We present a linear optimal perturbation analysis of streamwise invariant disturbances evolving in parallel round jets. The potential for transient energy growth of perturbations with azimuthal wavenumber m1 is analyzed for different values of Reynolds number Re. Two families of steady (frozen) and unsteady (diffusing) base flow velocity profiles have been used, for different aspect ratios α = R/θ, where R is the jet radius and θ is the shear layer momentum thickness. Optimal initial conditions correspond to infinitesimal streamwise vortices, which evolve transiently to produce axial velocity streaks, whose spatial structure and intensity depend on base flow and perturbation parameters. Their dynamics can be characterized by a maximum optimal value of the energy gain Gopt, reached at an optimal time τopt after which the perturbations eventually decay. Optimal energy gain and time are shown to be, respectively, proportional to Re2 and Re, regardless of the frozen or diffusing nature of the base flow. Besides, it is found that the optimal gain scales like Gopt1/m3 for all m except m = 1. This quantitative difference for azimuthal wavenumber m = 1 is shown to be based on the nature of transient mechanisms. For m = 1 perturbations, the shift-up effect [J. I. Jiménez-González et al., “Modal and non-modal evolution of perturbations for parallel round jets,” Phys. Fluids 27, 044105-1–044105-19 (2015)] is active: an initial streamwise vorticity dipole induces a nearly uniform velocity flow in the jet core, which shifts the whole jet radially. By contrast, optimal perturbations with m2 are concentrated along the shear layer, in a way that resembles the classical lift-up mechanism in wall-shear flows. The m = 1 shift-up effect is more energetic than the m2 lift-up, but it is slower, with optimal times considerably shorter in the case of m2 disturbances. This suggests that these perturbations may emerge very quickly in the flow when injected as initial conditions. When the base flow diffuses, the large time scale for m = 1 disturbances allows the shear layer to spread and the jet core velocity to decrease substantially, thus lowering the values of corresponding optimal gain and time. For m2, results are less affected, since the shorter transient dynamics does not leave room for significant modifications of the base flow velocity profiles, and the scaling laws obtained in the frozen case are recovered. Nevertheless, base flow diffusion hinders the transient growth, as a consequence of a weaker component-wise non-normality and a smoother, radially spread structure of optimal disturbances.

1.
G. K.
Batchelor
and
E.
Gill
, “
Analysis of the stability of axisymmetric jets
,”
J. Fluid Mech.
14
,
529
551
(
1962
).
2.
A.
Michalke
, “
On the inviscid instability of the hyperbolic-tangent velocity profile
,”
J. Fluid Mech.
19
(
4
),
543
556
(
1964
).
3.
D. G.
Crighton
and
M.
Gaster
, “
Stability of slowly diverging jet flow
,”
J. Fluid Mech.
77
(
2
),
397
413
(
1976
).
4.
P. J.
Morris
, “
The spatial viscous instability of axisymmetric jets
,”
J. Fluid Mech.
77
,
511
529
(
1976
).
5.
A.
Michalke
and
G.
Hermann
, “
On the inviscid instability of a circular jet with external flow
,”
J. Fluid Mech.
114
,
343
359
(
1982
).
6.
A.
Michalke
, “
Survey on jet instability theory
,”
Prog. Aerosp. Sci.
21
,
159
199
(
1984
).
7.
M.
Abid
,
M.
Brachet
, and
P.
Huerre
, “
Linear hydrodynamic instability of circular jets with thin shear layers
,”
Eur. J. Mech., B: Fluids
12
(
5
),
683
693
(
1993
).
8.
M. P.
Lessen
and
P. J.
Singh
, “
The stability of axisymmetric free shear layers
,”
J. Fluid Mech.
60
,
433
457
(
1973
).
9.
P.
Plaschko
, “
Helical instabilities of slowly divergent jets
,”
J. Fluid Mech.
92
(
2
),
209
215
(
1979
).
10.
P. J.
Schmid
and
D. S.
Henningson
,
Stability and Transition in Shear Flows
(
Springer-Verlag
,
2001
).
11.
T.
Ellingsen
and
E.
Palm
, “
Stability of linear flow
,”
Phys. Fluids
18
(
4
),
487
488
(
1975
).
12.
M. T.
Landahl
, “
A note on an algebraic instability of inviscid parallel shear flows
,”
J. Fluid Mech.
98
(
2
),
243
251
(
1980
).
13.
X.
Garnaud
,
L.
Lesshafft
,
P. J.
Schmid
, and
P.
Huerre
, “
Modal and transient dynamics of jet flows
,”
Phys. Fluids
25
,
044103
(
2013
).
14.
S. A.
Boronin
,
J. J.
Healey
, and
S. S.
Sazhin
, “
Non-modal stability of round viscous jets
,”
J. Fluid Mech.
716
,
96
119
(
2013
).
15.
L.
Brandt
, “
The lift-up effect: The linear mechanism behind transition and turbulence in shear flows
,”
Eur. J. Mech., B: Fluids
47
,
80
96
(
2014
).
16.
W. M. F.
Orr
, “
The stability or instability of the steady motions of a perfect liquid and of viscous liquid. Part I: A perfect liquid
,”
Proc. R. Ir. Acad., Sect. A
27
,
9
68
(
1907–1909
), available at http://www.jstor.org/stable/20490590;
W. M. F.
Orr
, “
The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid
,” ibid.
27
,
69
138
(
1907–1909
); available at http://www.jstor.org/stable/20490591.
17.
X.
Garnaud
,
L.
Lesshafft
,
P. J.
Schmid
, and
P.
Huerre
, “
The preferred mode of incompressible jets: Linear frequency response analysis
,”
J. Fluid Mech.
716
,
189
202
(
2013
).
18.
J. I.
Jiménez-González
,
P.
Brancher
, and
C.
Martínez-Bazán
, “
Modal and non-modal evolution of perturbations for parallel round jets
,”
Phys. Fluids
27
,
044105-1
044105-19
(
2015
).
19.
Y.
Detandt
, “
Numerical simulation of aerodynamic noise in low Mach number flows
,” Ph.D. thesis,
Université Libre de Bruxelles
,
Belgium
,
2007
.
20.
T. H.
New
and
W. L.
Tay
, “
Effects of cross-stream radial injections on a round jet
,”
J. Turbul.
7
,
1
20
(
2006
).
21.
M. B.
Alkislar
,
A.
Krothapalli
, and
G. W.
Butler
, “
The effect of streamwise vortices on the aeroacoustics of a Mach 0.9 jet
,”
J. Fluid Mech.
578
,
139
169
(
2007
).
22.
P.
Zhang
, “
Active control of a turbulent round jet based on unsteady microjets
,” Ph.D. thesis,
The Hong Kong Polytechnic University
,
2014
.
23.
H.
Yang
,
Y.
Zhou
,
R. M. C.
So
, and
Y.
Liu
, “
Turbulent jet manipulation using two unsteady azimuthally separated radial minijets
,”
Proc. R. Soc. A
472
,
20160417
(
2016
).
24.
N. A.
Bakas
and
P. J.
Ioannou
, “
Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface
,”
Phys. Fluids
21
,
024102
(
2009
).
25.
C.
Cossu
and
L.
Brandt
, “
Stabilization of Tollmien-Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer
,”
Phys. Fluids
14
(
8
),
L57
L60
(
2002
).
26.
C.
Cossu
and
L.
Brandt
, “
On Tollmien–Schlichting-like waves in streaky boundary layers
,”
Eur. J. Mech., B: Fluids
23
,
815
833
(
2004
).
27.
G.
Del Guercio
,
C.
Cossu
, and
G.
Pujals
, “
Stabilizing effect of optimally amplified streaks in parallel wakes
,”
J. Fluid Mech.
739
,
37
56
(
2014
).
28.
G.
Del Guercio
,
C.
Cossu
, and
G.
Pujals
, “
Optimal streaks in the circular cylinder wake and suppression of the global instability
,”
J. Fluid Mech.
752
,
572
588
(
2014
).
29.
G.
Del Guercio
,
C.
Cossu
, and
G.
Pujals
, “
Optimal perturbations of non-parallel wakes and their stabilizing effect on the global instability
,”
Phys. Fluids
26
,
024110-1
024110-14
(
2014
).
30.
M.
Marant
,
C.
Cossu
, and
G.
Pujals
, “
Optimal streaks in the wake of a blunt-based axisymmetric bluff body and their influence on vortex shedding
,”
C. R. Mec.
345
,
378
385
(
2017
).
31.
L. H.
Gustavsson
, “
Energy growth of three-dimensional disturbances in plane Poiseuille flow
,”
J. Fluid Mech.
224
,
241
260
(
1991
).
32.
S. S.
Sazhin
,
S. B.
Martynov
,
T.
Kristyadi
,
C.
Crua
, and
M. R.
Heikal
, “
Diesel fuel spray penetration, heating, evaporation and ignition: Modeling versus experimentation
,”
Int. J. Eng. Syst. Modell. Simul.
1
,
1
19
(
2008
).
33.
S. S.
Sazhin
,
S. A.
Boronin
,
S.
Begg
,
C.
Crua
,
J.
Healey
,
N. A.
Lebedeva
,
A. N.
Osiptsov
,
F.
Kaplanski
, and
M. R.
Heikal
, “
Jet and vortex ring-like structures in internal combustion engines: Stability analysis and analytical solutions
,”
Proc. IUTAM
8
,
196
204
(
2013
).
34.
P.
Marmottant
and
E.
Villermaux
, “
On spray formation
,”
J. Fluid Mech.
498
,
73
111
(
2004
).
35.
A.
Michalke
, “
Instabilität eines kompressiblen runden freistrahls unter berücksichtigung des einflusses der strahlgrenzschichtdicke
,”
Z. Flugwiss.
8
-9 ,
319
328
(
1971
).
36.
P.
Corbett
and
A.
Bottaro
, “
Optimal linear growth in swept boundary layers
,”
J. Fluid Mech.
435
,
1
23
(
2001
).
37.
A.
Antkowiak
and
P.
Brancher
, “
Transient energy growth for the Lamb-Oseen vortex
,”
Phys. Fluids
16
(
1
),
L1
L4
(
2004
).
38.
A.
Antkowiak
, “
Dynamique aux temps courts d’un tourbillon isolé
,” Ph.D. thesis,
Université Paul Sabatier de Toulouse
,
France
,
2005
.
39.
B.
Fornberg
,
A Practical Guide to Pseudospectral Methods
(
Cambridge University Press
,
1995
).
40.
C.
Canuto
,
M. Y.
Hussaini
,
A.
Quarteroni
, and
T.
Zhang
,
Spectral Methods in Fluid Dynamics
(
Springer
,
1988
).
41.
J. A. C.
Weideman
and
S. C.
Reddy
, “
A MATLAB differentiation matrix suite
,”
ACM Trans. Math. Software
26
(
4
),
465
519
(
2000
).
42.
R. R.
Kerswell
and
A.
Davey
, “
On the linear instability of elliptic pipe flow
,”
J. Fluid Mech.
316
,
307
324
(
1996
).
43.
P. J.
Schmid
and
D. S.
Henningson
, “
Optimal energy density growth in Hagen-Poiseuille flow
,”
J. Fluid Mech.
277
,
197
225
(
1994
).
44.
K. M.
Butler
and
B. F.
Farrell
, “
Three-dimensional optimal perturbations in viscous shear flow
,”
Phys. Fluids A
4
(
8
),
1637
16509
(
1992
).
45.
J.-M.
Chomaz
, “
Global instabilities in spatially developing flows: Non-normality and nonlinearity
,”
Annu. Rev. Fluid Mech.
37
,
357
392
(
2005
).
46.
N.
Osizik
,
Heat Conduction
, 2nd ed. (
John Wiley & Sons
,
1993
).
47.

As underlined by one referee, it should be noted that slightly larger growths are observed at m>3 for the diffusing base flow, which is counter-intuitive. The diffusion of the base flow is associated with the diffusion of the jet momentum thickness and therefore a decrease of the aspect ratio α. As a first approximation, one could extrapolate the analysis done in the frozen case by applying these results to an effective aspect ratio, which is smaller than the initial one due to diffusion. In terms of energy gain, this approach would predict a wrong trend, namely, a lower growth, since the gain decreases for decreasing α [see Fig. 3(a)]. Nevertheless, it turns out that the optimal perturbation in the diffusing case has a radial extension which is larger than in the frozen case [data not shown for m>3 but this trend is already visible for m = 2, see Figs. 12(i) and 12(l)]. As suggested by the referee, it is as if the optimal perturbation anticipates the diffusion of the shear layer and adapts itself with a wider spatial extent in order to resist better to its own viscous diffusion. As a consequence, the optimal perturbation is expected to be less affected by viscous diffusion than in the frozen case and therefore to better benefit from the energy growth mechanism at play in the shear layer, hence an eventual higher growth.

You do not currently have access to this content.