We study the laminar and turbulent channel flow over a viscous hyper-elastic wall and show that it is possible to sustain an unsteady chaotic turbulent-like flow at any Reynolds number by properly choosing the wall elastic modulus. We propose a physical explanation for this effect by evaluating the shear stress and the turbulent kinetic energy budget in the fluid and elastic layer. We vary the bulk Reynolds number from 2800 to 10 and identify two distinct mechanisms for turbulence production. At moderate and high Reynolds numbers, turbulent fluctuations activate the wall oscillations, which, in turn, amplify the turbulent Reynolds stresses in the fluid. At a very low Reynolds number, the only production term is due to the energy input from the elastic wall, which increases with the wall elasticity. This mechanism may be exploited to passively enhance mixing in microfluidic devices.

1.
D.
Alghalibi
,
M. E.
Rosti
, and
L.
Brandt
, “
Inertial migration of a deformable particle in pipe flow
,”
Phys. Rev. Fluids
4
(
10
),
104201
(
2019
).
2.
M. Z.
Bazant
and
T. M.
Squires
, “
Induced-charge electrokinetic phenomena: Theory and microfluidic applications
,”
Phys. Rev. Lett.
92
(
6
),
066101
(
2004
).
3.
S.
Berti
and
G.
Boffetta
, “
Elastic waves and transition to elastic turbulence in a two-dimensional viscoelastic Kolmogorov flow
,”
Phys. Rev. E
82
(
3
),
036314
(
2010
).
4.
F. G.
Bessoth
,
A. J.
Manz
 et al, “
Microstructure for efficient continuous flow mixing
,”
Anal. Commun.
36
(
6
),
213
215
(
1999
).
5.
K.
Gardner
,
E. R.
Pike
,
M. J.
Miles
,
A.
Keller
, and
K.
Tanaka
, “
Photon-correlation velocimetry of polystyrene solutions in extensional flow fields
,”
Polymer
23
(
10
),
1435
1442
(
1982
).
6.
I.
Glasgow
and
N.
Aubry
, “
Enhancement of microfluidic mixing using time pulsing
,”
Lab Chip
3
(
2
),
114
120
(
2003
).
7.
A.
Groisman
and
V.
Steinberg
, “
Elastic turbulence in a polymer solution flow
,”
Nature
405
(
6782
),
53
55
(
2000
).
8.
S. J.
Haward
,
G. H.
McKinley
, and
A. Q.
Shen
, “
Elastic instabilities in planar elongational flow of monodisperse polymer solutions
,”
Sci. Rep.
6
,
33029
(
2016
).
9.
C. C.
Hopkins
,
S. J.
Haward
, and
A. Q.
Shen
, “
Purely elastic fluid-structure interactions in microfluidics: Implications for mucociliary flows
,”
Small
16
(
9
),
1903872
(
2020
).
10.
D.
Izbassarov
,
M. E.
Rosti
,
M. N.
Ardekani
,
M.
Sarabian
,
S.
Hormozi
,
L.
Brandt
, and
O.
Tammisola
, “
Computational modeling of multiphase viscoelastic and elastoviscoplastic flows
,”
Int. J. Numer. Methods Fluids
88
(
12
),
521
543
(
2018
).
11.
J.
Jiménez
and
A.
Pinelli
, “
The autonomous cycle of near-wall turbulence
,”
J. Fluid Mech.
389
,
335359
(
1999
).
12.
J.
Jiménez
,
M.
Uhlmann
,
A.
Pinelli
, and
G.
Kawahara
, “
Turbulent shear flow over active and passive porous surfaces
,”
J. Fluid Mech.
442
,
89
117
(
2001
).
13.
D.
Kawale
,
E.
Marques
,
P. L. J.
Zitha
,
M. T.
Kreutzer
,
W. R.
Rossen
, and
P. E.
Boukany
, “
Elastic instabilities during the flow of hydrolyzed polyacrylamide solution in porous media: Effect of pore-shape and salt
,”
Soft Matter
13
(
4
),
765
775
(
2017
).
14.
S.
Kazemi
,
V.
Nourian
,
M. R. H.
Nobari
, and
S.
Movahed
, “
Two dimensional numerical study on mixing enhancement in micro-channel due to induced charge electrophoresis
,”
Chem. Eng. Process.
120
,
241
250
(
2017
).
15.
B.
Keshavarzian
,
M.
Shamshiri
,
M.
Charmiyan
, and
A.
Moaveni
, “
Optimization of an active electrokinetic micromixer based on the number and arrangement of microelectrodes
,”
J. Appl. Fluid Mech.
11
(
6
),
1531
1541
(
2018
).
16.
J.
Kim
and
P.
Moin
, “
Application of a fractional-step method to incompressible Navier–Stokes equations
,”
J. Comput. Phys.
59
(
2
),
308
323
(
1985
).
17.
J.
Kim
,
P.
Moin
, and
R.
Moser
, “
Turbulence statistics in fully developed channel flow at low Reynolds number
,”
J. Fluid Mech.
177
,
133
166
(
1987
).
18.
J. B.
Knight
,
A.
Vishwanath
,
J. P.
Brody
, and
R. H.
Austin
, “
Hydrodynamic focusing on a silicon chip: Mixing nanoliters in microseconds
,”
Phys. Rev. Lett.
80
(
17
),
3863
(
1998
).
19.
V.
Kumaran
, “
Stability of inviscid flow in a flexible tube
,”
J. Fluid Mech.
320
,
1
17
(
1996
).
20.
V.
Kumaran
and
R.
Muralikrishnan
, “
Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface
,”
Phys. Rev. Lett.
84
(
15
),
3310
(
2000
).
21.
Y.
Kuwata
and
K.
Suga
, “
Transport mechanism of interface turbulence over porous and rough walls
,”
Flow, Turbul. Combust.
97
(
4
),
1071
1093
(
2016
).
22.
Y.
Kuwata
and
K.
Suga
, “
Extensive investigation of the influence of wall permeability on turbulence
,”
Int. J. Heat Fluid Flow
80
,
108465
(
2019
).
23.
R. G.
Larson
, “
Instabilities in viscoelastic flows
,”
Rheol. Acta
31
(
3
),
213
263
(
1992
).
24.
C.-Y.
Lee
,
C.-L.
Chang
,
Y.-N.
Wang
, and
L.-M.
Fu
, “
Microfluidic mixing: A review
,”
Int. J. Mol. Sci.
12
(
5
),
3263
3287
(
2011
).
25.
C.-Y.
Lee
,
W.-T.
Wang
,
C.-C.
Liu
, and
L.-M.
Fu
, “
Passive mixers in microfluidic systems: A review
,”
Chem. Eng. J.
288
,
146
160
(
2016
).
26.
E. J.
Lim
,
T. J.
Ober
,
J. F.
Edd
,
S. P.
Desai
,
D.
Neal
,
K. W.
Bong
,
P. S.
Doyle
,
G. H.
McKinley
, and
M.
Toner
, “
Inertio-elastic focusing of bioparticles in microchannels at high throughput
,”
Nat. Commun.
5
(
1
),
1
9
(
2014
).
27.
G. H.
McKinley
,
P.
Pakdel
, and
A.
Öztekin
, “
Rheological and geometric scaling of purely elastic flow instabilities
,”
J. Non-Newtonian Fluid Mech.
67
,
19
47
(
1996
).
28.
G. A.
Mensing
,
T. M.
Pearce
,
M. D.
Graham
, and
D. J.
Beebe
, “
An externally driven magnetic microstirrer
,”
Philos. Trans. R. Soc., A
362
(
1818
),
1059
1068
(
2004
).
29.
T.
Min
,
J. Y.
Yoo
, and
H.
Choi
, “
Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows
,”
J. Non-Newtonian Fluid Mech.
100
(
1
),
27
47
(
2001
).
30.
A.
Monti
,
M.
Omidyeganeh
,
B.
Eckhardt
, and
A.
Pinelli
, “
On the genesis of different regimes in canopy flows—A numerical investigation
,”
J. Fluid Mech.
891
,
A9
(
2020
).
31.
A.
Monti
,
M.
Omidyeganeh
, and
A.
Pinelli
, “
Large-eddy simulation of an open-channel flow bounded by a semi-dense rigid filamentous canopy: Scaling and flow structure
,”
Phys. Fluids
31
(
6
),
065108
(
2019
).
32.
S. B.
Pope
,
Turbulent Flows
(
Cambridge University Press
,
2001
).
33.
M. E.
Rosti
and
L.
Brandt
, “
Numerical simulation of turbulent channel flow over a viscous hyper-elastic wall
,”
J. Fluid Mech.
830
,
708
735
(
2017
).
34.
M. E.
Rosti
and
L.
Brandt
, “
Suspensions of deformable particles in a Couette flow
,”
J. Non-Newtonian Fluid Mech.
262
(
C
),
3
11
(
2018
).
35.
M. E.
Rosti
,
L.
Brandt
, and
D.
Mitra
, “
Rheology of suspensions of viscoelastic spheres: Deformability as an effective volume fraction
,”
Phys. Rev. Fluids
3
(
1
),
012301(R)
.
36.
M. E.
Rosti
,
L.
Brandt
, and
A.
Pinelli
, “
Turbulent channel flow over an anisotropic porous wall - drag increase and reduction
,”
J. Fluid Mech.
842
,
381
394
(
2018
).
37.
M. E.
Rosti
,
L.
Cortelezzi
, and
M.
Quadrio
, “
Direct numerical simulation of turbulent channel flow over porous walls
,”
J. Fluid Mech.
784
,
396
442
(
2015
).
38.
M. E.
Rosti
,
S.
Pramanik
,
L.
Brandt
, and
D.
Mitra
, “
The breakdown of Darcy’s law in a soft porous material
,”
Soft Matter
16
,
939
(
2020
).
39.
V.
Shankar
and
V.
Kumaran
, “
Stability of non-parabolic flow in a flexible tube
,”
J. Fluid Mech.
395
,
211
236
(
1999
).
40.
E. S. G.
Shaqfeh
, “
Purely elastic instabilities in viscometric flows
,”
Annu. Rev. Fluid Mech.
28
(
1
),
129
185
(
1996
).
41.
V.
Steinberg
, “
Scaling relations in elastic turbulence
,”
Phys. Rev. Lett.
123
(
23
),
234501
(
2019
).
42.
A. D.
Stroock
,
S. K. W.
Dertinger
,
A.
Ajdari
,
I.
Mezic
,
H. A.
Stone
, and
G. M.
Whitesides
, “
Chaotic mixer for microchannels
,”
Science
295
(
5555
),
647
651
(
2002
).
43.
K.
Sugiyama
,
S.
Ii
,
S.
Takeuchi
,
S.
Takagi
, and
Y.
Matsumoto
, “
A full Eulerian finite difference approach for solving fluid-structure coupling problems
,”
J. Comput. Phys.
230
(
3
),
596
627
(
2011
).
44.
G.
Tryggvason
,
M.
Sussman
, and
M. Y.
Hussaini
, “
Immersed boundary methods for fluid interfaces
,”
Comput. Methods Multiphase Flow
3
,
37
77
(
2007
).
45.
T.
Tsukahara
,
Y.
Seki
,
H.
Kawamura
, and
D.
Tochio
, “
DNS of turbulent channel flow at very low Reynolds numbers
,” in
TSFP Digital Library Online
(
Begel House, Inc.
,
2005
).
46.
M. K. S.
Verma
and
V.
Kumaran
, “
A dynamical instability due to fluid-wall coupling lowers the transition Reynolds number in the flow through a flexible tube
,”
J. Fluid Mech.
705
,
322
347
(
2012
).
47.
M. K. S.
Verma
and
V.
Kumaran
, “
A multifold reduction in the transition Reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall
,”
J. Fluid Mech.
727
,
407
455
(
2013
).
You do not currently have access to this content.