Full text of DOI:10.1063/1.870409

Viscosity influence on the stability of a swirling jet with nonrotating core
L. G. Sarasúa, and A. C. Sicardi Schifino
Citation: Physics of Fluids 12, 1607 (2000); doi: 10.1063/1.870409
View online: https://doi.org/10.1063/1.870409
View Table of Contents: http://aip.scitation.org/toc/phf/12/6
Published by the American Institute of Physics

PHYSICS OF FLUIDS
VOLUME 12, NUMBER 6
JUNE 2000
Viscosity influence on the stability of a swirling jet with nonrotating core
a)
L. G. Sarasua and A. C. Sicardi Schifinob)
´
´
Facultad de Ciencias, Universidad de la Republica, Montevideo, Uruguay
͑Received 20 October 1998; accepted 12 January 2000͒
The effect of the viscosity on the linear stability of a swirling top hat jet with nonrotating core is
studied analytically. Numerical values of the growth rate are obtained. It is shown that the viscosity
induces instabilities in the region of wave numbers where no inviscid instability occurs. In the short
wave limit, the asymptotic solution reveals that the situation is quite different for the strictly inviscid
and the viscous cases. While in the inviscid fluid the growth rate is proportional to the wave number,
in the viscous case the growth rate approaches a constant value for large wave numbers. © 2000
American Institute of Physics. ͓S1070-6631͑00͒00405-0͔
We study the influence of the viscosity on the stability to
axisymmetric disturbances of a swirling top hat jet with non-
rotating core of radius R. The velocity is assumed to have a
constant circulation ⌫ outside of the core, and zero circula-
tion in the core of the jet. The axial component of the veloc-
We assume that the perturbations are of the form
u , ,w ,p ϭ f r͒,g r͒,h r͒,␲ r͒
vj
͑
͑
͑
j
͑
͖
͕
͖
͕
j
j
j
j
ϫexp i␣ zϪct͒
͔
͓
͑
͑where jϭi, ͒. Substituting these expressions into Eqs. ͑2͒,
v
˜
ity has a constant value W in the inner region and zero value
we obtain the modified Bessel equations
outside. The components for the basic flow U are explicitly
given by
1 d
dhv
dr
1 d
dhi
dr
r
Ϫ␮2h ϭ0,
r
Ϫ␣2h ϭ0,
i
ͩ
ͪ
ͩ
ͪ
v
r dr
r dr
0,⌫/2␲r,0 for rϾR
͕
͖
U,V,W ϭ
,
͕
͖
ͭ
˜
2
2
0,0W for rϽR
͕
͖
˜
with ␮ ϭ␣ Ϫi␣(cϪW)/␯. By requiring the solution to be
bounded at the origin, we obtain the solutions hv
ϭAI0(␮r) and hiϭBI0(␣r), and therefore the complete ex-
pression of the disturbances in the viscous region are
in (r,␪,z) cylindrical components. Note that there is a veloc-
ity jump in both the azimuthal and the axial velocity.
A similar model, considering the fluid as inviscid, has
been investigated by Martin and Meiburg,1 and previously by
Rotunno.2 In the former, the capacity of the increasing cir-
culation to stabilize the Kelvin–Helmholtz instability3 is in-
vestigated.
i d
␣ d
fϭϪB
I0 ␣r͒ϪiA
͑
I0 ␮r͒,
͑
2
␣ dr
␮ dr
hϭBI0 ␣r͒ϩAI ␮r͒,
͑
͑
0
In our model, it is not possible to take the viscosity to be
nonzero everywhere in the fluid. If the viscosity is taken to
be different to zero everywhere, the stress tensor is infinite at
rϭR by the presence of the velocity jump. In the present
model we consider that the viscosity is small but different to
zero in the inner region, while the fluid is inviscid in the
outer region.
˜
␲ϭB cϪW͒I ␣r͒.
͑
͑
0
In the outer region, the flow can be described by the
potential theory.1 Requiring the solution to be bounded at
infinity, we find the expressions for the perturbations in the
outer region,
The linearized equations governing the perturbations v
in the inner region are
i d
hϭCK0 ␣r͒, fϭϪC
K0 ␣r͒, ␲ϭCcK ␣r͒.
͑ ͑
0
͑
␣ dr
ץv
ץt
1
ϩU•ٌvϭϪ ٌpϩ␯⌬2v,
͑1͒
We now impose boundary conditions at the separation
surface of the two regions, which can be expressed as ␩
ϭRϩ␦, where ␦ϭE exp(ik(zϪct)). The kinematical condi-
tion that the surface moves with the fluid gives D␩/Dt
ϭd␩/dtϩ(U•ٌ)␩ϭu, that is,
␳
with ٌ•vϭ0, where p is the perturbation on the pressure, ␳
the density, and ␯ the viscosity of the fluid. In order to solve
these equations, we write the disturbance as the sum vϭvi
ϩvv ,4 where the first term is irrotational ٌϫviϭ0. By using
these properties, we obtain that ͑1͒ is satisfied if v verifies
␣
˜
␣ cϪW EϭϪAIЈ ␣R Ϫ BIЈ ␮R ,
͒
͒
͒
͑3͒
͑
͑
͑
0
0
␮
ץ
ץvi
ץt
1
ϩU•ٌ v ϭ␯ٌ2v ,
ϩU•ٌviϭϪ ٌp.
͑2͒
␣cdϭϪCKЈ ␣R͒.
͑4͒
͑
ͩ
ͪ
v
v
0
ץt
␳
The dynamical boundary condition is that the stress of
the fluid must be continuous across the surface,
a͒
Electronic mail: sarasua@fisica.edu.uy
b͒
Electronic mail: asicardi@fisica.edu.uy
␴
ij,2(Rϩ␦)ϭ␴ij,1(Rϩ␦). Linearizing, these read
1070-6631/2000/12(6)/1607/4/$17.00
1607
© 2000 American Institute of Physics

´
L. G. Sarasua and A. C. Sicardi Schifino
1608
Phys. Fluids, Vol. 12, No. 6, June 2000
* *
FIG. 1. Growth rate ␴ϭ␣ c for ͑a͒
⌬ϭ0. The flow is unstable for all val-
ues of Re. The growth rate ␴ ap-
*
proaches a constant value as ␣ tends
to infinity. ͑b͒ ⌬ϭ10. In this case, the
circulation existing around the jet sta-
bilizes the flow for small wave num-
bers. However, the stabilization of the
increasing circulation is less effective
for the viscous fluid.
␶
R͒ϭ0
͑
*
IЉ ␣ ͒
͑
2i
2i
rz,2
0
*
*
*
*
␣
c Ϫ1ϩ
␣
c Ϫ1ϩ
ͩ
ͪ
ͩ
ͪ
*
Re
I0 ␣ ͒
Re
͑
*
*
IЈ ␣ ͒
͑
0
IЈ ␣ ͒
͑
*
K ␣ ͒
͑
ץP02
ץP01
0
2Ϫ⌬
*
Ϫ
ϩ
c
␲
R͒ϩ
rϭRϪ␶
R͒ϭ␲ R͒ϩ
,
͑
͑
rr,2
͑
ͯ
ͯ
2
1
*
*
*
*
␣ I0 ␣ ͒
͑
I0 ␣ ͒ K ␣ ͒
͑
͑
Ј
ץ␶
ץr
rϭR
*
*
͑
IЉ ␮ ͒ IЈ ␣ ͒
0
2
͑
4
0
2
*
*
␣
␮
ϭ0,
͑5͒
Re2
I0 ␣ ͒
*
*
͑
IЈ ␮ ͒
͑
where ␶ is the deviatoric term of the stress tensor and P0 is
0
ij
the pressure of the unperturbed flow. Substituting for the
expressions of the disturbances we get two equations which
together with Eqs. ͑3͒ and ͑4͒ constitute a linear system for
the constants A, B, C, E. Elimination of these constants give
the following eigenvalue equation:
˜
˜
*
*
*
␮
where
ReϭWR/␯,
c ϭc/W,
␣ ϭ␣R,
˜
2
2
ϭͱ␣
*
*
*
Ϫi(c Ϫ1)␣ Re, and ⌬ϭ(⌫/2␲WR) . The sign of
the imaginary part of the eigenvalue cϭcrϩici determines

Phys. Fluids, Vol. 12, No. 6, June 2000
Viscosity influence on the stability of a swirling jet . . .
1609
FIG. 2. The variation of the imaginary
*
part of the velocity ci as a function of
*
*
log Re, for ␣ ϭ12 and ␣ ϭ24. In the
*
first case (␣ Ͻ2⌬), the growth rate
has maximum at the value Re
ϭ14.6, while in the second one (␣
a
*
Ͼ2⌬), the mode is most unstable as
Re→ϱ.
the stability of the jet. When ci is positive, the disturbance
grows exponentially and the flow is unstable.
The stabilization which takes place for ⌬Ͼ0 appears to
be related to the Rayleigh stability,3 since this case corre-
sponds to the situation in which the outer circulation is
greater than the circulation in the core ͑which is null in fact͒.
We show here that the Rayleigh stability remains dominant
at small wave numbers even in the presence of viscosity.
This is, for all values of Re as in the inviscid case.1
Let us consider Re to be the Reynolds number of the
basic flow. As Re tends to infinity, the eigenvalue equation
for the inviscid case of Martin and Meiburg1 is recovered
from ͑5͒ for the particular case of zero circulation in the core.
We solve the eigenvalue equation by use of the Newton
*
method to determine c for different given values of Re and
For arbitrary Re the flow is unstable to sufficiently large
wave number. Moreover, Fig. 1 shows that the growth rate of
the disturbances with wave number between ⌬ and 2⌬ has a
maximum for a finite value of Re. In fact, there are modes
which are stable in the inviscid case, but become unstable in
the viscous case. To study in detail this property, let us con-
sider that ⌬ӷ1 and Re is very high but finite. Employing the
asymptotic expression for the modified Bessel functions for
*
the wave number ␣ . Figure 1 shows the nondimensional
*
*
velocity growth ␴ϭ␣ ci for the most unstable solution,
i.e., the eigenvalue with maximum imaginary part, as a func-
tion of ␣ for different values of Re.
*
This figure reveals that when ⌬ is different to zero, the
flow is stable for perturbations with small wave numbers, but
unstable for sufficient large wave numbers at all values of
Re. However, with ⌬ϭ0 the flow is unstable for all wave
numbers at any value of Re.
*
large values of the argument, Eq. ͑5͒ becomes (c Ϫ1
2
2
2
*
*
*
* *
ϩi2␣ /Re) ϩc Ϫ⌬/␣ ϩ4␣ ␮ /Re ϭ0.
Let us consider the limit of small wave numbers. The
Since we assumed Re to be very high, we consistently
consider only the least order terms ͑in 1/Re͒. Then, the fol-
*
asymptotic expression of Eq. ͑5͒ ͑as ␣ →0͒ results in the
*
following eigenvalue equation:
lowing approximate expression for c is obtained:
*
i
i2␣
␣
⌬
2
*
␣
1
*
*
*
*
ln ␣ Ϫ
c Ϫ1ϩ
c Ϫ1ϩ
Ϫc
ͩ
ͪͩ
ͪ
1Ϯͱ2⌬/␣ Ϫ1 ϩ Ϫ1Ϯ1/ 1Ϫ2⌬/␣
*
͒͒
*
c ϭ
*
Re
Re
2
2
i.
͑
͒
͑
͑
2
Re
͑6͒
2
*
2␣
ϩ
ϭ0.
Re2
*
From the instability condition ci Ͼ0, we find that the
flow is unstable for all disturbances with ␣ Ͼ⌬, for finite
To solve this quadratic equation, we conserve lower or-
der terms in ␣ , considering in separate form those terms
depending on Re and those independent of it. Hence, the
expression of c for small wave numbers is
*
*
Re. This last result is unexpected since usually it has been
accepted that the viscosity acts purely as a stabilizing agent
for swirling flows in the absence of rigid boundaries.5 Re-
cently, Mayer and Powell6 and Khorrami7 reported viscous
instabilities in their study of the trailing vortex stability.
They found that a stable range of wave numbers in the in-
viscid case was destabilized by increasing viscosity.
It can be pointed out that this destabilizing influence of
viscosity can appear in other flows like boundary layer or
parallel flows.3,8,9
*
i
2
1Ϯͱ1Ϫ2⌬ Re Ϫ2Re ␣
2
2
*
*
*
ln ␣ ͒.
c ϭ1Ϫ
͑
2 Re
When ⌬Ͼ0, the last term under the square root is negligible
and then the flow is stable for all values of Re. With ⌬ϭ0,
the term under the square root is greater than 1 and then the
basic flow is unstable for long waves.

´
1610
Phys. Fluids, Vol. 12, No. 6, June 2000
L. G. Sarasua and A. C. Sicardi Schifino
*
The dependence of ci as a function of log Re is shown
phenomenon that a swirling flow can be destabilized by the
presence of the viscosity, without the presence of rigid walls.
Rayleigh stability predominates in the region of small wave
numbers. However, the stable region is reduced in the pres-
ence of viscosity.
in Fig. 2. For wave numbers less than 2⌬, the growth rate has
a maximum for a relative low Re, and tends to zero as Re
→ϱ, decreasing with 1/Re.
*
On the other hand, for a fixed wave number ␣ Ͼ2⌬,
the disturbance is most unstable as Re tends to infinity.
It is interesting to compare these results with those ob-
tained in recent works by Mayer and Powell6 and Khorrami,7
in which the stability of the trailing vortex is studied by
numerical methods. Results from these earlier investigations
have several similar features to what we have observed. In
particular, in Ref. 6 the authors report instabilities with
growth rate decreasing as 1/Re at a fixed wave number, in
These results show that the effects of the viscosity are
drastic in the range of large wave numbers. The conclusions
drawn from inviscid models require great care in the short
wave region.
The present model is restrictive since it takes the fluid to
be viscous in a limited zone of the volume. Analytically, we
have obtained results which are in agreement with the more
computationally intensive investigations performed in Refs.
6 and 7.
*
agreement with the expression for c in Eq. ͑6͒. These co-
incidences are remarkable, taking into account the differ-
ences in the models.
ACKNOWLEDGMENTS
In the limit of large ␣ ͑finite Re͒, the approximation
This work was supported in part by the Programa de
Desarrollo de las Ciencias Basicas ͑PEDECIBA, Uruguay͒
and by a project CONICYT-Clemente Estable ͑No. 3046͒ of
the Consejo Nacional de Investigacion Cientıficas y tecnicas
*
*
*
␮ Ӎ␣ Ϫ(1/2)i(c Ϫ1)Re holds and hence Eq. ͑5͒ be-
´
comes a quadratic equation with solutions
*
*
c ϭ1/2 1Ϫi␣ /Re͒
͑
´
´
´
͑CONICYT—Uruguay͒.
2
Ϯϩ1/2ͱ i␣ /Reϩ1͒ Ϫ2ϩ2⌬/␣ .
*
*
͑
*
Taking the limit ␣ →ϱ, we find the two solutions for
1J. E. Martin and E. Meiburg, ‘‘On the stability of the swirling jet shear
layer,’’ Phys. Fluids 6, 424 ͑1994͒.
*
*
*
the complex velocity approach c →1, c →Ϫi␣ /Re. The
2R. Rotunno, ‘‘A note on the stability of a cylindrical vortex sheet,’’ J.
Fluid Mech. 87, 761 ͑1978͒.
first one is the unstable solution, and its imaginary part tends
*
*
to zero as 1/␣ in the limit ␣ →ϱ. Then the growth rate
approaches a constant value for large wave numbers, for all
values of Re ͑see Fig. 1͒. The situation is quite different from
3F. G. Drazin and W. H. Reid, Hydrodynamic Stability ͑Cambridge Uni-
versity, Cambridge, 1981͒.
4H. Lamb, Hydrodynamics ͑Dover, New York, 1945͒.
5G. K. Batchelor and A. E. Gill, ‘‘Analysis of the stability of axisymmetric
jets,’’ J. Fluid Mech. 14, 529 ͑1962͒.
*
the strictly inviscid model case, for which c →1/2(1Ϯi) as
*
␣ →ϱ, and as a consequence, the growth rate tends to in-
6E. W. Mayer and K. G. Powell, ‘‘Viscous and inviscid instabilities of a
trailing vortex,’’ J. Fluid Mech. 245 ͑1992͒.
finity in the limit of large wave numbers. Viscous analysis
thus demonstrates that although large wave number modes
are the most unstable in the strictly inviscid case, they are
also the most susceptible to the stabilizing action of the vis-
cosity.
7M. R. Khorrami, ‘‘On the viscous modes of instability of a trailing vor-
tex,’’ J. Fluid Mech. 255 ͑1991͒.
8L. N. Howard, ‘‘Hydrodynamic stability of a jet,’’ J. Math. Phys. 37
͑1959͒.
9X. Li and R. S. Tankin, ‘‘On a temporal instability of a two dimensional
viscous liquid sheet,’’ J. Fluid Mech. 104 ͑1981͒.
In this work, we present analytical evidence of the novel