This document numerically simulates instabilities in counter flows of viscoelastic fluids using two rheological models, Oldroyd-B and De Witt. Two instability mechanisms are observed: with low Reynolds numbers (Re < 0.1), circular vortex structures arise near the central point; with moderate Reynolds numbers (Re > 3), the instability onset resembles classical inertial instabilities. Very similar results are obtained for both fluid models, unlike in other flow problems. The instabilities are attributed to elastic energy accumulating and dominating over viscous forces at high Weissenberg numbers.

1. NUMERICAL OBSERVATION OF INSTABILITIES
IN COUNTER FLOWS OF VISCOELASTIC FLUID
Igor A. Mackarov
Mackarov@gmx.net
Abstract
Counter flows of the Oldroyd-B viscoelastic fluid within cross channels are described numerically. The simulation is based on the pressure-correction method in the form involving a simple grid topology whose
convergence has been earlier formally proved by the report author.
The process has been calculated from the rest state until the reach of the flow stationary conditions. It is found that on the stabilization phase with some threshold Weissenberg numbers the form of the flows undergoes
essential restructuring on two found mechanisms: with moderate Reynolds numbers the change of the flows is nearly stochastic and close to the loss of stability by a Newtonian fluid, whereas with smaller Reynolds
numbers (less than 0.1) circular vortex-like structures arise in the vicinity of the flows central point where anomalies of the pressure and the normal stress distributions are also observed. With extremely high Weissenberg
numbers (more than 100) such vortexes periodically rise and damp, with smaller Wi the flows, as a rule, become stable with the reach of the stationary phase of the counter flows.
These structures are shown to be close to the instability phenomena observed before in the experimental study of such kind of flows. Also, similarity is pointed out of the observed large-scale vortex-like structures with
other elastic instability features described hitherto, such as solitary vortex pairs ("diwhirls"), wavy cells etc.
Again, a set of calculations has been performed with another kind of fluids from the generalized Maxwell family: the De Witt fluid. Very close behaviors of the two fluids have been reported, unlike the earlier presented
case of the Couette flow, where the De Witt fluid, having non-zero normal stresses in this case, turns more instability-prone than the Oldroyd-B fluid under the same flow conditions.
In the context of the problem investigated, common features of the purely elastic instabilities, in comparison with the classical inertial instabilities, are discussed.
BACKGROUND
The elastic instabilities are known to depend on a fluid elastic en-
ergy which may be accumulated in a flow involving high stretch.
An example is the flow within cross microchannels experimentally
observed in [1] (flows of viscoelastic polyacrylamide+glycerol so-
lutions). The elastic instabilities there were observed: in the form
of loss of the symmetry, and as large-scale circular structures
(fluctuations). It was therefore of interest to numerically inves-
tigate this king of flow and such features using known rheological
models.
PROBLEM STATEMENT,
AND SOLUTION TECHNIQUE
Due to the domain symmetry, the problem was first solved in the
quarter domain (later the solution in the whole domain will be
shown and compared with this one).
PSfrag
x
y
Vertical wall
Horizontal wall
pinlet = t
1+t
poutlet = 0
0 1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
4
5
6
7
8
9
10
11
FIGURE 1: General layout of counter streams (a quarter do-
main)
Together with the momentum and continuity equations, each of the
two rheological model equations was solved:
θ
Dσ
Dt
+σ = 2µd (1)
Oldroyd-B
DOσi j
DOt
≡
∂σi j
∂t
+vk
∂σi j
∂xk
−
∂vi
∂xk
σk j −
∂vj
∂xk
σik (2)
De Witt (with Jauman derivative)
DJσi j
DJt
≡
∂σi j
∂ t
+vk
∂σi j
∂ xk
−ωikσjk −ωjkσik (3)
where σ,d,ω are stress, deformation rate, and vorticity tensors
respectively.
Very close solutions for the 2 models were obtained (all the results
below are shown for Oldroyd-B). This similarity, absent in other
problems, is discussed at the end.
Features of numerical procedure
• Pressure-correction method according to [2].
• Upwind differences for the convective terms.
• Elastic grid, more fine closer to the central point and the walls
corner, varied from 300 up to 4000 nodes, the time step - from
10−3
up to 10−5
.
NORMAL STRESSES
IN THE HIGH-STRETCH REGION
The normal stresses sharply grow by the absolute value in the
flows turning zone, especially near the central and the walls cor-
ner points, as in the case of σxx shown below:
xy
σxx
Central point of counter flows
Walls corner
inlet
outlet
0
2
4
6
8
10
12
0
2
4
6
8
10
12
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
FIGURE 2: Normal stress distribution of the counter flows.
Re=0.1, Wi=4, t∼=3, the grid has 675 nodes
INSTABILITIES
Overall, two different mechanisms of the flow perturbations were
distinguished. Here is the picture typical for the first one. There
is a group consisting of 3 vortexes with opposite directions of
rotation (cf. ’diwhirls [3], i.e., pairs rotating in opposite directions)
x
y
Group of 3
circular structures
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
FIGURE 3: Large-scale instability specific for low (≤0.1)
Reynolds numbers. Re=0.05, Wi=4, t∼=2.7, the grid has 1200
nodes
With the vortexes intensity rising and damping periodically, it
turned out convenient to describe the degree of stability (or per-
turbations) of the whole of the flow by a "stability functional":
F(t) =
1
N ∑
i, j
∂ui j(t)
∂t
+
∂vi j(t)
∂t
(4)
where summation is taken over the grid, and N is the number of
nodes in the domain.
t
StabilityfunctionalF
0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025
FIGURE 4: Time dependencies of stability functional (4) for
Reynolds number 0.05, and Weissenberg numbers 4, 16, and
64. Highlighted is the time moment of the flow conditions
stabilization (pinlet ≈ 1), the grid has 300 nodes
The second mechanism is specific for higher Reynolds numbers:
x
y
Locus of instability growth
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
FIGURE 5: Beginning of sharp instability development spe-
cific for moderate (≥3) Reynolds numbers. Re=5, Wi=3,
t∼=5.3, the grid consists of 300 nodes
Evidently, the situation is close to the classical Reynolds instabil-
ity, though this instability may first affect some local zone only (as
a rule the region between the central and the wall corner points),
the rest of the flow being regular for some time.
t
StabilityfunctionalF
cf. previous figure
"Bursts" of instabilities -
0 5 10 15 20 25 30
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
FIGURE 6: Time dependencies of the stability functional (4)
for Reynolds number 5, and Weissenberg numbers 0.5, 1, and
3. Highlighted is the time moment of the flow conditions
stabilization (pinlet ≈ 1), the grid has 300 nodes
FULL-DOMAIN SOLUTION
In all aspects, the full-domain and the one-quadrant solutions
proved very close. Meanwhile, some violations of the symmetry
on top of the vortexes intensity were noticed (cf. [1]).
x
y
8 8.5 9 9.5 10 10.5 11 11.5 12
8
8.5
9
9.5
10
10.5
11
11.5
12
FIGURE 7: Full-domain observation of the large-scale insta-
bility, specific for low (≤0.1) Reynolds numbers. Marked are
four triple vortex structures. Re=0.1, Wi=4, t∼=2.7, the grid
consists of 4800 nodes
HIGH WEISSENBERG NUMBERS
Calculations (in both full and quarter domains) were made with
Wi up to 150 with Re such that Wi·Re∼0.5 (cf. [4] with Wi=100,
Re=0.01). Periodically rising circular structures, like on FIGURE
4, were observed too but their amplitude didn’t damp up to t∼30.
Cf. [5] in regard with the conclusion that with extremely high
Weissenberg numbers the oscillations initiated at the stabilization
of the Couette flow may never finish (within the used model).
DISCUSSION
• Two used models reveal very close characteristics of the flows
in the problem considered, including the normal stress distribu-
tions, and the flow restructuring/instability phenomena - unlike
acceleration phase or the Cuette flow where the De Witt fluid
flows turn much less stable which is likely due to the presence
of the normal stresses, absent in the case of Oldroyd-B [5, 6].
This indirectly points to the crucial role of the normal stresses
in the elastic instability development.
• Figures 3, 7 demonstrate non-stochastic flows - why instability?
- Like with the classical Reynolds instability (when the convec-
tive energy begins to dominate over the viscous factor, case of
Fig. 6), the essential flow restructuring comes up when the ac-
cumulated elastic energy (controlled by Wi) begins to dominate
over the viscous factor [7].
References
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