Center
Ell Turkel
Tel Avlv University
Abstract
The fluctuating field of a jet excited by transient mass injection is
simulated numerically. The model is developed by expanding the state vector
as a mean state plus a fluctuating state. Nonlinear terms are not neglected
and the effect of nonlinearity is studied. The results show a significant
spectral broadening in the flow field due to the nonlinearity. In addition,
large scale structures are broken down into smaller scales.
*This work was partially supported by NASA Contracts No. NASI-14472 and NASI-
16394 for the first and third authors were in residence at ICASE, NASA Langley
Research Center, Hampton, VA. Additional support for the first author was
provided by the U. S. Air Force Contract No. AFOSR-76-2881 and U. S. Depart-
ment of Energy Contract No. DE-AC02-76ER03077.
Introduction
A numerical method for simulating the axisymmetric fluctuating field of a
forced circular Jet has been developed. The method is based on a solution of
the Euler equations for homentropic flow in cylindrical coordinates. The Jet
is forced by transient point mass injection. The source strength is scaled by
a parameter _ and the state vector W = (p,p,u,v) is expanded as ! mean
state plus a f__luctuati__ngstate of order _. Here p is the pressure, p the
density and u and v are the axial and radial velocity components respec- .
tively.
The solution is expanded in the form P = P0 + cp, etc. where the mean
state (wlth subscript "0"5 is assumed to satisfy the unforced Euler
equations. The variables (p,p,u,v) will represent the fluctuating field in
response to the Jet forcing. New variables e = (po+_p)u and
f = (p0+EP)v are introduced for simplicity. Wi[h cylindrical spatial co-
ordinates z(axial) and r(radial) the Euler equations for the fluctuating field
become
(PVo+PoV)
Pt+( u0+ 0u)z+ ( V0+P0V)r+ mr
et + (e(U0+EU))z + (e(V0+EV))r + Pz + fU0r - eV0r Efu _ emu 415r
ft + (f(U0+mU))z + (f(V0+_V) )r + Pr + eU0z fV0z Efv-- -- _ = _mv.r
In the development of system (I) it is assumed that the derivatives of
the mean pressure and density in space can be neglected. It is an exact con-
sequence of the full Euler equations with a source of mass injection, and of
the expansion into mean and fluctuating states. The source is cm (units
density/timeS. The mean velocities U0 and V0 are taken from measurements
of a spreading Jet. The system is solved in a cylindrical rectangle including
a seml-infinite pipe from which the Jet exits. The computational domain is
illustrated in Fig. i.
The use of the quasi-momentum variables e and f gives a system from
which the fluctuating field can be computed directly rather than as a small
part of the total field. In addition the linear limit can be recovered by
simply setting _ = 0 in (I). The nonlinear terms are explicitly
exhibited. The fluctuating pressure p is obtained from the density p by
the homentropic relation
= AT = P0(l + m _---)Y (2)
P0 '
where y = 1.4 in air.
The fundamental assumption entering into the derivation of 41) is that
the mean state is a solution to the unforced Euler equations. This is not
true for a state determined from experimental measurements. Numerical experi-
ments have verified however, that the qualitative features of the fluctuating
solution are insensitive to small changes in the mean state and thus we
believe that the solution to (15 qualitatively represents the fluctuating
field in response to a a given source.
The source is assumed to be a delta function in space (modelled by a
sharp Gaussian) with a pulse-like time dependence. The source location, Zs,
-2-
is on the Jet centerline approximately 1.2 jet diameters downstream of the
nozzle exit. Specifically
m(t,z,r) = f(t)6Ir2+(Z-Zs)2), (3)
where the function f(t) is
• f(t) = e-(at2+bt-2), (4)
for suitable constants a and b. The use of (4) permits the investigation
of a broad band spectrum.
The fluctuating field described by (i) reduces to the acoustic field for
large distances. The near field and flow field are dominated by instability
waves which are generated because the mean flow is linearly unstable. The
pulse is assumed to dominate the natural sources of Jet noise. These natural
sources are both the linear and nonlinear terms in (i). In real Jet these
terms are determined from the turbulent fluctuations whereas in the numerical
model these natural sources are excited by the pulse. The important physical
effect is the generation of packets of instability waves in the flow. Large
scale structures which are related to mean flow instabilities have been ob-
served experimentally in both forced and unforced Jets [1,2]. These struc-
tures interact with and modify the resulting acoustic fields. The model per-
mits this interaction to be studied in both the linear (E = 0) and nonlinear
(E # 0) regimes. Calculations with the linear model and comparison with
experiments are described elsewhere [3_4]. In the rest of this paper we
describe the numerical requirements in order to compute with this model and
the effects of the nonlinear terms on the fluctuating field.
Numerical Model
In order to numerically integrate (I) it is necessary to resolve the so-
lution over large length scales (far field, near field, and flow field). This
necessitates the use of a higher order difference scheme. The system can be
written in the form
wt + F + G = H, (5)z r
where w is the vector (_,e,f) and F, G, and H are appropriate
functions. Equation (5) is split into two one-dimensional operators in z
and r. Each l-d system is integrated by using a fourth order version of the
MacCormack scheme [5]
_n+l n At [,_TFi+SFi+l_Fi_2)wi = wi +_-_ + AtH 1
n+l I/ofw_n n At (TFi-SFi l+gi 2) + AtHl) (6)wi wi _ _ ,
together with a symmetric variant (HI is obtained from some splitting of
H). Typical grids require of the order of 40,000 grid points over distances
of the order of 50 Jet diameters in all directions. Our experience has been
that on problems of this size, second order schemes are not sufficient to ob-
tain accurate solutions with the amount of resolution that is feasible. The
explicit scheme (6) naturally lends itself to vectorization and has been
implementedon the CDC CYBER-203with great efficiencies.
-3-
In addition it is necessary to impose boundary conditions which accurate-
ly simulate outgoing radiation at the far field boundaries. A family of radia-
tion boundary conditions has been developed which provide increasingly
accurate approximations to outgoing radiation. The leading member of this
family is
Pt
w ere c _s the ambient sound speed and p_ the ambient density. Here •
dR = r2 _ z and u is the outgoing radial velocity based on a spherical co-
ordinate system near the source m.
System (I) includes terms proportional to r-I. This singularity at the
axis is resolved by including these terms in the flux vector G when r=0.
In addition it is necessary to modify the difference formula (6) at
boundaries. This is accomplished by introducing fictitious grid points
outside of the computational domain and using a third order extrapolation of
the flux function (F or G). This approach was found to be the most readily
vectorlzable. It has been verified that the resulting scheme is fourth order
accurate.
Nonlinear Results
We next describe results illustrating the effect of the nonlinear terms
on the fluctuating field. In Figs. 2 and 3 the fluctuating pressure is shown
as a function of axial location z/D (D is the Jet diameter) and non-dimen-
sional time tc/D for two different radial positions and
for _ = 0.00 and c = 0.05. All figures show an acoustic wave (speed of
sound normalized to unity) in the downstream direction trailed by several much
larger waves. These are instability waves which travel with a speed of
approximately .TUj where _= .66c is the Jet exit velocity. A series ofacoustic ripples can also seen propagating upstream. These are due to dif-
fraction of the upstream acoustic wave by the nozzle lip.
The figures indicate that the nonlinear terms have little effect on the
primary acoustic pulse and on the acoustic diffraction from the nozzle lip.
The nonlinearity has a pronounced effect on the instability waves. Increasing
the nonlinearity causes these predominantly large scale structures to break up
into smaller scale structures. This can be seen in both the additional
ripples which trail the instability waves and a sharpening of the individual
pulses indicating an enhanced high frequency content. It is also evident that
for increasing r/D these smaller scale structures are comparable in ampli-
tude to the primary instability waves.
In Figs. 4 and 5 the fluctuating vorticity is shown for two fixed times
and for _ = 0.00 and _ = 0.05. The intense vortices in Fig. 4 correspond
to the instability waves in Figs. 2 and 3 while the vortices at the later time
in Fig. 5 represent a residual shedding of vorticity from the nozzle lip. It
is apparent from the figures that nonlinear effects tend to slow down the vor-
tices as they propagate downstream. Thus the trailing vortices catch up with
the leading vortices and a possible pairing of vortices can be observed. True
vortex merging, which has been observed experimentally [6], depends heavily on
viscosity as well as nonlinearity and is not simulated here.
-4-
In Figs. 6a and 6b the normalized power spectral densities (PSD) for the
fluctuatingaxial velocity and pressure are plotted for the linear and non-
linear cases as a function of the Helmholtz number fD/c (f is the
frequency). These figures clearly indicate the shift into higher frequencies
and the overall broadening of the spectra caused by the nonlinear effects.
Then results are typical for the fluctuating field at all points.
o
Discussion
The fundamentaldifferencebetween the nonlinearand linear computations
is the breakdown of the large scale structures into smaller scale
structures. This is associatedwith a transferof energy into higher frequen-
cies or equivalentlya broadeningof the spectral content of the fluctuating
field. It is also illustratedby the increasedinteractionbetween the dif-
ferent vortices. In real Jets this is a fundamentalstep in the transitionto
fully developed turbulence. The results indicate that at leastthe initial
stages of this energy cascadeinto smaller scales can be simulatedJust by the
nonlinearterms of an axisymmetrlcand invlscldcalculation.
The generationof smaller scale fluctuationsfully Justifies the use of
fourth order spatialdiscretizations.The numericalscheme is accurateand is
in general stable. Higher values of € can be readilycomputedalthoughthe
Gausslan approximationto the 6 function source will have to be smoothed
OUt.
-5-
References
[I] Crow, S. C. and Champagne,F. H.: Orderly Structurein Jet Turbulence,
J. Fluid Mech., Vol. 48, Part 3, 1971, pp. 547-591.
[2] Maestrello,L. and Fung, Y. T.: Quasi-PerlodlcStructureof a Turbulent
Jet, J. Sound & Vib., Vol. 64, 1979, pp. 107-122.
Maestrello, L., Bayliss, A., and Turkel, E.: On the Interactionof a
Sound Pulse with the Shear Layer of an AxlsymmetricJet, J. Sound &
Vi____b.,Vol. 74, 1974, pp. 281-301.
[4] Bayllss, A. and Maestrello,L.: Simulationof Instabilitiesand Sound
Radiationin a Jet, AIAA J., Vol. 19, No. 7, 1981, pp. 835-841.
[5] Gottlleb, D. and Turkel, E.: Dissipative Two-Four Methods for
Time-DependentProblems,Math. Comput.,Vol. 30, 1976, pp. 703-723.
[6] Ho, Chl-ming and Huang, Lein-saing: Subharmonlcs and Vortex Merging in
Mixing Layers, to appear in J. Fluid. Mech., 1982.
-6-
FAR-FIELDBOUNDARY
r
NOZZLE JETSPREADING I
PIPE
I
DI2
z SOURCE CENTERLINE7-
!--UPSTREAMNOZZLEBOUNDARY
Figure i. Computational Domain
NONACOUSTICFIELD BREAKDOWNOF
NONACOUSTIC FIELD
ACOUSTIC
FIELD
UPSTREAM
tc
oo
D
____ACOUSTIC
€ = 0 FIELD E = 0.05
0 I I I I r'---.._ENDNOZZLE l
-I0 0 20 -I0 0 20
z/D FLOWDIRECTION z/D
Figure 2. Three-dimensional plots of the fluctuating pressure r/D = .29.
BREAKDOWNOF
NONACOUSTICFIELD
20 m __
ACOUSTIC _E = 0 ACOUSTIC -E = 0.05
FIELD FIELD
UPSTR UPSTREAM
tc
oo
D
COUSTIC
0 FIELD
l l I I I_-END NOZZLE
-I0 0 20 -I0 0 20
z/D _ z/D
FLOWDIRECTION
Figure 3. Three-dimensional plots of the fluctuating pressure r/D = .61.
E=0 _ E = 0.05
[
/" , /
Q "'_::' ::" I
' COUNTERCLOCKWISEROIAIION_ -] I I I I I I I I I I I 1 I
• 0 I 2 3 4 5 6 7 -I 0 I 2 3 4 5 6 7
zID zID
Figure 4. Fluctuating vorticity at te /D = i0.
E=0 E = 0.05
riD
I I I I I I I I I I I I I I I
-I 0 I 2 3 4 5 6 7 -I 0 1 2 3 4 .5 6 7
z/D zlD
Figure 5. Fluctuating vorticity at tc_/D = 30.
I I I I .. _ __L_
0 .05 .II .16 .22 .27 .32 .38 .43 .49
fD/U.]
Figure 6a. Flow field velocity spectra z/D = 7.3, r/D = 1.5.
.9- E=0 E = 0.05
o I I I I
0 .05 .II .16 .22 .27 .32 .38 .43 .49
fD/U.
J . ,,_
Figure 6b. Acoustic far-field spectra z/D = 38, r/D = 20.
1. Report No. 2. GovernmentAccessionNo. 3. Recipient'sCatalogNo.
NASA TM-84506
l' 4. Title and Subtitle 5. Report Date
Simulationof the FluctuatingField of A Forced jet August 1982
6. Performing Organization Code
505-31-33
7. Author(s) 8, Performing Organization Report No. ;"
Alvin Bayliss, Lucio Maestrello,Eli Turkel
10. Work Unit No.
9. Performing Organization Name and Address
NASA Langley ResearchCenter '11.Contractor GrantNo.
Hampton, VA 23665
13. Type of Report and PeriodCovered
12. Sponsoring Agency Name and Address Technical Memorandum
NationalAeronauticsand Space Administration
Washington,DC 20546 14SponsoringAgencyCode
15. Supplementary Notes
16. Abstract
The fluctuatingfield of a jet excitedby transientmass injectionis
simulatednumerically. The model is developedby expandingthe state vector
as a mean state plus a fluctuatingstate. Nonlinearterms are not neglected
and the effect of nonlinearityis studied. The results show a significant
spectral broadeningin the flow field due to the nonlinearity. In addition,
large scale structuresare brokendown into smallerscales.
17. Key Words (Suggested by Author(s)) 18. Distribution Statement
Jet Noise
Jet Flow
Stability Unclassified- Unlimited
Free shear layer Subject Category71
19. Security Qassif.(of thisreport) 20. SecurityCla.if. (of this _ge) 21. No. of Pages 22. Dice
Unclassified Unclassified 14 A02
N-3os ForsalebytheNationalTechnicalInformationService,Sprinsfield,Virginia2216t

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Simulation of the fluctuating field of a forced jet

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