P-49
9June 30 - July 3, 2015 Melbourne, Australia

Transitional structures and sound emission in a supersonic round jet forced bya pair of helical modes

Daisuke Watanabe
Graduate School of Engineering
Toyama University
3190 Gofuku, Toyama-shi, Toyama 930-8555, Japandwata@eng.u-toyama.ac.jp
Hiroshi Maekawa
Graduate School of Informatics and Engineering
The university of electro-communications
1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japanmaekawa@mce.uec.ac.jp
using either active or passive turbulence control measures(see Seiner 1992). Progress in this area has been limited
by unclear understanding of the physical supersonic jet
noise source mechanism. These mechanisms have been
extensively studied using round jets. Non-round geometryof the jet exit has been experimentally studied forbeneficial noise reduction relative to the round jets. Machwave emission and shock noise are the dominant acoustic
sources of supersonic jet noise. Improperly expanded
nozzles produce shock noise that dominates acoustic
emission in the jet forward quadrant. Shock noise,
however, could be minimized through appropriate designof nozzle geometry. It has been observed experimentallythat the acoustic radiation from jets is dominated by Machwaves (see review by Tam 1995). Turbulent structures
traveling at supersonic speed within the jet are generally
thought to be responsible for Mach waves, and they havebeen modelled as a combination of linearly unstablemodes. Reduction of Mach wave emission represents themost serious challenge to the successful design of a
suppressor nozzle. The understanding of jet turbulence
and noise emission is very crucial for jet investigations.
DNS of supersonic round jet was performed and broughtmuch light to these mechanisms. Freund et al. (2000)
simulated a perfectly expanded Mach 1.92 jet at
ReD=2000 and its sound field. Watanabe et al. (2003,
2006) performed linear stability analysis and DNS of asupersonic plane jet. Their results show that the plane jet(Mc =1.17) forced by a pair of oblique modes (the obliquemode is most unstable at high convective Mach numbers)
with the subsonic phase speed suppresses the emitted
Mach wave intensity. On the other hand, in thighconvective Mach number, the most unstable (first helicalmode) modes have the supersonic phase speed in a roundjet (Luo et al., 1997, Parras et al., 2010 and Watanabe2014). The supersonic phase speed of the most unstablemode in a round jet makes the suppression of Mach waves
more difficult.

ABSTRACTThe three-dimensional compressible Navier-Stokes
equations are numerically solved to study the transitional
structure and the resultant acoustic emission in asupersonic round jet at high convective Mach numbers. A
5th-order compact upwind algorithm developed by Denget al. (1996) is used for spatial derivatives and a 4th-order
Runge-Kutta scheme for time advancement. The NavierStokes characteristic boundary conditions are used in thestreamwise and radial directions and periodic boundary
conditions in the azimuthal direction. Numerical results
for the convective Mach number Mc = 1.00 (Mj=2.0) and
Rer0=1000 based on the jet nozzle radius r0 are presented.
Four different cases were investigated. The first case is thejet forced randomly (random case). The other cases are thejet flow forced by the linear unstable modes. These cases
are the jet flow forced by a pair of first helicalmodes(m=±1 case), second helical modes(m=±2 case) andthird helical modes(m=±3 case), respectively. Growth of apair of third helical modes (m=±3) causes the early decayof jet centerline velocity. Furthermore, radiated pressure
fluctuations generated by the growth of a pair of third
helical modes (m=±3) which have subsonic phase velocity
are suppressed compared with turbulent jet.

INTRODUCTION
Compressible jets, which can be found in many
applications such as rocket, scramjet, ramjet and turbojet
engines, have been of fundamental importance in the
study of compressible free shear flows. With new noise
regulations, reducing of acoustic noise is the one of key
technological challenges facing proposed supersoniccommercial aircraft. The numerical investigation ofsupersonic jets is expected to guide such technologicalprogress on aircraft where the jet exhaust velocity issupersonic. On the other hand, methods have long beensought to find an efficient means for reduction of jet noise1 u  1  ru r u  1  u u 


t
r
r
r

 u u x  u r u


r
x
2
1 p  1  r  r
1    x  , (3)





r   r 2 r
r 
x 
 u x  1  ru r u x  1  u u x 


r

t
r
r
 u x u x xp  1  r xr  1  x  xx 
, (4)
 


x  r r
x 
r 

The linear inviscid instability wave model andLighthill acoustic analogy approach have provided
invaluable guidance to the development of the jet noise
prediction (see Zorumski 1982). It is generally acceptedthat the mechanism for Mach wave radiation (Tam et al.,1984) (as well as growth of compressible mixing layers,
Morris et al., 1990) is strongly connected with theamplification rate of instability modes and that the
instability wave scales with the convective Mach number.
The real physical process, however, involves nonlinear
interaction of various instability modes in the flow filed.
DNS researches of the nonlinear interaction for thesupersonic jets in the transition regime are important forunderstanding of these mechanisms.



We investigate, by means of DNS, the spatiallydevelopment of high Mach number round jet forced withthe unstable/random disturbances. Both the fluid dynamic
structures and noise of round jets are studied. In thepresent study, we focus on the effect of inflow
disturbances on the Mach wave emission from the roundjet using spatial 3-D DNS. The round jet is of interest, as
described above, due to practical importance of supersonic
combustion and jet noise generation. The sound fields
related to the energy nonlinear evolution of various a pairof linear unstable modes are examined at high convective
Mach numbers. In this pair, the effects of growth of a pairof first, second and third helical modes (m=±1, m=±2 andm=±3; m indicates the azimuthal wave number) on the jetsound fields and flow field in a round jet are presented.

Et 1 r Et  p u r  1 Et  p u 


t r
r
r

Et  p u x x
1  ru i ir  1  u i i   u i ix 



r
r 
x
r
1   T  1   T    T  , (5)

 r


  r r  r  r 2     x  x 
where

Et 
NUMERICAL METHODSIn the direct numerical simulation, the nondimensional equations governing the conservation laws ofmass, momentum, and energy for a compressible
Newtonian fluid are solved using a fifth-order compact
upwind algorithm for conservation form [1] with a time
integration 4th-order Runge-Kutta algorithm. The
governing equations in cylindrical coordinates are given
as follows; xx 

 1  ru r  1  u   u x  0 , (1)
t r rx
r 

ρ u x u x ρ u r u r ρu  u 
p
, (6)


(γ 1)
22
2

  u x  2  1 ru r  1 u u x  



 2
,
x  Re  z  3  r rr  rr   u r  2  1 ru r  1 u u x  



 2
,
r 
x  Re  r  3  r r     1 u u r 
 
2
r 
Re   r 

2  1 ru r  1 u u x  


 
,
x  r 
3  r r u r  1  ru r u r  1  u r u 


r

t
r
r
 u r u x  u uxrp  1  r rr  1  r    rx 
, (2)
 



r  r r
r 
r
x 

 xr   rx   u r u x 


,
r 
Re  x

 r   r     u  1 u r r   ,Re  r  r  r  

 x   x 

  1 u x u 


,
x 
Re  r 
(7)

2

 


(  1)M RePr
2T

By using equation (14), the convective Mach number forthe present study is Mc = 0.97 (Mj= 1.89).
For 3-D spatial DNS, after a grid-convergence
analysis, the computational mesh was Nx×Nr×N=
801×150×128. Mesh points were clustered toward the jet
shear layer in the normal direction and downstream
streamwise locations near the end of the potential core.
NSCBC (the Navier-Stokes Characteristic Boundary
Conditions, Poinsot et. al., 1992) were implemented in thetreatment of the boundaries at the in/outflow and far
normal regions. Periodic boundary conditions were
implemented in the azimuthal direction. Outflow
boundaries were located at x=35r0 in the streamwise
direction and at r = 20r0. For a pair of helical modes case(m=±1, m=±2 and m=±3 case), the inlet disturbance vector~
d in  ~in , u~x in , u~r in , u~ in , ~pin  can be written as(8),γM 2 p,ρ(9)μ T 2 / 3 ,(10)where M  U j / c  and c is the speed of sound at
the ambient fluid. The all variables, in all the following
discussions, are non-dimensionalized by the characteristic
physical scales such as U j ,   , T and the jet nozzle
radius of r0 where the subscripts j and  indicate the jet
centerline and the ambient fluid, respectively. The
Reynolds number isRe 

U j r0~d in   Am dˆ m. ( r ) expim  (2St ) t  ,

.

(11)
where

U j 
12.5  r r0     .
1  tanh 2 
 4  r0 r  d̂ m is the eigenfunction of corresponding

instability mode calculated by linear stability analysis. m
and St indicate the corresponding azimuthal wave number
and nondimensional frequency. The amplitude of eachmode is 2% of the jet and the Strouhal numbers are set to
be St=0.1 closely corresponding to the maximum growthrate. On the other hand, the magnitude of the forced
streamwise velocity disturbance is chosen to be 10% of
the jet centerline velocity for the random case.

The flow field of a jet issuing from circular nozzle
into an ambient fluid can be divided into a potential coreregion, a transition region, and a fully developed region.
The "top-hat" jet profile belongs to the potential coreregion with a thin but finite shear layer. We employed themean velocity profile U (r ) given by

U (r ) 

(15)

m

(12)
RESULTS

The mean temperature T (r ) was calculated with the
Crocco-Busemann relation for unity Prandtl number;

(a)

(b)

(c)

(d)

  1 (U jU (r )  U (r ) )2

T (r )  M

2


2

Uj


T (U j  U (r ))UjT jU (r ) (13),Ujwhere  is the ratio of specific heats. M   U j / c andc is the speed of sound at the quiescent stream. The jet
centerline Mach number M j  U j / c j is obtained fromthe relation M j  M  c  / c j . The jet was heated withexit temperature ratio T j / T  1.12 . This ratio was used
for the simulation of a perfectly expanded Mach 1.92 jet(Watanabe et al., 2002). A jet convective Mach number
Mc may be defined byMc 

M j T j / T
1  T j / T.

Figure 1. Second invariant Q structure (iso-surfaces of
Q=0.01) for a) random case, b) m=±1 case, c) m=±2case and d) m=±3 case.

(14)

Figure 1 shows the vortex structures visualized with
iso-surfaces of the second invariant Q of the velocity
3

gradient tensor. Typical vortex structures appear in the jet
due to the growth of each inlet disturbance. In the random
case, the vortex structures spread almost in all azimuthal
directions (the cross-section view at the downstream
location shows almost a circle shape from the jet center).
In the m=±1 case, m=±2 case and m=±3 case, the vortex
structures extend radially in the two, fore, six distinct
azimuthal directions from the jet center line, respectively.
In these cases, the vortex structures extend far away ascompared with the random case. These characteristic
expansions of vortex structures affect the jet velocity.
Figure 2 shows the contour plots of instantaneous
streamwise velocity at x=32r0 where y=r/r0 cos andz=r/r0 sin. In these figures, the same characteristic
distributions as the vortex structures are also seen in thejet stream wise velocity.
(a)center line velocity distribution, as shown in Figure 4. In
the m=±3 case, the jet center line velocity decreases
sooner to uc=0.2 at the downstream location of x=35r0,
though the jet center line velocity of the random casedecreases to uc=0.4 at the same location.
(a)

(b)

(b)

(c)

(c)

(d)

(d)

Figure 2.
The contour plots of
instantaneous
streamwise velocity for a) randomcase, a pair of b)
first helical modes case (m=±1), c) second helicalmodes case (m=±2) and d) third helical modes case(m=±3) at x=32r0.

Figure 3. Downstream evolution of mean streamwise
velocity (contour plots); a) random case, a pair of b)
first helical modes case (m=±1), c) second helicalmodes case (m=±2) and d) third helical modes case(m=±3) at x=32r0.

Figures 3(a), (b), (c) and (d) show the downstream
evolution of mean streamwise velocity for random case,m=±1 case, m=±2 case and m=±3 case, respectively. In the
random case, the contour plots of mean streamwise
velocity spread gradually towards the downstream (Figure3a). On the otherhand, in the m=±1 case, m=±2 case andm=±3 case, growth of a pair of helical modes leads tosooner shear layers expansions around the end of jet
potential region. Figure 3(b), (c) and (d) indicate that theshear layers spread rapidly around x=22r0, x=15r0, x=20r0,
for m=±1 case, m=±2 case and m=±3 case, respectively.
Around these points correspond with the point thatcharacteristic expansions of vortex structures appear.
The centerline streamwise velocity shown in figure 4
clearly indicates the location where the potential corecloses. The inflow disturbance also influences on the jet

1

uc

0.8
0.6
0.4
0.2
0
0

random
m=±1
m=±2
m=±3

10

20
x/r0

30

Figure 4. Mean jet centerline velocity.
4

(a)
(b)

0.2

i max

(a)

(c)

0.1

0 3
10

(d)

m=1
m=2
m=3
m=4

104

105

106

Re

(b)
1.2

cr/c∞

1.1

1

Figure 5. RMS of overall fluctuating pressure for a)randomcase, a pair of b) first helical modes case(m=±1), c) second helical modes case (m=±2) and d)
third helical modes case (m=±3) at x=32r0.

0.9 3
10

m=1
m=2
m=3

104

105

106

Re

Figures 5 indicates RMS of overall fluctuatingpressure for the each case at x=32r0. In the m=±1 case,m=±2 case and m=±3 case, the distributions of pressure
fluctuation magnitudes show the different geometricpatterns. These patterns are caused by interfering between
the pair of helical modes with the same frequency
propagating mutually but in the opposite directions. Note
that, with the increase of the azimuthal wave number m,
the pressure fluctuation magnitudes become smaller at thejet far field. In the m=±3 case, radiated pressure
fluctuations are especially smaller compared with theother cases.
In order to find the reason of the pressure fluctuationsin the jet mentioned the above, the linear stability analyses
are carried out. Turbulent structures traveling at
supersonic speed in the jet are generally thought to be
responsible for Mach waves (see Papamoschou, 1997,
Morris 2010 and Kearney-Fischer et al., 2011). Figure 6
shows the Reynolds number dependence of the maximum
linear growthrate i and the phase velocity cr correspondswith the maximum growth rate of the each helical mode.
With the increase of the azimuthal wave number m, themaximum linear growth rate and phase velocity becomes
smaller over the Reynolds number range shown in the
figure. For m=3 at Re=1000, the phase velocity is lower
than acoustic velocity of the ambient fluid c∞, and cr/c∞remain near unity at high Reynolds number. This result
suggests that the third helical mode may much lesscontribute to radiation of Mach waves. As figures 2 and 3
show, the third helical mode is a good candidate toeliminate the acoustic waves in the round jet. DNS of thejet effectively forced by the third helical mode will be
performed to achieve the Mach wave elimination at highMach number.

Figure 6.
Reynolds number dependence of a)
maximum linear groth rate and b) phase velocity of thehelical mode at maximum growth rate.

CONCLUSIONS
Spatial DNS of a supersonic round jet for Mc=2.0 has
been performed. The numerical results provide newphysical insights into jet expansion and noise generation
in a round jet. Upstream disturbance conditions play animportant role for the evolution of the downstream
structure, such as development of shear layers andtransition process in a jet. Growth of a pair of helicalmodes is responsible for the characteristic expansions ofvortex structure and streamwise velocity in a round jet.
Especially growth of a pair of third helical modes (m=±3)
causes the early decay of jet centerline velocity.
Furthermore, radiated pressure fluctuations generated by
the growth of a pair of third helical modes (m=±3) whichhave subsonic phase velocity are suppressed comparedwith turbulent jet.

ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI Grant
Number 24560189 and 15K05787. Some of thesimulations were executed at the Information Technology
Center of the University of Tokyo.

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