ht.1.EngngSci.Vol.17.pp.U93 0 Pergamon Prers Ltd., 1979. Printed in Great Britain
UNSTEADY VISCOUS FLOW IN THE VICINITY OF AN AXISYMMETRIC STAGNATION POINT ON A CIRCULAR CYLINDER RAMA SUBBA REDDY GORLAt Departmentof MechanicalEngineering,ClevelandState University,Cleveland,OH 44115,U.S.A. AbstractAn analysis is presentedto investigatethe unsteady fluiddynamic characteristicsof an axisymmetric stagnationflow on a circular cylinder performinga harmonicmotion in its own plane. Different solutions are presentedfor the small and high values of the reducedfrequencyof oscillation.The rangeof Reynolds numbersconsidered was from 0.01 to 100. Numericalsolutions for the velocity functions are presentedand the wall values of the velocity gradientsare tabulated. INTRODUCTION AN EXACTsolution of the NavierStokes equations governing the problem of 2dimensional stagnation flow against a flat plate has been given by Hiemenz[l]. The problem of the axisymmetric stagnation flow against a flat plate was later on studied by Homann[2]. Wang[3] obtained a similarity solution for the axisymmetric stagnation flow on an infinite circular cylinder. The transient response behavior of an axisymmetric stagnation flow on a circular cylinder due to unsteady free stream velocity has been investigated by Gorla[4]. Quite recently, Gorla[5] has studied the steady, laminar, incompressible boundary layer flow at an axisymmetric stagnation point over a circular cylinder moving with a uniform velocity. The present work is undertaken in order to investigate the transient viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder performing a harmonic motion in its own plane. Solutions are constructed for small and high values of the frequency parameter. An estimate has been made of the ratio of the out of phase and the inphase components of the wall shear stress as a function of the reduced frequency parameter. GOVERNING EQUATIONS
Let us consider a laminar, incompressible flow at an axisymmetric stagnation point on a circular cylinder performing a harmonic motion in its own plane, namely, in the zdirection, while the flow at T+ m remains steady. A model of’ the flow with the coordinate system is shown in Fig. 1. The appropriate governing equations within boundary layer approximation are given by: Mass
(1) Momentum
The boundary conditions are given by
u=o
r= a: w =
VW
.
&or
where
w= W,=2Az. tAssociatc Professor. 87
i=t/q
(4)
R. S. R. GORLA
88
t
z
_____
20
1. Fig. 1. Coordinate system and flow development.
COORDINATE Proceeding
with
TRANSFORMATION
AND SOLUTION
the analysis, we define
u = Aaq“*
. f( 7)
w = 2Af’(q) + VW. ein’ . g’(v) k = (Q/2A).
(5)
It may be verified that the continuity equation is automatically satisfied. After substituting the expressions in (5) into eqns (2) and (3), we have qr+f”+Re[l+ff”Cf’)*]=O
(6)
T&”+ g”Relfg”  f’g’  ikg’] = 0.
(7)
The primes above denote differentiation with respect to 77 only. The transformed boundary conditions are given by f( 1) = f’( 1) = 0 and f’(m) = 1 g(1) = 0, g’(1) = 1 and g’(m)= 0.
(8)
It may be noted that the problem corresponding to the steady motion of the cylinder with a uniform velocity, namely, k = 0, has been solved by Gorla[S]. In the oscillating cylinder case, two limiting cases corresponding to k 4 1 and k + 1 will be considered. LOW FREQUENCY
CASE
For this case, we assume that g(q) = go(q) + (ik) . gdv) + W* * e(v) + . . .
(9)
substituting (9) into (7) and collecting like powers of (ik), one obtains vg’t;+g;+Re.(fgI;f’gh}=O
(10)
vg’t+g;+Re*(fg;‘f’glgh}=O
(11)
89
Unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder ~g~+g;‘+Re.(fg’z’f’gig;}=O
(12)
vg’;‘+g’3’+Re.(fg’;f’g;g;}=O
(13)
qgy+gz+Re.(fgif’g;g;}=O
(14)
etc. The boundary conditions are given by go(l)=& gj(l)=O,
g&(l)= 1,
[email protected])=O
g)(l) =O, gj(O”)= 0 for
_i* 1.
(15)
Equations (10)(14) are solved in conjunction with (6), on the computer, by means of the fourth order Runge Kutta numerical procedure. In the above equations, Re was chosen to be a prescribable parameter and was varied from 0.01 to 100. The numerical results obtained for the velocity functions f’(n), g;(n), g;(n), g;(n), g;(n) and g;(n) are shown in Figs. 26 for a range of Re values. The values of f”(l), g;(q), g;(q), g;(q) and g!(n) for the same range of Reynolds numbers have been tabulated in Table 1. In many practical applications, it is usually the surface characteristics such as the local wall shear that are of importance, The evaluation of the wall shear stress requires only the information contained in Table 1. The local wall shear stress can be written as
Fig. 2. Distribution off’, g;, g;, gi. g; and g; for Re = 0.01. Table 1. Values of Re“*f”(l) and Re“*gxl) for various values of Re Re 0.01
0.10 1.00 10.00 100.00
Re“*f”(l) 3.155182 1.946369 1.484185 1.316427 1.259642
Re“*gXl)
Re“‘g#l)
Re“*ggl)
2.708642 1.517035  1.065cr77 0.8%923 0.839077
0.509410 0.504376 0.490761 0.491026 0.492325
0.0768662 0.0946052 0.0946999 0.0942794 0.0944939 
Re
“‘g;( 1)
0.0173569 0.0338370 0.0318336 0.0312977 0.031177~ _..___
Re“*gXl) o.OO4oOs43 0.0132088 0.0123787 0.011%05 0.0117816 ._._. ____ ,.__
R. S. R. GORLA
Fig, 3. Distribution off’, g& g;, gi, g; and g; for Re = 0.1
7
6
5
4
\ ;;\ 2 0
0.5
1.0
Fig. 4. Distribution off’, g;, g;, g& g; and gi for Re = 1.0.
1.5
Unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder 30
Re
= 10
2.6I
2.6
24
Fig. 5. Distribution off’, g& g;, g;, g$ and g; for Re = 10.
1.50
Re = 100
1.45.
1.40.
1.35
1.30 1) 1.25
1.20
1.15
1.10
1.05
1.0 0
0.5
1.0
Fig. 6. Distribution off’, g& g;. gi, g; and g4 for Re = 100.
1.5
91
R.S.R.GORLA
92
The shear stress at the wall due to the gflow is then given by 2/.Lvw Tw=  a . eln’ [g;( 1) + (ik) . g;( 1) + (i/c)’  gI( 1) + * * 1. The ratio of the out of phase shearing stress 7wito the inphase component T,,,,is given, to the first order in k, by (17)
. HIGH FREQUENCYCASE
For this case, we assume that
(18)
g’= exp (I’Sdn).
Substituting the above expression into eqn (7), we have q[S’+S’]+S+ReWf’ik]=O we
(19)
now set
S(V)= Ok)“*e S,,(q) + S,(q) + (ik)“* * S*(v) + (ik)’ *$(v) + ( ik)3’2*S4(q) + *. . .
(20)
That is S(T)
=

(gz

i[(t)“*
* So(v)
So(q)
+
S,(7j)
(&)I”
(&)“*
*
* Sz(T7)
+;
* (&)“*
* s4.
S*(v)+ G>. &(Tj)i
($)“*’
* *
S4(71)
”
*I*
(21)
Substituting (20) into eqn (19) and upon equating like powers of (ik)“*, we have So(T/)=
($)I’*
Sdd =  &[;+Ref] {; Ref’  Ref. S,  qS:> l/2 S3(7I)
=
. [(

1 + Ref)s2 + s;v + 2sls2n1
I/2
>
.
[(l
+
Ref)s3+ s;q + 2s1s3q + &I
It may be verified that the boundary condition g’(1) = 1 is satisfied by the choice of limits in the integral in eqn (18). For this case, the ratio of the out of phase shearing stress r,,+to the inphase component rW, is given by 1 ._._=__[ll~+Re.~(l))] 32 VRe k +Re 64 4
(22)
Unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder
93
I I
K Approrimotlon
+Small I
Lar90
K Approximation
OY 0
I
2
3
4
5
6
7
6
9
IO
II
K Fig. 7. r,Jr,, vs K for Re = 100.
In Fig. 7, the limiting cases for small and large k, eqns (17) and (22) are plotted and a smooth curve joining these asymptotes is presented for Re = 100. AcknowledgementThe author is thankful to Dr. Vincent H. Larson, Professor and Chairman of the Department of Mechanical Engineering at the Cleveland State University for his interest in this work. A portion of this work has been completed at the NASA Langley Research Center in Hampton, Virginia while the author worked there as a summer research fellow. NOMENCLATURE constant used in eqn (4) radius of cylinder velocity protile functions reduced frequency parameter (fW2A) pressure Reynolds number (Aa*/2v) coordinate normal to the cylindrical surface temperature velocity component in rdirection velocity of the wall velocity component in zdirection coordinate parallel 13 the wall dimensionless coordinate dynamic viscosity kinematic viscosity fluid density frequency of oscillation Subscripts w conditions at the wall
m conditions far away from the wall REFERENCES [l] K. HIEMRNZ, LXnglersJ. 326,321 (1911). [2] F. HOMANN, 2. Angew. Moth. Med. 16, 153(1936). [3] W. WANG, Q. Appl. Math. 32, 207 (1974). [4] R. S. R. GORLA, ht. J. Engng Sci. 16,493 (1978). [5] R. S. R. GORLA, ht. L Engng Sci. 16, 397 (1978). (Received 24 March 1978)