Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

References

References
1.
Leibovich
,
S.
,
1984
, “
Vortex Stability and Breakdown: Survey and Extension
,”
AIAA J.
,
22
(9), pp.
1192
1206
.
2.
Sarpkaya
,
T.
,
1995
, “
Turbulent Vortex Breakdown
,”
Phys. Fluids
,
7
(
10
), pp.
2301
2303
.
3.
Novak
,
F.
, and
Sarpkaya
,
T.
,
2000
, “
Turbulent Vortex Breakdown at High Reynolds Numbers
,”
AIAA J.
,
38
(
5
), pp.
825
834
.
4.
Faler
,
J. H.
, and
Leibovich
,
S.
,
1977
, “
Disrupted States of Vortex Flow and Vortex Breakdown
,”
Phys. Fluids
,
20
(
9
), pp.
1385
1400
.
5.
Grabowski
,
W. J.
, and
Berger
,
S. A.
,
1976
, “
Solutions of the Navier–Stokes Equations for Vortex Breakdown
,”
J. Fluid Mech.
,
75
(
3
), pp.
525
544
.
6.
Spall
,
R. E.
,
Gatski
,
T. B.
, and
Grosch
,
C. E.
,
1987
, “
A Criterion for Vortex Breakdown
,”
Phys. Fluids
,
30
(
11
), pp.
3434
3440
.
7.
Spall
,
R. E.
,
Gatski
,
T. B.
, and
Ash
,
R. L.
,
1990
, “
The Structure and Dynamics of Bubble-Type Vortex Breakdown
,”
Proc. R. Soc. London, Ser. A
,
429
(
1877
), pp.
613
637
.
8.
Darmofal
,
D. L.
,
1996
, “
Comparisons of Experimental and Numerical Results for Axisymmetric Vortex Breakdown in Pipes
,”
Comput. Fluids
,
25
(
4
), pp.
353
371
.
9.
Beran
,
P. S.
, and
Culick
,
F. E. C.
,
1992
, “
The Role of Non-Uniqueness in the Development of Vortex Breakdown in Tubes
,”
J. Fluid Mech.
,
242
, pp.
491
527
.
10.
Beran
,
P. S.
,
1994
, “
The Time-Asymptotic Behavior of Vortex Breakdown in Tubes
,”
Comput. Fluids
,
23
(
7
), pp.
913
937
.
11.
Lopez
,
J. M.
,
1994
, “
On the Bifurcation Structure of Axisymmetric Vortex Breakdown in a Constricted Pipe
,”
Phys. Fluids
,
6
(
11
), pp.
3683
3693
.
12.
Leibovich
,
S.
, and
Kribus
,
A.
,
1990
, “
Large Amplitude Wave Trains and Solitary Waves in Vortices
,”
J. Fluid Mech.
,
216
, pp.
459
504
.
13.
Vyazmina
,
E.
,
Nichols
,
J. W.
,
Chomaz
,
J. M.
, and
Schmid
,
P. J.
,
2009
, “
The Bifurcation Structure of Viscous Steady Axisymmetric Vortex Breakdown With Open Lateral Boundaries
,”
Phys. Fluids
,
21
(
7
), p.
074107
.
14.
Meliga
,
P.
, and
Gallaire
,
F.
,
2011
, “
Control of Axisymmetric Vortex Breakdown in a Constricted Pipe: Nonlinear Steady States and Weakly Nonlinear Asymptotic Expansions
,”
Phys. Fluids
,
23
(
8
), p.
084102
.
15.
Snyder
,
D. O.
, and
Spall
,
R. E.
,
2000
, “
Numerical Simulation of Bubble-Type Vortex Breakdown Within a Tube-and-Vane Apparatus
,”
Phys. Fluids
,
12
(
3
), pp.
603
608
.
16.
Wang
,
S.
, and
Rusak
,
Z.
,
1996
, “
On the Stability of an Axisymmetric Rotating Flow in a Pipe
,”
Phys. Fluids
,
8
(
4
), pp.
1007
1016
.
17.
Wang
,
S.
, and
Rusak
,
Z.
,
1997
, “
The Dynamics of a Swirling Flow in a Pipe and Transition to Axisymmetric Vortex Breakdown
,”
J. Fluid Mech.
,
340
, pp.
177
223
.
18.
Liang
,
H.
, and
Maxworthy
,
T.
,
2005
, “
An Experimental Investigation of Swirling Jets
,”
J. Fluid Mech
,
525
, pp.
115
159
.
19.
Benjamin
,
T. B.
,
1962
, “
Theory of the Vortex Breakdown Phenomenon
,”
J. Fluid Mech.
,
14
(
04
), pp.
593
629
.
20.
Gallaire
,
F.
,
Chomaz
,
J.-M.
, and
Huerre
,
P.
,
2004
, “
Closed-Loop Control of Vortex Breakdown: A Model Study
,”
J. Fluid Mech.
,
511
, pp.
67
93
.
21.
Rusak
,
Z.
,
Wang
,
S.
,
Xu
,
L.
, and
Taylor
,
S.
,
2012
, “
On the Global Nonlinear Stability of a Near-Critical Swirling Flow in a Long Finite-Length Pipe and the Path to Vortex Breakdown
,”
J. Fluid Mech.
,
712
, pp.
295
326
.
22.
Wang
,
S.
, and
Rusak
,
Z.
,
1997
, “
The Effect of Slight Viscosity on a Near-Critical Swirling Flow in a Pipe
,”
Phys. Fluids
,
9
(
7
), pp.
1914
1927
.
23.
Keller
,
J. J.
,
Egli
,
W.
, and
Exley
,
W.
,
1985
, “
Force- and Loss-Free Transitions Between Flow States
,”
Z. Angew. Math. Phys.
,
36
(
6
), pp.
854
889
.
24.
Rusak
,
Z.
,
1996
, “
Axisymmetric Swirling Flow Around a Vortex Breakdown Point
,”
J. Fluid Mech.
,
323
, pp.
79
105
.
25.
Mattner
,
T. W.
,
Joubert
,
P. N.
, and
Chong
,
M. S.
,
2002
, “
Vortical Flow, Part 1: Flow Through a Constant Diameter Pipe
,”
J. Fluid Mech.
,
463
, pp.
259
291
.
26.
Umeh
,
C. O. U.
,
Rusak
,
Z.
,
Gutmark
,
E.
,
Villalva
,
R.
, and
Cha
,
D. J.
,
2010
, “
Experimental and Computational Study of Nonreacting Vortex Breakdown in a Swirl-Stabilized Combustor
,”
AIAA J.
,
48
(
11
), pp.
2576
2585
.
27.
Rusak
,
Z.
, and
Lamb
,
D.
,
1999
, “
Prediction of Vortex Breakdown in Leading-Edge Vortices Above Slender Delta Wings
,”
J. Aircr.
,
36
(
4
), pp.
659
667
.
28.
Batchelor
,
G. K.
,
1956
, “
On Steady Laminar Flow With Closed Streamlines at Large Reynolds Number
,”
J. Fluid Mech.
,
1
(
02
), pp.
177
190
.
29.
Squire
,
H. B.
,
1956
, “
Rotating Fluids
,”
Surveys in Mechanics
, G. K. Batchelor and R. M. Davies,
Cambridge University Press
,
Cambridge, UK
, pp.
139
161
.
30.
Long
,
R. R.
,
1953
, “
Steady Motion Around a Symmetrical Obstacle Moving Along the Axis of a Rotating Liquid
,”
J. Meteorol.
,
10
(
3
), pp.
197
203
.
31.
Rusak
,
Z.
,
1998
, “
The Interaction of Near-Critical Swirling Flows in a Pipe With Inlet Azimuthal Vorticity Perturbations
,”
Phys. Fluids
,
10
(
7
), pp.
1672
1684
.
32.
Gallaire
,
F.
, and
Chomaz
,
J.-M.
,
2004
, “
The Role of Boundary Conditions in a Simple Model of Incipient Vortex Breakdown
,”
Phys. Fluids
,
16
(
2
), pp.
274
286
.
33.
Leclaire
,
B.
,
Sipp
,
D.
, and
Jacquin
,
L.
,
2007
, “
Near-Critical Swirling Flow in a Contracting Duct: The Case of Plug Axial Flow With Solid-Body Rotation
,”
Phys. Fluids
,
19
(
9
), p.
091701
.
34.
Leclaire
,
B.
, and
Sipp
,
D.
,
2010
, “
A Sensitivity Study of Vortex Breakdown Onset to Upstream Boundary Conditions
,”
J. Fluid Mech.
,
645
, pp.
81
119
.
35.
Buntine
,
J. D.
, and
Saffman
,
P. G.
,
1995
, “
Inviscid Swirling Flows and Vortex Breakdown
,”
Proc. R. Soc. London
, Ser. A,
449
(
1935
), pp.
139
153
.
36.
Xu
,
L.
,
2012
, “
Vortex Flow Stability, Dynamics and Feedback Stabilization
,” Ph.D. dissertation,
Rensselaer Polytechnic Institute
,
Troy, NY
.
37.
Granata
,
J.
,
2014
, “
An Active Feedback Flow Control Theory of the Vortex Breakdown Process
,”
Ph.D. dissertation
,
Rensselaer Polytechnic Institute
,
Troy, NY
.
38.
Rusak
,
Z.
,
Granata
,
J.
, and
Wang
,
S.
,
2015
, “
An Active Feedback Flow Control Theory of the Axisymmetric Vortex Breakdown Process
,”
J. Fluid Mech.
,
774
, pp.
488
528
.
39.
Webster
,
B. E.
,
Shephard
,
M. S.
,
Rusak
,
Z.
, and
Flaherty
,
J. E.
,
1994
, “
Automated Adaptive Time-Discontinuous Finite-Element Method for Unsteady Compressible Airfoil Aerodynamics
,”
AIAA J.
,
32
(
4
), pp.
748
757
.
You do not currently have access to this content.