The steady fluid flow that is initiated by a cascade of cylinders, which oscillates harmonically in an unbounded, incompressible, viscous fluid that is otherwise at rest, is investigated, both numerically and experimentally. Finite‐difference techniques are used to obtain the numerical solutions, and the streaming flow results show a reasonable agreement with the experimental data. It is found, both numerically and experimentally, that for large values of the streaming Reynolds number, Rs, the flow is not symmetrical about the axis of the oscillation, i.e., there exists a critical value of Rs, Rs0 say, such that the streaming flows for RsRs0 are unstable and those for Rs<Rs0 are stable. In order to understand this asymmetry, a stability analysis is performed, and the agreement between the theory and the experimental results is encouraging.

1.
H.
Schlichting
, “
Berechnung ebener periodischer grenzschichtströmungen
,”
Phys. Z
3
,
327
(
1932
).
2.
J. M.
Andres
and
U.
Ingard
, “
Acoustic streaming at high (low) Reynolds number
,”
J. Acoust. Soc. Am.
25
,
928
(
1953
).
3.
N.
Riley
, “
Oscillating viscous flows
,”
Mathematika
12
,
161
(
1965
).
4.
N.
Riley
, “
Oscillatory viscous flows, review & extension
,”
J. Instrum. Math. Appl.
3
,
419
(
1965
).
5.
J. T. Stuart, “Unsteady boundary layer” in Laminar Boundary Layers (Oxford University Press, London, 1963).
6.
J. T.
Stuart
, “
Double boundary layers in oscillating viscous flows
,”
J. Fluid Mech.
24
,
673
(
1966
).
7.
C. Y.
Wang
, “
On high frequency oscillatory viscous flows
,”
J. Fluid Mech.
32
,
55
(
1967
).
8.
E. W.
Haddon
and
N.
Riley
, “
The steady streaming induced between oscillating circular cylinders
,”
Q. J. Mech. Appl. Math.
32
,
265
(
1979
).
9.
A.
Bertelsen
,
A.
Svardal
, and
S.
Tjotta
, “
Nonlinear streaming effects associated with oscillating cylinders
,”
J. Fluid Mech.
59
,
493
(
1973
).
10.
A. F.
Bertelsen
, “
An experimental investigation of high Reynolds-number steady streaming generated by oscillating cylinders
,”
J. Fluid Mech.
64
,
589
(
1974
).
11.
S. K.
Kim
and
A. W.
Troesch
, “
Streaming flows generated by high-frequency small-amplitude oscillations of arbitrarily shaped cylinders
,”
Phys. Fluids A
1
,
975
(
1989
).
12.
S.
Tabakova
and
Z.
Zapryanov
, “
On the hydrodynamic interaction of two spheres oscillating in a viscous fluid I. Axisymmetrical case
,”
Z. Angew. Math. Phys.
33
,
344
(
1982
).
13.
Z.
Zapryanov
,
Zh.
Kozhoukharova
, and
A.
Iordanova
, “
On the hydrodynamic interaction of two circular cylinders oscillating in a viscous fluid
,”
Z. Angew. Math. Phys.
39
,
204
220
(
1988
).
14.
D. B.
Ingham
and
B.
Yan
, “
Fluid flows around cascades
,”
Z. Angew. Math. Phys.
44
,
54
(
1993
).
15.
B.
Yan
,
D. B.
Ingham
, and
B. R.
Morton
, “
Streaming flow induced by an oscillating cascade of cylinders
,”
J. Fluid Mech.
252
,
147
(
1993
).
16.
R. M.
Fearn
,
T.
Mullin
, and
K. A.
Cliffe
, “
Nonlinear flow phenomena in a symmetric sudden expansion
,”
J. Fluid Mech.
211
,
595
(
1990
).
17.
W. Tollmien, “General instability criterion of laminar velocity distributions,” Technical Memorandum of Nat. Advances and Communications in Aeronautics, Washington, DC, No. 792, 1936.
18.
P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981).
19.
S. C.
Reddy
,
P. J.
Schmid
, and
D. S.
Henningson
, “
Pseudospectra of the Orr-Sommerfeld operator
,”
SIAM J. Appl. Math.
53
,
15
(
1993
).
20.
S. C.
Reddy
and
D. S.
Henningson
, “
Energy growth in viscous channel flows
,”
J. Fluid Meth.
252
,
209
(
1993
).
21.
R. Betchov and W. O. Criminate, Stability of Parallel Flows (Academic, New York, 1967).
22.
I. J.
Sobey
and
O. G.
Drazin
, “
Bifurcations of two-dimensional channel flows
,”
J. Fluid Mech.
171
,
263
(
1986
).
23.
P. John, “Plane-sudden expansion flows and their stability,” Ph.D. thesis, Imperial College, London, England, 1984.
24.
L. H.
Thomas
, “
The stability of plane Poiseuille flow
,”
Phys. Rev.
91
,
78
(
1953
).
25.
M. R.
Osborne
, “
Numerical methods for hydrodynamic stability problems
,”
SIAM J. Appl. Math.
15
,
539
(
1967
).
26.
J.
Gary
and
R.
Helgason
, “
A matrix method for ordinary differential eigenvalue problems
,”
J. Comput. Phys.
5
,
169
(
1970
).
27.
L. N. Trefethen, “Approximation theory and numerical linear algebra,” in Algorithms for Approximation II, edited by J. Mason and M. Cox (Chapman and Hall, London, 1990).
28.
L. N. Trefethen, “Pseudospectra of matrices,” in Numerical Analysis 1991, edited by D. F. Griffiths and G. A. Watson (Longman, White Plains, NY, 1992).
29.
L. M.
Mack
, “
A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer
,”
J. Fluid Mech.
73
,
497
(
1976
).
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