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A note on the steady streaming induced around a circular cylinder in an oscillatory flow field
Journal of Sound and Vibration (1980) 73(2), 316-320
A NOTE ON THE STEADY STREAMING INDUCED AROUND CYLINDER IN AN OSCILLATORY FLOW FIELD 1. INTRODUCTION
AND FORMULATION
A CIRCULAR
OF THE PROBLEM
A feature of particular interest of the flow induced around a fixed cylinder (of radius a) in a viscous fluid (of kinematic viscosity V) which oscillates as Uo cos wf (U, being a fixed vector, see Figure 1) far from the cylinder, is the persistence of a steady streaming outside
Figure 1. The two dimensional polar co-ordinates referred to in the text.
the oscillatory boundary layer (Stokes layer). The flow is assumed incompressible. The vorticity equation expressed in terms of the non-dimensional stream function P(r, f3,7) is an appropriate basis for a theoretical approach to this two-dimensional problem: i.e., P) ;ozY+E;1 a(P;v’ a(r, 8)
1 =@iV,
(1)
where E = &/wa, M = a J/o/v,r = wt,P = Pph/ Uoa(P,,, being the physical stream function). The dimensionless velocity field (U, V) is related to the stream function by
(v,
v)=(-;$g
where (r, 0) refer to the polar co-ordinates sketched in Figure 1. In terms of a small parameter E <<1 the solution of equation (1) is usually written as
and an inner (Pg’) and outer (W$“) solution for !POcan easily be constructed subject to [l]). The equation for Pi is obtained by inserting P0 into the non-linear term of equation (1) and an inner (Y?’ ) and outer (P:“’ ) solution for Yy, can also be constructed. It turns out (see reference [2]) that outside the Stokes layer the steady part of the flow is governed by the equation
M >>1 (see, for example, reference
@)V2Y:“ 1 alu1,s, , ,‘) 1 r 0022-460X/80/220316
+ 05 $02.00/O
a(r, 0)
= RV41y::,
316 @ 1980 Academic Press Inc. (London) Limited
LE-ITERS (R
TO
THE
317
EDITOR
subject to the boundary conditions
= e*M*),
a&2N+1
1
(3)
imposed on Yft! by the steady part of !I”;“. When the notation
is introduced into equation (2), the following well known asymptotic $(l, 8; R) can be carried out: U,‘J;
R) -_
rLo(L s)+&l(i,
expansion
of
(4)
0)+* . . .
R-too
This expansion, after partial integration and some manipulations are carried out, leads to
a3*o w. a*? :wo a2rjlo
at3=--ae
a[
(5)
a[ ayae’
a3dh a*th ah a2tioag, a2tio a31Lo =_-w. a2r/llw. -__
2+Xj-
a0 ag
-g
arae ae
z+- a(
alao
5---
’
al3
(6)
(higher order terms of $ are not considered), with the boundary conditions, $o(O, 8) = 0,
=e+ 2 [!3+ N=l a( 1 g=.
2Ntl aNo
$*(0,8)=0,
,
[
w.1 a5
=
t=OD
(7)
0,
= NEo bfi2N+‘,
(8)
where of course the flow outside the boundary layer must be calculated before {bN} is known (N = 0, 1,2, . . .). Riley [3] studied the problem stated by equations (5) and (7) and solved for the three first terms in the Blasius series expansion (9) In what follows here a similar expansion of +r(l, 0) is discussed: namely,
N=O
These expansions give the equations
(11)
z
kzl
wN-k(2k
+
l){-+O,k$iiN-k
+
$Lb,kt,bb,N
k}, N = 2, 3,4,.
(12) . ’ t
318
LETTERS
TO THE EDITOR
-f&,N=O 0 LNlL1.N
=
$
,it,
akbN-k{-(pk
+
l)$O.k$tN-k
+2(N+
1)$b,khN-k
-[2(N-k)+1]~;;,,9,,,-,}-~~~~N,
N=
(13)
,
1, 2, 3,. . . i
N
LN=-$+(l-e-‘ &(N+l)e-+N+L)e-‘ , d5 dt
(14)
where the solution fio,o = 1 -em6 (see reference [3], equation (27), in which the notation is different) has been used to obtain expressions (12), (13) and (14). The boundary conditions are $O.N
(0)
=
0,
[&,Nlr=O
=
1,
[tib,Nl~=m
=
0,
(15)
Ijl1.N
(0)
=
0,
[$;.Nlt=O
=
0,
[$‘,,N]~=m
=
1.
(16)
Inspection of equations (12) and (13) reveals that 4 0.N and +i,N have identical general homogeneous solutions for N 2 1. 2.
SOLUTION
An important step in achieving the general solution for an arbitrary term of the Blasius series is to establish the general homogeneous solution of the equation concerned. The general homogeneous solution JIb:&(S) of (1&N([) can be constructed by superposition of terms amJm e-“‘. Some details of the calculations for determining {a,,,} are given in the Appendix. The results of these calculations are
+C
2N+1(2N + 1)2N(2N - 1) . . . (2N - n +2) -nL e 2 I (n!)*(n - 1) n=2
4
@I) $o,N([)=&e-‘ +B
1+(2N+l)[e-‘-
-(6N+4)+[+
2N+l ____ 2
52 e-c
4
+ 2N+l~ n=2
+(2N+
K _(2N+1)2N(2N-1). (
1)((2N)!) ,;, (-l)k-’
. . (2N-n+2)
(n!)*(n
n
((2N+k$o;+k
-
1)
s>
_ l)e-‘2N+k”],
e-“’
(17)
whereK2=-(2N+1)(4N+1)/4andforna3 K
n
=(2N-n+2)(n-2)K n3-n2
_ ” 1
+ (2N + 1)(2N)(2N - 1) . . . (2N-n ((n -1)!)*(n3-n2)
+3)
2(N-n [
n-2
+2)
n(3n -2)(2N-n n2(n-1) -
+2) I *
According to equations (12) and (13) (18)
&k = rl% whenNsl.ForN=Oonefinds *.~~=Aoe-t+Bo[l+~e-C]+Co
(k -2)! -4+~+~~2e~r+k~2(-l)k~1(k~)2(k_l)e~kL
1 .
(19)
LETTERS
TO
THE
EDITOR
The particular solutions can now be found either by variation of the parameters, matching the residual terms R,,,,,U; N) = Lv(GJ’ to the inhomogeneity
e-“0
319 or by (20)
terms of equations (12) and (13) by choosing special values of {a,,}. 3.
NUMERICAL
EXAMPLE
AND
CONCLUSION
The results obtained in the preceding section have been used [4] to establish a four term Blasius series expansion of t,k The results are presented here as a streamline diagram in Figure 2 and velocity profiles for different values of 8 in Figure 3, where also a three term
Figure 2. Streamline diagram based on a four term Blasius series of the stream function rY
Figure 3. Dimensionless velocity (V) profiles for different values of 0 us. stretched boundary layer co-ordinate (C). Full and broken curves based on three and four term Blasius series for the stream function, respectively.
320
LETTERS TO THE EDITOR
Blasius solution is plotted. The curves in this figure indicate a significant difference between a three and a four term Blasius series for 0 2 7r/3. A three term Blasius series seems to give the velocity with reasonable accuracy for (816 7r/3 while by the application of a four term series the range can be extended to 101s (5/12)r.t Department of Mechanics, University of Oslo, Oslo, Norway (Received 7 July
A.
F. BERTELSEN
1980) REFERENCES
1966 Journal of Fluid Mechanics 24, 673-687. Double boundary layers in oscillatory viscous flows. N. RILEY 1967 Journal of the Institute of Mathematics and its Applications 3, 419-434. Oscillatory viscous flows. Review and extension. N. RILEY 1965 Mathematika 12, 161-175. Oscillating viscous flows. A. F. BERTEL~EN 1980 Preprint Series (Applied Mathematics) No. 10, University of Oslo, Norway. A note on the solution of the two-dimensional slip-boundary layer problem. E. W. HADDON and N. RILEY 1979 The Quarterly Journal of Mechanics and Applied
1. J. T. STUART
2. 3. 4. 5.
Mathematics 32,265282.
The steady streaming induced between oscillating circular cylinders.
APPENDIX The general %,,(5;
residual
term
N) = L~h&“’
of a test
solution
a,,&“’
eWnr is
e-“‘I
= a,,,{[m(m
-
l)(m - 2)J”-3 + m(m - l)(l - 3n)f”-‘+
mn(3n - 2)[“-’
+(n2-n3)~“‘]e-“C+[-m(m-1)~“-2+2m(n-1-2N)~”-’ + (-n + 1 + 2N)(n - l)l”‘] e-(“+l)[} = amn{P,,,([;
N) e-“’ + Q,,,(f;
N) e-‘“C1”},
(Al)
and has the following properties:
R1,2N+1([;
RcLI(~; N) = 0,
(A2)
Ro,2nr+I([; N) = -2ao,2nr+lN(2N + l)* e-(2Nt1)C,
(A3)
N) =
a1v2N+1[(2N + l)(6N + 1) -2N(2N
+ l)*[] e-(2N+1)C.
(A4)
Equation (A2) indicates that e-’ is a homogeneous solution for every N. The construction of the other homogeneous solutions consists of choosing numerical values of am.nsuch that
m.n+~Pm.n+~(l; N) + am,nQm.n(z; WI = 0 CCa m for every II. The simplest expressions are obtained when the properties (A3) and (A4) are utilized.
t Note added in proof: For a numerical approach to the problem stated by equations
PI.
(2) and (3), see reference
A NOTE ON THE STEADY STREAMING INDUCED AROUND CYLINDER IN AN OSCILLATORY FLOW FIELD 1. INTRODUCTION
AND FORMULATION
A CIRCULAR
OF THE PROBLEM
A feature of particular interest of the flow induced around a fixed cylinder (of radius a) in a viscous fluid (of kinematic viscosity V) which oscillates as Uo cos wf (U, being a fixed vector, see Figure 1) far from the cylinder, is the persistence of a steady streaming outside
Figure 1. The two dimensional polar co-ordinates referred to in the text.
the oscillatory boundary layer (Stokes layer). The flow is assumed incompressible. The vorticity equation expressed in terms of the non-dimensional stream function P(r, f3,7) is an appropriate basis for a theoretical approach to this two-dimensional problem: i.e., P) ;ozY+E;1 a(P;v’ a(r, 8)
1 =@iV,
(1)
where E = &/wa, M = a J/o/v,r = wt,P = Pph/ Uoa(P,,, being the physical stream function). The dimensionless velocity field (U, V) is related to the stream function by
(v,
v)=(-;$g
where (r, 0) refer to the polar co-ordinates sketched in Figure 1. In terms of a small parameter E <<1 the solution of equation (1) is usually written as
and an inner (Pg’) and outer (W$“) solution for !POcan easily be constructed subject to [l]). The equation for Pi is obtained by inserting P0 into the non-linear term of equation (1) and an inner (Y?’ ) and outer (P:“’ ) solution for Yy, can also be constructed. It turns out (see reference [2]) that outside the Stokes layer the steady part of the flow is governed by the equation
M >>1 (see, for example, reference
@)V2Y:“ 1 alu1,s, , ,‘) 1 r 0022-460X/80/220316
+ 05 $02.00/O
a(r, 0)
= RV41y::,
316 @ 1980 Academic Press Inc. (London) Limited
LE-ITERS (R
TO
THE
317
EDITOR
subject to the boundary conditions
= e*M*),
a&2N+1
1
(3)
imposed on Yft! by the steady part of !I”;“. When the notation
is introduced into equation (2), the following well known asymptotic $(l, 8; R) can be carried out: U,‘J;
R) -_
rLo(L s)+&l(i,
expansion
of
(4)
0)+* . . .
R-too
This expansion, after partial integration and some manipulations are carried out, leads to
a3*o w. a*? :wo a2rjlo
at3=--ae
a[
(5)
a[ ayae’
a3dh a*th ah a2tioag, a2tio a31Lo =_-w. a2r/llw. -__
2+Xj-
a0 ag
-g
arae ae
z+- a(
alao
5---
’
al3
(6)
(higher order terms of $ are not considered), with the boundary conditions, $o(O, 8) = 0,
=e+ 2 [!3+ N=l a( 1 g=.
2Ntl aNo
$*(0,8)=0,
,
[
w.1 a5
=
t=OD
(7)
0,
= NEo bfi2N+‘,
(8)
where of course the flow outside the boundary layer must be calculated before {bN} is known (N = 0, 1,2, . . .). Riley [3] studied the problem stated by equations (5) and (7) and solved for the three first terms in the Blasius series expansion (9) In what follows here a similar expansion of +r(l, 0) is discussed: namely,
N=O
These expansions give the equations
(11)
z
kzl
wN-k(2k
+
l){-+O,k$iiN-k
+
$Lb,kt,bb,N
k}, N = 2, 3,4,.
(12) . ’ t
318
LETTERS
TO THE EDITOR
-f&,N=O 0 LNlL1.N
=
$
,it,
akbN-k{-(pk
+
l)$O.k$tN-k
+2(N+
1)$b,khN-k
-[2(N-k)+1]~;;,,9,,,-,}-~~~~N,
N=
(13)
,
1, 2, 3,. . . i
N
LN=-$+(l-e-‘ &(N+l)e-+N+L)e-‘ , d5 dt
(14)
where the solution fio,o = 1 -em6 (see reference [3], equation (27), in which the notation is different) has been used to obtain expressions (12), (13) and (14). The boundary conditions are $O.N
(0)
=
0,
[&,Nlr=O
=
1,
[tib,Nl~=m
=
0,
(15)
Ijl1.N
(0)
=
0,
[$;.Nlt=O
=
0,
[$‘,,N]~=m
=
1.
(16)
Inspection of equations (12) and (13) reveals that 4 0.N and +i,N have identical general homogeneous solutions for N 2 1. 2.
SOLUTION
An important step in achieving the general solution for an arbitrary term of the Blasius series is to establish the general homogeneous solution of the equation concerned. The general homogeneous solution JIb:&(S) of (1&N([) can be constructed by superposition of terms amJm e-“‘. Some details of the calculations for determining {a,,,} are given in the Appendix. The results of these calculations are
+C
2N+1(2N + 1)2N(2N - 1) . . . (2N - n +2) -nL e 2 I (n!)*(n - 1) n=2
4
@I) $o,N([)=&e-‘ +B
1+(2N+l)[e-‘-
-(6N+4)+[+
2N+l ____ 2
52 e-c
4
+ 2N+l~ n=2
+(2N+
K _(2N+1)2N(2N-1). (
1)((2N)!) ,;, (-l)k-’
. . (2N-n+2)
(n!)*(n
n
((2N+k$o;+k
-
1)
s>
_ l)e-‘2N+k”],
e-“’
(17)
whereK2=-(2N+1)(4N+1)/4andforna3 K
n
=(2N-n+2)(n-2)K n3-n2
_ ” 1
+ (2N + 1)(2N)(2N - 1) . . . (2N-n ((n -1)!)*(n3-n2)
+3)
2(N-n [
n-2
+2)
n(3n -2)(2N-n n2(n-1) -
+2) I *
According to equations (12) and (13) (18)
&k = rl% whenNsl.ForN=Oonefinds *.~~=Aoe-t+Bo[l+~e-C]+Co
(k -2)! -4+~+~~2e~r+k~2(-l)k~1(k~)2(k_l)e~kL
1 .
(19)
LETTERS
TO
THE
EDITOR
The particular solutions can now be found either by variation of the parameters, matching the residual terms R,,,,,U; N) = Lv(GJ’ to the inhomogeneity
e-“0
319 or by (20)
terms of equations (12) and (13) by choosing special values of {a,,}. 3.
NUMERICAL
EXAMPLE
AND
CONCLUSION
The results obtained in the preceding section have been used [4] to establish a four term Blasius series expansion of t,k The results are presented here as a streamline diagram in Figure 2 and velocity profiles for different values of 8 in Figure 3, where also a three term
Figure 2. Streamline diagram based on a four term Blasius series of the stream function rY
Figure 3. Dimensionless velocity (V) profiles for different values of 0 us. stretched boundary layer co-ordinate (C). Full and broken curves based on three and four term Blasius series for the stream function, respectively.
320
LETTERS TO THE EDITOR
Blasius solution is plotted. The curves in this figure indicate a significant difference between a three and a four term Blasius series for 0 2 7r/3. A three term Blasius series seems to give the velocity with reasonable accuracy for (816 7r/3 while by the application of a four term series the range can be extended to 101s (5/12)r.t Department of Mechanics, University of Oslo, Oslo, Norway (Received 7 July
A.
F. BERTELSEN
1980) REFERENCES
1966 Journal of Fluid Mechanics 24, 673-687. Double boundary layers in oscillatory viscous flows. N. RILEY 1967 Journal of the Institute of Mathematics and its Applications 3, 419-434. Oscillatory viscous flows. Review and extension. N. RILEY 1965 Mathematika 12, 161-175. Oscillating viscous flows. A. F. BERTEL~EN 1980 Preprint Series (Applied Mathematics) No. 10, University of Oslo, Norway. A note on the solution of the two-dimensional slip-boundary layer problem. E. W. HADDON and N. RILEY 1979 The Quarterly Journal of Mechanics and Applied
1. J. T. STUART
2. 3. 4. 5.
Mathematics 32,265282.
The steady streaming induced between oscillating circular cylinders.
APPENDIX The general %,,(5;
residual
term
N) = L~h&“’
of a test
solution
a,,&“’
eWnr is
e-“‘I
= a,,,{[m(m
-
l)(m - 2)J”-3 + m(m - l)(l - 3n)f”-‘+
mn(3n - 2)[“-’
+(n2-n3)~“‘]e-“C+[-m(m-1)~“-2+2m(n-1-2N)~”-’ + (-n + 1 + 2N)(n - l)l”‘] e-(“+l)[} = amn{P,,,([;
N) e-“’ + Q,,,(f;
N) e-‘“C1”},
(Al)
and has the following properties:
R1,2N+1([;
RcLI(~; N) = 0,
(A2)
Ro,2nr+I([; N) = -2ao,2nr+lN(2N + l)* e-(2Nt1)C,
(A3)
N) =
a1v2N+1[(2N + l)(6N + 1) -2N(2N
+ l)*[] e-(2N+1)C.
(A4)
Equation (A2) indicates that e-’ is a homogeneous solution for every N. The construction of the other homogeneous solutions consists of choosing numerical values of am.nsuch that
m.n+~Pm.n+~(l; N) + am,nQm.n(z; WI = 0 CCa m for every II. The simplest expressions are obtained when the properties (A3) and (A4) are utilized.
t Note added in proof: For a numerical approach to the problem stated by equations
PI.
(2) and (3), see reference