Understanding the principles of jet noise propagation is an essential ingredient of systematic noise reduction research. High speed computer methods offer a unique potential for dealing with complex real life physical systems whereas analytical solutions are restricted to sophisticated idealized models. The classical formulation of sound propagation through a jet flow was found to be inadequate for computer solutions and a more suitable approach was needed. Previous investigations selected the phase and amplitude of the acoustic pressure as dependent variables requiring the solution of a system of nonlinear algebraic equations. The nonlinearities complicated both the analysis and the computation. A reformulation of the convective wave equation in terms of a new set of dependent variables is developed with a special emphasis on its suitability for numerical solutions on fast computers. The technique is very attractive because the resulting equations are linear in nonwaving variables. The computer solution to such a linear system of algebraic equations may be obtained by well-defined and direct means which are conservative of computer time and storage space. Typical examples are illustrated and computational results are compared with available numerical and experimental data

Liu, C. H.

Padula, S. L.

English

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NASA TECHNICAL NASA TM X-71941
MEMORANDUM COPYNO.
(NASA-TM-X-71941) A COMPUTING. METHOD FOR N75-103591
cm SOUND PROPAGATION THROUGH A NONUNIFORM
JET STREAM (NASA) 19 p BC $3,25
, CSCL 20D Unclas
G3/34 _53168
A COMPUTING METHOD FOR SOUND PROPAGATION
- THROUGH A NONUNIFORM JET STREAM
21C:a:
Sharon L. Padula and C. H. Liu
December 1974
This Informal documentation medium Is used to provide accelerated or
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NATIONAL AERONAUTCS AND SPACE ADMINISTRATION
LANGLEY RESEARCH CENTER, HAMPTON, VRGINIA 23665
https://ntrs.nasa.gov/search.jsp?R=19750002287 2020-03-23T03:26:59+00:00Z
1. Report No. 2. Government Accessimon No. 3. Recipient's Catalog No.
TM X 71941
4. Title and Subtitle 6. Report Oate
A Computing Method for Sound Propagation Through a December 1974
Nonuniform Jet Stream e. Porming Orgenizatic Cde
26.200
7. Author(s) 8. Performing Organization Report No.
Sharon L. Padula and Chen-Huei Liu
10. Work Unit No.
9. Performing Organization Name and Address 505-03-12-02
NASA Langley Research Center
Hampton, Virginia 23665 11. Contract or Grant No.Hampton, Virginia 23665
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address NASA Technical Memorandum
National Aeronautics & Space Administration 14. Sponsoring Agency Code
Washington, DC 20546
15. Supplementary Notes
Interim technical information release, subject to possible revision and/or
later formal publication
16. Abstract Understanding the principles of jet noise propagation is an essential
ingredient of systematic noise reduction research. High speed computer methods
offer a unique potential for dealing with complex real life physical systems
whereas analytical solutions are restricted to sophisticated idealized models. Th4
classical formulation 6f sound propagation through a jet flow was found to be in-
adequate for computer solutions and a more suitable approach was needed. Previousinvestigations selected the phase and amplitude of the acoustic pressure as
dependent variables requiring the solution of a system of nonlinear algebraic equa
tions. The nonlinearities complicated both the analysis and the computation. A
reformulation of the convective wave equation in terms of a new set of dependent
variables is developed with a special emphasis on its suitability for numerical
solutions on fast computers. The technique is very attractive because the re-
sulting equations are linear in nonwaving variables. The computer solution to
such a linear system of algebraic equations may be obtained by well-defined anddirect means which are conservative of computer time and storage space. Typical
examples are illustrated and computational results are compared with available
numerical and experimental data.
17. Key Words (Suggested by Author(s)) (STAR category underlined) 18. Distribution Statement
Sound propagation, jet noise, numerical
solution, harmonic point source, non-
uniform flow field.
19. Security Qailf. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*
Unclassified Unclassified 18 $3.00
The National Technical information Service, Springfield, Virginia 22151
Available fromlity, P.O. Box 33, College Park, MD 20740
STIF/NASA Scientific and Technical Information Facility, P.O. Box 33. College Park, MD 20740
A COMPUTING METHOD FOR SOUND PROPAGATION
THROUGH A NONUNIFORM JET STREAM
Sharon L. Padula and C. H. Liu*
NASA Langley Research Center
Hampton, Virginia
ABSTRACT
Understanding the principles of jet noise propagation is an
essential ingredient of systematic noise reduction research. High speed
computer methods offer a unique potential for dealing with complex real
life physical systems whereas analytical solutions are restricted to
sophisticated idealized models. The classical formulation of sound
propagation through a jet flow was found to be inadequate for computer
solutions and a more suitable approach was needed. Previous investigations
selected the phase and amplitude of the acoustic pressure as dependent
variables requiring the solution of a system of nonlinear algebraic equations.
The nonlinearities complicated both the analysis and the computation. A
reformulation of the convective wave equation in terms of a new set of
dcDendent variables is developed with a special emphasis on its suitability
for numerical solutions on fast computers. The technique is very attractive
because the resulting equations are linear in nonwaving variables. The
computer solution to such a linear system of algebraic equations may be
obtained by well-defined and direct means which are conservative of computer
time and storage space. Typical examples are illustrated and computational
results are compared with available numerical and experimental data.
* NAS-NRC/NASA Research Associate g
2
Symbols
a Ambient speed of sound
A Amplitude
d Jet exit diameter, 0.01905
f Frequency in Hertz
M Flow Mach number; U i/a
r,e,o Spherical polar coordinates as in figure 1
r Non-dimensional r w.r.t. d; r/d
t Time
U Longitudinal component of flow velocity
U Jet exit velocity
W Non-dimensional angular frequency; wd/a
x Cartesian coordinate in the windward direction
of the jet
Z Complex variable; Aei'I
Constant in equation 3; a2 < 1
Coordinate transformation; tan-1 (a- 2 tan 6)
n Coordinate transformation; ln r
I Quasi velocity potential
Yi, To Phase; T = Y@ + / + o(r2)
w Angular frequency
3INTRODUCTION
This work is an extension of a program at NASA Langley Research Center
concerning the investigation of noise generated by a jet. It improves upon
previous work by Schubert1 and by Liu and Maestrello2 in which finite
difference equations appropriate to the propagation of sound are solved by
iterative methods for a sinusoidal point source on the axis of the potential
flow region of a subsonic jet. A computer-oriented method is developed
which solves typical sound propagation problems while using a minimum amount
of costly computer resources. The method is tested for cases equivalent
to those presented in reference 1 so that comparisons can be made with
numerical as well as experimental results. The present results compare
favorably with experimental measurements by Grande , whereas the numerical
computations in reference 1 overpredict the depth of the downstream "valley"
in jet noise directivity. More notably, the present method drastically
reduces the requisite computer time, computer storage space and peripheral
usage. Furthermore, the current formulation can be solved by direct methods
as opposed to the iterative procedure employed by Schubert. Direct methods
are desirable because they are straightforward, they guarantee results and
because suitable computer routines are often available as standard library
subroutines.
The method presented here has been thoroughly tested using the idealized
model outlined in reference 1. Immediate extension to more realistic models,
e.g.,reference 2, is possible and would presumably net even more reliable
results.
I4
Formulation of Problem
The Nonuniform Jet Flow Field.- The present method of solution is
tested for the case of an acoustic point source located on the centerline
and contained within the potential core of a spreading jet, figure 1. The
jet nozzle is cylindrical with diameter, d. The jet flow is axisymmetric but
nonuniform such that the velocity profiles at any cross-section are determined
from experimental and empirical data. The origin of the coordinate system
is fixed at the source while the coordinate of the flow field is fixed at
the virtual origin of the jet. Most of the calculations are done in the
spherical polar coordinates, (r, e, 0).
The Convective Wave Equation.- Sound propagation through a flow field
can be described most simply by the following convective wave equation:
) 8 2 2]+- V =0 (1)
Here a is the ambient speed of sound, U is flow velocity, and N is Obukhov's
"quasi-potential" variablel . The formulation in terms of H is selected over
th _suual pressure fourmaLtion since in this way relatively good results are
possible without the addition of corrective terms to the right-hand side of
the equation.
Assuming a harmonic source, H can be expressed as = A exp I((r - Wt)
where A & T are the amplitude and phase of the radiated sound waves and r
denotes the nondimensional radial distance with respect to the nozzle
diameter, d. Substituting H into equation 1 results in a time independent
partial differential equation which is nonlinear in A and T. Schubert1
chooses to solve this equation by application of the finite difference
5method. This necessitates solving a system of simultaneous nonlinear
algebraic equations.
There is no direct method for solving such a system of nonlinear equations.
Iterative computer methods are available but they have a number of drawbacks.
First, they are highly specific. The coding is dependent on the problem
to be solved and must be rewritten and retested for each new equation.
Secondly, the accuracy of the results depends upon how well and how quickly
the process converges. In the case of sound propagation through a flow,
it remains to be shown that the process does in fact converge for all
instances.
The problems inherent in iterative methods can be avoided. The variable
T can be represented by the expansion, T = {o + I/r + o(r 2) so that T
approaches T0 as r approaches infinity. Then H = Z exp i(rPo - t) .
Thus, Z is a single complex variable and equation 1 can be rewritten
as a linear partial differential equation in Z. In this way, the solution by
finite difference method involves merely the solution of a system of
simultaneous linear algebraic equations. Most computer subroutine libraries
contain routines adequately suited to this task.
Numerical Solution
Underlying Assumptions.- The basic assumptions and basic scheme for
solving the problem follow directly from previous authors1 '2 . Figure 2
shows the spreading jet superimposed on a polar grid. An antijet is assumed
in order to avoid problematic boundary conditions along the rigid walls of
the jet nozzle.
The value of the complex variable, Z, changes most rapidly near the
point source and in the region of significant flow. It is desirable to
apply the finite difference method to an unequally spaced grid which has a
6concentration of points in the areas of greatest change, figure 2. To avoid
the difficulties posed by an uneven grid, an even grid is specified in some
new coordinates, ' and , figure 3. This new grid in (nz) maps onto the
desired uneven grid in (r,e) according to the following transformation:
S= Iln r
1 -2 (2)
= tan-1 (a-2 tan e).(2)
For illustration purposes, the polar grid in (n,C) is thought of as a
rectangular grid with n x m intersecting lines.
The Finite Difference Method.- The partial differential equation in Z
should be satisfied at each interior point of the region of interest. If
the partial derivatives in this equation are replaced by their central
(ifference approximations, a nine-point difference equation results. Writing
this difference equation at each interior grid point gives (n-l) x (m-l)
equations in n x m unknowns. Boundary conditions are the additional equations
needed to specify a unique solution.
Boundary Conditions.- The equation in Z is an elliptic type partial
differential equation, thus,the boundary value problem is well-posed.
The region, D., ib bounded by two concentric half circles centered on
the point source and by two line segments on the axis of symmetry, figure 4.
The inner circle has a radius of k d and the outer circle has a radius of
100 d, where d is.the diameter of the jet nozzle. In this way, the inner
boundary is contained in the potential core of the jet, and the outer boundary
approximates the far field.
The inner boundary conditions come from a solution derived by Moretti
and Slutky4 The solution i specified for a oint source in a uniformand Slutsky .The solution is specified for a point source in a uniform
REPRODUCIBILITY OF THE
ORIGINAL PAGE IS POOR 7
flow. It gives the sound pressure level for points a small distance from
the source. Since the acoustic source and,the inner boundary are contained
in the potential core of the Jet, they are in a locally uniform flow and
the Moretti and Slutsky solution is approximately valid.
The outer boundary conditions at the far field are essentially the
same as the Sommerfeld radiation conditions. Along the centerline, three-point
symmetry conditions are used.
Results and Discussion
Figures 5-7, show sample results at a variety of flow velocities,
acoustic source frequencies and radial distances from the source. In each
graph, sound pressure level relative to SPL at 0 = 900 is plotted against
0. These figures facilitate comparisons between the experimental data of
reference 3 and the numerical results of this paper and reference 1.
Figure 5 shows comparable results for the flow velocities Mach 0.3,
Mach 0.5, and Mach .0.9. Here radial distance from the source and source
frequency remain constant at 100 diameters and 3000 Hertz, respectively.
Notice that the corre3pondence between experimental results and the current
numerical results is esvecially goou at the lower Mach numbers. Also notice
the discrepancies between the numerical results of reference 1 and experimental
results at these same Mach numbers.
Figure 6 shows graphs, similar to those above, where radial distance
and flow velocity are constant (r = 100 d and M = 0.3) while frequency changes
from 3000 Hz to 5000 Hz to 7000 Hz. In all cases, numerical results obtained
by the present method are superior to those obtained in reference 1 with
respect to their agreement with experiment.
Figure 7 illustrates the differences between the current method and
REPRODUCIILITY u - -
ORIGINAL PAGE IS POOR
the previous one. Sound pressure levels for Mach 0.7 and frequency 3000 Hz
are plotted at a variety of radial distances from the source. It is clear
that the differences between the graphs increase with the radial distance.
Even greater discrepancies are evident when phase is plotted against e as in
figure 8. Agai., results for Mach 0.7 and frequency 3000 Hz are plotted at
a variety of rw,±1l distances from the source. In this case, the graphs
from reference 1 are markedly different from those produced by the present
method.
It should be emphasized that the present method and the method of
reference 1 use the same sound propagation model for testing purposes.
T~eoretically, they should produce identical results. Differences arise
when theory is translated into practical computer programs. Iterative
methods, such as the one described in reference 1, are often inaccurate
because the iterative process must be terminated after a reasonable amount of
computer time has elapsed. Furthermore, an iterative method is more
susceptible to round-off errors which tend to multiply as computing time
incmr ses. Direc "cthods, such as the method presented here, largely avoid
these problems and generally produce more reliable results. This explains why
graphs of the direct method's results match graphs of experimental data so
closely.
The present method has other advantages besides accuracy. It is quite
straightforward and relatively easy to program. A typical program will
execute in well under 1 minute of central memory time in a CDC 6600
computer. The computer storage space requirements are reasonable so that
transfer of data to and from peripheral storage devices is unnecessary.
None of these things can be said about the method of reference 1. Far from
guaranteeing results in under 1 minute, the iterative method can not
9guarantee convergence to a solution for the general case. Furthermore, the
algorithm for solving a system of nonlinear equations is quite complex and
requires large amounts of storage space.
Conclusion
It has been shown that the present method for sound propagation
problems has distinct advantages over previous methods. It is a direct
rather than iterative method and, consequently, it is faster, more accurate,
and less complicated. The method is conservative of computer resources,
relatively easy to program, and makes use of standard library subroutines.
Since the method itself is relatively simple, increasingly complex and
realistic models can be tested.
References
1. Schubert, L. K.; "Numerical Study of Sound Refraction by a Jet Flow II,
Wave Acoustic," The Journal of the Acoustical Society of America,
vol. 51, no. 2, part 1, 1972, pp. 447-463.
2. Liu, C. H. and Maestrello, L.: "Propagation of Sound Through a Real Jet
Flowfield," AIAA Paper, no. 74-5, AIAA 12th Aerospace Sciences Meeting,
Washington, D.' , January 1974. To appear in AIAA Journal.
3. Grande, E.: "Refraction of Sound by Jet Flow and Jet Temperature II"
NASA CR-840, 1967.
4. Moretti, G. and Slutsky, S.: "The Noise Field of a Subsonic Jet,"
General Applied Science Laboratory, GASL Technical Report 150, 1959,
(AFOSR TN-59-1310).
yI
U (x,\ z)
Figure 1. Jet flow configuration.
CONSTANT
INCREMENTS
IN 2
NS -N---
Ew . ' ifrncoe grd 2-s sece"n in h hsc an
D C
IB
,77
D A F A
- (n-1)( )
Figure 3. A grid for the finite difference method
in the (n, ) plane
INNER BOUNDARY:
MORETTI AND SLUTSKY'S SOLUTION (1959)-h=~--
OUTER BOUNDARY:
SOMMERFELD RADIATION CONDITIONS
CENTER LINE:
THREE-POINT SYMMETRY CONDITIONS
r =100 d
C u
POIT SOURCE-
fogurc- 4. Bogn;ry condiFerons
e. deg - e., deg-
0 10 20 30 40 0 10 20 30 40
M =0.3 M= 0.9
-10 "/ -10-
-20 -20- f = 30U Hz
SPL d= 314"
(re e= 90 ) 0  10 20 30 40 // a = 1.055
dB - - a
/ - M = 0.5 /
-10 / -40'
-- PRESENT ANALYS IS
S.---SCHUBERT (1969)
-20 / 
-50 --- EXPERIMENT-GRANDE (1967)
SPL 0 20 log 71
-30- (re 8= 90) = 90
Figure 5. Sound pressure level at 100 d from the source
for several Mach numbers. Source at 2 d.
! 0 0 E 40 0 10 2 30 4
-10 055 -10 2-46
3 =0 Hz 700 Hz
Kj 
-201
SM. = 0.3SPL -3-1
SLir " -30 'd = 3/4'
0 wd(re O= 901 w
d8 -10 - -40
0 W= 1.76 -50-20 f M1 -50-
f = 500 Hz / -- PRESENT ANALYSIS
=--SCHUBERT (1969)
-30 -60 --- EXPERIMENT-GRANDE (1967)
-40
Figure 6. Sound pressure level at 100 d from the source
for several frequencies. Source at 2 d.
SPL
(re 0= 900) e, deg----
dB 0 10 20 30
-10 ""
I
-20
• I
-30 " PRESENT ANALYSIS
d ----- SCHUBERT (1969)
o 100 d
-40 O 51.4 d
x 26.4d
A 13.6 d
+ 7.0 d
Figure 7. Sound pressure level at various distances from the source.MO.7, W-1i.055 (3000 Hz for a 0.75" jet ), source at 2 d.
I K - 8T deg
10 K, 30 40 50 60 70 8 0
Os
7 f
-.02-o .
-.03'
PRESENT ANALYSIS
----- SCHUBERT (1969)
-.04 O 100 d
0 51.4d
x 26.4 d
A 13.6 d
Figure 8. Phase a various distances from the source.
M-0.7 , W=1.055 ( 3000 Hz for a 0.75" jet) , source at 2 d.


A Computing Method for Sound Propagation Through a Nonuniform Jet Stream

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