In Part I [Wei et al., 2004, 2004 ASME Int. Mech. Eng. Conference], we presented the experimental results for swirling flows of water and cetyltrimethyl ammonium chloride (CTAC) surfactant solution in a cylindrical vessel with a rotating disk located at the bottom for a Reynolds number of around $4.3×104$ based on the viscosity of solvent. For the large Reynolds number, violent irregular instantaneous secondary flows at the meridional plane were observed by use of a particle image velocimetry system. Because of the limitations of our computer resources, we did not carry out direct numerical simulation for such a large Reynolds number. The LES and turbulence model are alternative methods, but a viscoelastic LES/turbulence model has not yet been developed for the surfactant solution. In this study, therefore, we limited our simulations to a laminar flow. The marker-and-cell method proposed for Newtonian flow was extended to the viscoelastic flow to track the free surface, and the effects of Weissenberg number and Froude number on the flow pattern and surface shape were studied. Although the Reynolds number is much smaller than that of the experiment, the major experimental observations, such as the inhibition of primary and secondary flows and the decrease of the dip of the free surface by the elasticity of the solution, were qualitatively reproduced in the numerical simulations.

1.
Wei
,
J. J.
,
Li
,
F. C.
,
Yu
,
B.
, and
Kawaguchi
,
Y.
, 2006, “
Swirling Flow of a Viscoelastic Fluid With Free Surface, Part I, Experimental Analysis of Vortex Motion by PIV
,”
ASME J. Fluids Eng.
0098-2202,
128
, pp.
69
76
.
2.
Bohme
,
G.
,
Voss
,
R.
, and
Warnercke
,
W.
, 1985, “
Die Frei Obserflache Einer Flussigkeit Uber Einer Riiterenden Scheibe
,”
Rheol. Acta
0035-4511,
24
, pp.
22
23
.
3.
Debbaut
,
B.
, and
Hocq
,
B.
, 1992, “
On The Numerical Simulation of Axisymmetric Swirling Flows of Differential Viscoelastic Liquids: The Rod-Climbing Effect and the Quelleffekt
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
43
, pp.
103
126
.
4.
Xue
,
S. C.
,
Phan-Thien
,
N.
, and
Tanner
,
R. I.
, 1999, “
Fully Three-Dimensional Time-Dependent Numerical Simulations of Newtonian and Viscoelastic Swirling Flows in a Confined Cylinder, Part I. Method and Steady Flows
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
87
, pp.
337
367
.
5.
Siginer
,
A.
, 1984, “
General Weissenberg Effect in Free Surface Rheometry, Part I: Analytical Consideration
,”
ZAMP
0044-2275,
35
, pp.
545
558
.
6.
Siginer
,
A.
, 1984, “
General Weissenberg Effect in Free Surface Rheometry, Part II: Experiments
,”
ZAMP
0044-2275,
35
, pp.
618
633
.
7.
Siginer
,
A.
, 1984, “
Free Surface on a Simple Fluid Between Rotating Eccentric Cylinders, Part I: Analytical Solution
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
15
, pp.
93
108
.
8.
Siginer
,
A.
, and
Beavers
,
G. S.
, 1984, “
Free Surface on a Simple Fluid Between Rotating Eccentric Cylinders. Part II: Experiments
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
15
, pp.
109
126
.
9.
Siginer
,
D. A.
, 1986, “
Torsional Oscillations of a Rod in a Layered Medium of Simple Fluids
,”
Int. J. Eng. Sci.
0020-7225,
24
, pp.
631
640
.
10.
Siginer
,
D. A.
, 1989, “
Free Surface on a Viscoelastic Liquid in a Cylinder With Spinning Bottom
,”
Macromol. Chem.
,
23
, pp.
73
90
.
11.
Siginer
,
A.
, 1991, “
Viscoelastic Swirling Flow With Free Surface in Cylindrical Chambers
,”
Rheol. Acta
0035-4511,
30
, pp.
159
174
.
12.
Siginer
,
D. A.
, and
Knight
,
R. W.
, 1993, “
Swirling Free Surface Flow in Cylindrical Containers
,”
J. Eng. Math.
0022-0833,
27
, pp.
245
264
.
13.
Siginer
,
D. A.
, 2004, “
On the Nearly Viscometric Torsional Motion of Viscoelastic Liquids Between Shrouded Rotating Disks
,”
ASME J. Appl. Mech.
0021-8936,
71
, pp.
305
313
.
14.
Nichols
,
B. D.
, and
Hirt
,
C. W.
, 1971, “
Calculating Three-Dimensional Free Surface Flows in the Vicinity of Submerged and Exposed Structures
,”
J. Comput. Phys.
0021-9991,
12
, p.
234
.
15.
Nichols
,
B. D.
, and
Hirt
,
C. W.
, 1975, “
Methods for Calculating Multidimensional, Transient Free Surface Flows Past Bodies
,”
Proc. of the First International Conf. On Num. Ship Hydrodynamics
,
Gaithersburg
, MD, Oct.
20
23
.
16.
Hirt
,
C. W.
, and
Nichols
,
B. D.
, 1981, “
Volume of Fluid Method For the Dynamics of Free Boundaries
,”
J. Comput. Phys.
0021-9991,
39
, p.
201
.
17.
Koren
,
B.
,
Lewis
,
M. R.
,
van Brummelen
,
E. H.
,
van Leer
,
B.
, 2002, “
Riemann-Problem and Level-Set Approaches for Homentropic Two-Fluid Flow Computations
,”
J. Comput. Phys.
0021-9991,
181
, pp.
654
674
.
18.
Tomes
,
M. F.
,
Mangiavacchi
,
N.
,
Cuminato
,
J. A.
,
Castelo
,
A.
, and
McKee
,
S.
, 2002, “
A Finite Difference Technique for Simulating Unsteady Viscoelastic Free Surface Flows
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
106
, pp.
61
106
.
19.
Kawaguchi
,
Y.
,
Wei
,
J. J.
,
Yu
,
B.
, and
Feng
,
Z. P.
, 2003, “
Rheological Characterization of Drag-Reducing Cationic Surfactant Solution: Shear and Elongational Viscosities of Dilute Solution
,” in
Proc. of the 4th ASME/JSME Joint Fluids Engineering Conference
, Honolulu,
Hawaii
.
20.
Giesekus
,
H.
, 1982, “
A Simple Constitutive Equation for Polymer Fluids Based on the Concept Deformation-Dependent Tensorial Mobility
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
11
, pp.
69
109
.
21.
Phillips
,
T. N.
, and
Williams
,
A. J.
, 2002, “
Comparison of Creeping and Inertial Flow of an Oldroyd B Fluid Through Planar and Axisymmetric Contractions
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
108
, pp.
25
47
.
22.
Hill
,
C. T.
, 1972, “
Nearly Viscometric Flow of Viscoelastic Fluids in the Disk and Cylindrical Geometry, II: Experimental
,”
Trans. Soc. Rheol.
0038-0032,
16
, pp.
213
245
.
23.
Stokes
,
J. R.
,
Grahan
,
L. J. W.
,
Lawson
,
N. J.
, and
Boger
,
D. V.
, 2001, “
Swirling Flow of Viscoelastic Fluids. Part I, Interaction Between Inertia and Elasticity
,”
J. Fluid Mech.
0022-1120,
429
, pp.
67
115
.
24.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
, 1987,
Dynamics of Polymer Liquids, Volume 1, Fluid Mechanics
,
John Wiley & Sons, Inc.
, New York, pp.
64
65
.
25.
Dimitropoulos
,
C. D.
,
Sureshkumar
,
R.
, and
Beris
,
A. N.
, 1998, “
Direct Numerical Simulation of Viscoelastic Turbulent Channel Flow Exhibiting Drag-Reduction: Effect of the Variation of Rheological Parameters
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
79
, pp.
433
468
.
26.
Yu
,
B.
, and
Kawaguchi
,
Y.
, 2003, “
Effect of Weissenberg Number on the Flow Structure: DNS Study of Drag-Reducing Flow With Surfactant Additives
,”
Int. J. Heat Fluid Flow
0142-727X,
24
, pp.
491
499
.
27.
Yu
,
B.
, and
Kawaguchi
,
Y.
, 2004, “
Direct Numerical Simulation of Viscoelastic Drag-Reducing Flow: A Faithful Finite Difference Method
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
116
, pp.
431
466
.