This work carries earlier finite-difference calculations of the Widnall instability of vortex rings into the late non-linear stage. Plots of energy in azimuthal Fourier modes indicate that low-order modes dominate at large times; their structure and dynamics remain unexplored, however. An attempt was made to calculate the acoustic signal using the theory of Mohring (1978), valid for unbounded flow. This theory shows that only low-order azimuthal modes contribute to the sound. As a check on the effects of axial periodicity and a slip wall at large radius imposed by the numerical scheme, the acoustic integrals were also computed in a truncated region. Half of the terms contributing to the sound have large differences between the two regions, and the results are therefore unreliable. The error is less severe for a contribution involving only the m = 2 mode, and its low frequency is consistent with a free elliptic bending wave on a thin ring

Shariff, K.

Verzicco, R.

English

NASA Technical Reports Server

Center for Turbulence Research
Proceeding8 o] the Summer Program 199_
N95- 21048
Vortex ring instability and its
By R. Verzicco 1 AND K. Sharit_
f_ Y
sound
This work carries earlier finite-difference calculations of the Widnall instability of
vortex rings into the late non-linear stage. Plots of energy in azimuthal Fourier
modes indicate that low-order modes dominate at large times; their structure and
dynamics remain unexplored, however. An attempt was made to calculate the
acoustic signal using the theory of M/ghring (1978), valid for unbounded flow. This
theory shows that only low-order azimuthal modes contribute to the sound. As
a check on the effects of axial periodicity and a slip wall at large radius imposed
by the numerical scheme, the acoustic integrals were also computed in a truncated
region. Half of the terms contributing to the sound have large differences between
the two regions, and the results are therefore unreliable. The error is less severe for
a contribution involving only the m = 2 mode, and its low frequency is consistent
with a free elliptic bending wave on a thin ring.
1. Introduction
Many shear flows of practical interest contain large-scale coherent motions, and
some aspects of their dynamics must be responsible for the generation of sound
(smaller scale motions affect the dynamics of the large scales but themselves radi-
ate as higher order poles). What these aspects might be is difficult to identify from
experiments since the most energetic or most visually apparent motions may not be
the most radiative and a detailed knowledge of time-dependence is required. It is
a usual procedure in fluid dynamics to extract from a flow the most representative
structure with the hope that its evolution alone still retains the important features
of the whole flow. In the present case the evolution of azimuthal instabilities on an
isolated vortex ring is numerically simulated into the late non-linear stage. We are
motivated by the following: (i) Observations using phased arrays of microphones
which indicate that the acoustic source location in jets is near the end of the po-
tential core where vortex ring structures break down (Bridges & Hussain 1987, p.
309). Implicit in the consideration of a single ring is the premise that the actual
breakdown process does not involve interactions between rings. Factors which may
be important in reality are enhanced growth-rate and mode selection due to mu-
tual straining and tearing or pairing. (ii) A good fit (Shariff & Leonard 1992, p.
275) of the acoustic frequency of a turbulent vortex ring measured by Zaitsev et
al. (1990) to the axisymmetric elliptic core ring but a poor fit to the elliptic bending
mode. Further studies along these lines were carried out in Zaitsev & Ko'pev (1993)
1 Universit_ di Roma, "La Sapienza", Dipartimento di Meccaniea e Aeronautiea
2 NASA Ames Research Center
PfIIII]i_I_IQPAGE _.__ niUT FtLr_
https://ntrs.nasa.gov/search.jsp?R=19950014631 2020-06-16T08:27:54+00:00Z
222 R. Verzicco fd K. Shariff
FIGURE 1.
--, F/v = 1200; ....
.............. i............... i............... :."..............._...............
-0.010
t
Acoustic signal due to axisymmetric elliptically distorted vortex ring.
, F/_ = 3000.
0.020"
0.015"
0.010"
0.005"
0.000"
-0.005"
and Kop'ev & Chernyshev (1993). (iii) The observations of Maxworthy (1977) of
the development of swirling flow and a solitary bulge wave following the Widnall
instability.
2. Methods
Calculations were performed using a second order finite difference code in cylin-
drical coordinates (x, r, ¢) assuming axial periodicity (with length Lx) and a slip
wall at r = Ro as described in Verzicco & Orlandi (1993). The spurious straining
due to the infinite row of rings and due to the wall is estimated to be less than 2%
of the strain (due to ring curvature) which drives the instability. Units are chosen
so that the circulation, F, and ring radius, R, are unity. To avoid effects of axisym-
metric unsteadiness, a ring with an initially Ganssian core (of radius a = .4131)
and F/v = 3000 was axisymmetrically relaxed to a quasi steady state in which w¢/r
becomes some function of the streamfunction. The number of grid intervals in each
direction is 128 with a smaller radial spacing, obtained by a non uniform mesh,
near the vortex core. The domain size is Ro = L_ = 6. A random divergence-free
perturbation is applied to the basic-state velocity.
The acoustic pressure, po, is obtained from the theory of MShring (1978):
po xixj D'"ff 1 / xi(x x w)j d3x,p.- ° Q,i- (1)
where A -- xixi. The triple prime denotes the third time derivative and is numer-
ically evaluated using cubic splines. Due to the factor xixj only the symmetrized
Qij are required, and in cylindrical coordinates one obtains:
Vorte= ring instability 223
t=O t=15
i
i
i
i // .
/ /
J i/i /i {:/
i t / : ,;
i ]: 11
i i"/¢i //
I i illi :"/
i !/
i "1i /:
,!/:..
i
i
' / ,::'.,'.'-:,
i \ .....:::i
i
i
i
i
t----25
FIGURE 2. Contour-plots of azimuthal vorticity fon an eUiptically distorted Gaus-
sian core vortex ring at F/v = 3000. Contour increments are A = 0.1, -----, axis
of symmetry. The ring motion is downward.
I1 -- / xr2_¢(0) dxdr,
/4 = / xr2wr(2) dxdr,
Is = / xr2w_(2) dxdr,
J3 = f x_rw_(1) dxdr,
/f_ = / r3_(1) dxdr,
Qll = 2_rI1 (0 mode only)
7r (gx -/4 - 2Ia - Is) (0 and 2 modes)
=
7r
Qaa = _ (I4 - 2I_ + Is - g_) (0 and 2 modes) (2)
Q_2 + Q_a = _r(/2 - J_ - J2 + K_) (1 mode only)
qla -t- Q31 = 7r (,/4 - J3 -/3 + K2) (1 mode only)
q2a + Q3_ = _r(Ie - Iv - Ks) (2 mode only)
The various integrals in Eq. (2) are
I_ = / xr2_x(1) dxdr,
/6 = / xr_wr(2) dxdr,
Ja - / x_r_(1) dxdr, (3)
J4 = [ x2 r_r(1) dxdr,
where, for instance, _(m) and _(m) denote the ruth sin and cos azimuthal modes,
respectively, of the axial vorticity. Note that only the modes m < 2 are involved.
/3 = f xr_wx(1) dxdr,
/7 = f xr_(2) dxdr,
J2 = f x2r_(1) dxdr,
K_ = f r3_(2) dxdr,
224 R. Verzicco _: K. Shariff
(a)
I1
10t[J .... - z = : v. = :. _- -_ = __
16
1
1
l so' 15'0 200 250
(b)
1
l
1.
1
_} 50 1O0 150 200 250
FIGURE 3. Evolution of energy in azimuthal modes. Mode index m: • , 0; •, 1;
o, 2;[], 3; " , 4; o, 5; v, 6; +, 7. (a) Total energy. (b) Energy in azimuthal
velocity component. Certain modes have been omitted for clarity.
The theory of M6hring is valid for an unbounded domain in which the vorticity
decays exponentially at infinity. This holds provided the computational domain
is large compared with the vortical region. As the ring leaves and re-enters the
period, what its position would be in an unbounded domain is tracked in order to
obtain an x value in the unbounded domain for every grid point. The field is phase
shifted axially so that the vortical region is centered in the integration domain and
mid-point rule quadrature is employed. To assess errors due to out-lying vorticity,
integration is also performed in a smaller domain of radial and axial dimensions
RT = 5Ro/6 and LT = 5LJ6, respectively. As a test, the acoustic signal for an
axisymmetric ring with an elliptically distorted Gaussian core was obtained (Fig. 1).
Vortex ring instability 225
t = 104 t = 163
t = 156 t = 214
FIGURE 4. Contours of vorticity magnitude in the (r, 0) plane of maximum [ w [.
The spatial resolution of the simulation is twice the resolution of the figure.
It has the form of a damped cosine. The damping is due to the readjustment of the
initial vorticity distribution towards an equilibrium distribution (Fig. 2). For an
inviscid axisymmetric elliptic core ring with uniform vorticity, the acoustic signal is
easily worked out (Shariff et al. 1989, p. 112) from the dynamics of Moore (1980).
For the present case it is an undamped cosine with amplitude .022 which is close to
the initial amplitude computed. Also, the period of the acoustic signal agrees with
the theoretical prediction.
226 R. Verzicco _J K. Shariff
o.o|o
_00_
o.ooo
-O.OOS-
-0.0|G
,, F ,
i : _l +: IG : --
140
0.00el ¸
0.0_O
-0,00_ ¸
_,.0_ ¸
W
• i ,, [ ,.
H . i e**
_,.'. , '...,, 'I: , ...._.,_',,..,
..._ ........... +...._,÷.. ++................ _ ..................
_,* :
, .................i...................=:'+._................._..................
, ! ! !
80 100 120 140
0.004-
0,0_"
0._0"
_.005-
-0,004"
60
** ,: , . :
,:_ : , : , ,+
,_'",, " :":", ,. " : ...... " ....."'"i.
..... m ........... ;............. *.........
..................f .................] ...........
-F
0.0o_
0.006
0.004
O,OO2
G000
_t0_
...................+...............................+ ,,"t
: : ! i
_. o • ,o .: ,
-t-
0.001 .................... ".".................. ." .................. ne .................
JOLO04
-,F-
o.omo...................::......................................+"I:..............
• : I I
I+ : • ;
-o.m,_o...................+...................T..................:,+................
: : +
-o._I$ ! : :
dO _0 IO0 120 140
t
FIGURE 5. Acoustic components. --
t
, Full-domain; .... , Truncated domain.
3. Results
Applying the inviscid stability theory of Widnall et al. (1974), valid for thin
cores, to the Oaussian profile, one finds that the so-called 'non-rotating second
radial mode' corresponds to m = 2.26R/a = 5.47 so that either m = 5 or 6 should
be most unstable. The theory also predicts that there should be a band of growing
modes around the most unstable one. Fig. 3a shows that the numerical result is
consistent with this picture. Since the initial perturbation is not a combination of
eigenmodes, there is an initial transient, after which the m = 6 mode is dominant
showing a large range of linear growth. At t = 150 the m = 6 mode gives way to
m = 4, likely due to viscous spreading of a(t); this switching of the dominant mode
has also been previously observed by Hasselbrink (1992) and Shariff et aI. (1994).
At later times the non-linearly amplified m = 1 and 2 modes become dominant
and all modes saturate. That the overall ring shape follows the dominant mode is
shown in Fig. 4 where contour plots of Iwl in the (r, ¢) plane of maximum Iwl are
Vorfez ring instability 227
displayed.
Maxworthy (1977) used dye visualization to infer that following breaking of the
instability wave, a swirling flow developed in the azimuthal direction accompanied
by a propagating solitary bulge wave. Fig. 3b plots modal energies in the azimuthal
velocity component to illustrate the rapid development of a mean swirl (e). This
fact together with the nonlinear growth of the rn = 1 wave already shown in Fig. 3a
may be related to Maxworthy's observation.
Fig. 5 shows contributions to the acoustic signal during the linear instability
phase comparing the results for the full and truncated integration domains. For
the diagonal terms the two domains have differences comparable to the amplitude.
These terms involve the m = 0 and rn = 2 modes. The off-diagonal terms have
smaller differences mainly in the form of high frequency oscillations present in the
truncated domain. The particle turn-around time is about 12 initially, and it cor-
responds to the period of acoustic oscillations due to an elliptically distorted core.
On the other hand, the term Q2"a+ Q'a'_, which involves only the m = 2 mode, has a
larger period. This is consistent with the frequency ratio of the two modes for thin
uniform vorticity cores: using Eq. (35) in Widnall & Sullivan (1973) we obtain
A ip.o .2ring
= vf3-_-q log(a/R),
felliptic core r£-
which gives an estimate of 0.3 for the present case (a is the radius of the uniform
vorticity core).
Further work using the same method should either focus on the sound for axi-
ally periodic flow (cylindrical wavefronts at infinity) using the appropriate acoustic
analogy or employ a larger domain.
4. Conclusions
The time evolution of azimuthal instabilities on a thick isolated vortex ring into
the late non-linear stage was studied by a numerical simulation of the Navier-Stokes
equations. Initially the m = 6 mode is most amplified, consistent with the linear
theory of Widnall et al. (1974). Subsequently, due to the viscous spreading of the
ring, the rn = 6 gave way to m = 4. In the late non-linear stage low order modes
(m = 2 and m = 1) became dominant and a rapid growth of a mean swirl is present,
which may be related to some dye observations of Maxworthy (1977).
The acoustic signal computed using the theory of M6hring (1978) was presented
for the linear instability phase. Since this theory is valid for unbounded flow, each
contribution (Qij) was computed in a full and in a truncated domain to estimate
errors associated with outlying vorticity. The errors were quite large for the diagonal
terms. Further work is needed to eliminate these errors.
REFERENCES
BRIDGES, J. E. & HUSSAIN, A. K. M. F. 1987 Roles of initial condition and
vortex pairing in jet noise. J. Sound. Vib. 117, 289-311.
228 R. Verzicco _ K. ,qhariff
HASSELBRINK, E. F. 1992 Breakdown and mixing in a viscous circular vortex ring.
BS thesis, Dept. of Mech. Eng., Univ. Houston.
KOP'EV, V. F. _ CHERNYCHEV, S. A. 1993 Sound radiation by high frequency
oscillations of the vortex ring. AIAA Paper 93-4362. 15th Aeroacoustics Conf.
MAXWORTHY, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech.
81,465-95.
MOHRING, W. 1978 On vortex sound at low Math number. J. Fluid Mech. 85,
685-691.
MOORE, D. W. 1980 The velocity of a vortex ring with a thin core of elliptical
cross-section. Proc. Roy. Soc. Lo_d. A. 370, 407-415.
SHARIFF, K. & LEONARD, A. 1992 Vortex rings. Ann. Rev. Fluid Mech. 24, 235-
279.
SHARIFF, K., LEONARD, A. _ FERZIGER, J. H. 1989 Dynamics of a class of
vortex rings. NASA TM 102257
SHARIFF, K., VERZICCO, R. _: ORLANDI, P. 1994 A numerical study of three-
dimensional vortex ring instabilities: viscous corrections and early non-linear
stage. J. Fluid Mech. 279, 351-375.
VERZICCO, R. _ ORLANDI, P. 1993 A finite-difference scheme for three-dimen-
sionM incompressible flows in cylindrical coordinates. Submitted to J. Comput.
Phys.
WIDNALL, S. E., BLISS, D. B. & TSAI, C.-Y. 1974 The instability of short waves
on a vortex ring. J. Fluid Mech. 66_ 35-47.
WIDNALL, S. E. & SULLIVAN, J. P. 1973 On the stability of vortex rings. Proc.
Roy. Soc. Lond. A. 332, 335-353.
ZAITSEV, M. Y., KOP'EV, V. F., MUNIN, A. G. & POTOKIN, A. A. 1990 Sound
radiation by a turbulent vortex ring. Soy. Phys. Dokl. 35, 488-489.
ZAITSEV, M. Y. & KOP'EV, V. F. 1993 Mechanism of sound generation by a
turbulent vortex ring. Acoust. Phys. 39, 562-565.


Vortex ring instability and its sound

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server