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Rotationally symmetric fluctuating flow about a rotating disk
hf. J. Engng Sci. Vol. 22, No. 2, pp. 16s173, Printed in Great Britain.
1984 0
002s7225/84 1984 Pergamon
$3.00 + 03 Press Ltd.
ROTATIONALLY SYMMETRIC FLUCTUATING FLOW ABOUT A ROTATING DISK ARUN KUMAR VERMA Department of Mathematics, Indian Institute of Technology, Kharagpur, India Abstract-Superposed
mean flow due to fluid performing torsional oscillations at large from a rotating disk is analysed. For low frequency of oscillations an analytical-numerical is obtained for small values of the amplitude parameter, which agrees fairly well with the numerical results in the limit of zero frequency. Method of multiple scales is employed a solution for high frequency range and for finite values of the amplitude parameter.
distances solution available to derive
1. INTRODUCTION
motion induced by a rotating disk has been considered by a number of authors and is by now well understood. Rogers and Lance [ l] have considered the physical situation in which the fluid away from the disk is also rotating with an angular velocity which differs from that of the rotating disk but is in the same sense. A slightly different case, in which the fluid in the far field is rotating in the sense opposite to that of the disk, has been discussed by Ockendon[2], Evans[3] and more recently by Bodonyi[4]. The specific purpose of present paper is to analyse the fluid motion superposed by imposed torsional oscillations in the far field over that induced by a rotating disk. The resulting fluid motion fluctuates with a non-zero mean. The flow characteristics of such a fluid motion are studied separately for low and high frequencies of the imposed periodic motion. The Fettis-Benton method as employed by Chawla[5] enables us to obtain the low frequency solution as power series expansion. The high frequency solution, on the other hand, is analysed by introducing an additional length scale. We derive specific expressions for the radial and azimuthal components of the skin friction which are in good agreement with the available numerical results of Rogers and Lance[ l] and Evans[3] in the limit of zero angular frequency (corresponding to slow steady rotation in the far region).
THE FLUID
2. MATHEMATICAL
FORMULATION
We take the disk to be of infinite radius and coincident with the plane z = 0. The region z > 0 is occupied by a viscous incompressible fluid of semi-infinite extent. The disk is rotating with angular velocity R, in contact with the fluid which is performing torsional oscillations of angular frequency o and amplitude ER in the far off region. We take Cartesian coordinates (x, y, z) in a non-rotating frame of reference, with z as the time. In consistency with the continuity equation, it is appropriate to assume the velocity components (u, V, w) and the pressure p in the form u=R
, x(G+mm)--g 2x
-;f$-y(G+acosot)
w = (vQ$H(~, t),p = p@22(i(x2+ y2)c2 cos2 at - xyc sin at} - pvRP([, t),
(2.la) (2.lb)
with 1’2.z,t =Rz,a
=ojR,
(2.2)
where cr is a non-dimensional frequency parameter. In (2.1), flow components H and G are functions of the dimensionless variables [ and t. Introducing the above expressions in the Navier-Stokes’ equations and equating the terms proportional to x and the terms proportional to y separately to zero, we arrive at the following set of differential equations H,, = Hrir - HH,, + ;Ht2 - 2G2 - 4G cos at, G, = G,, - HG, + H(G -I- df, cm ot, 165
(2.3a)
166
A. K. VERMA
where a suffix denotes a partial derivative. The appropriate situation under consideration are
boundary conditions for the
H(O, t) = 0, H,(O,t) = 0, G(0,t) = 1 -E cos at, H,(cq t) = 0, G(co, t) = 0. (2.3b) We develop two separate solutions of the governing eqns (2.3) valid for small and large values of the frequency parameter cr. 3. LOW
FREQUENCY
SOLUTIONS
For 1~1< 1, we may develop the low frequency solution of the differential system (2.3) in the form H([, t) = Ho(<) + L{I?([) eio’+ H’*(l) e-‘“‘} + c2{H”([) + H2(c) e2ior+ H2*(c) e-2iuf}+ 0(c3), G(L t) =
(3.la)
Go(l) + c{G’(() eiat+ G'*(c) eCor) + t’{G”(() + G2(l) eziar+ G2*(t') e-2io’} + 0(c3),
(3.lb)
where an asterisk denotes the complex conjugate of the function under it. Substituting (3.1) in (2.3) and equating same order of 6, we obtain the following set of equations for (HO, GO), (H’, G’), (H”, G”), (H2, G’): H& - H”H$ + +H;’ - 2G02 = 0, G;[ - HOG; + H;G” = 0, Ho(O)= 0 =
H;(O), GO(O) = 1, H@)
Hi,, - jaHi - H’H:,
= 0 = Go(~),
+ HFH[ - HRH’ - 4G”G1 = 2G”,
(3.2a) (3.2b) (3.3a)
Gk - iaG’ - HoGi + HfG’ - GFH’ + GoHi = -iHF,
(3.3b)
H’(0) = 0 = Hi(O), G’(0) = -;, H;(m)
(3.3c)
H;,, - HOG”,,+ H;H;
- HRH” - 4GOG” = H’H:; + H’*H$
Gf, - HOG; + H;G” - G;H” + G”Hf = H’Gt* + H’*G;
- HiHi’+
- G’H;*
H”(O) = 0 = H;(O), G”(O) = 0, H;(m)
= 0 = G’(m),
4G’G’* + 2(G’ f G’*),
- G’*H;
- f(H;* _t Hi),
= 0 = GS(cn),
H& - 2iaHf: - H”Hfc + HFHt - H&H2 - 4G”G2 = H’H!, - iHi + 2Gi2 + 2G1,
(3.4a)
(3.4b) (3.4c) (3Sa)
Gtc - 2iaG’ - HOG: + HFG2 - G;H2 + GOH; = H’G; _ H;G I_ #I,
(3Sb)
H2(0) = 0 = H;(O), G2(0) = 0, H;(co) = 0 = G2(~).
(3.5c)
The steady state problem characterized by the set (3.2) has been solved by von K&r-man[6], Cochran [7], Fettis [8] and Benton [9]. The superimposed oscillatory and mean flow given by (3.3~(3.5) evidently depends on Ho and Go. For the subsequent analysis of (H’, G’), etc., we shall make use of the steady solution obtained by Benton[9]. For small values of a, we look for a solution of (3.3) in the form H’ = ‘f (ia)“HA, G’ = 2 (ia)“GA. n=O
IT=0
(3.6)
167
Symmetric flow about a rotating disk
Substituting (3.6) in (3.3), we obtain HA,,, - H’H;,,
+ H;H& - HicH; - 4G’G; = 2G”,
GA,,- HOG& + H;G; - G;H:, . + G”H& = -fH;, H;(O) = 0 = H&(O), G;(O) = -;, - H’H;,,
Hh,
+ H;H;,
H&(m) = 0 = G;(m),
- HicH; - 4G’G; = H&,
G;,, - HOG!, + H;G; - G;H; + G”H;[ = GA, H;(O) = 0 = H;,(O), G;(O) = 0, Hf,(co) = 0 = G:(m),
(3.7a) (3.7b) (3.7c)
(3.8a) (3.8b) (3.8~)
and so on. In order to solve (3.7) to (3.8) which gives the quasi-steady flow corresponding to (r = 0, we write the solution in the form H:, = C + CD + ;(H” + CH;), G; = E +
(3.9)
where C, D, E and F are given by D,, - HOD,, + H;D, - H;[D - 4G°F = 0, F,, - H’F, + HfF - GFD + G”Dc = 0, C,,, - H°CcI + H;C, - H$
- 4G”E = 2G” - 3D,, + 2H”D, - HFD,
EcI - HOE, + H;E - G;C + GOC, = -;H;
- 2F, + H°F - GOD,
(3.10)
(3.1 la) (3.11b)
with C(0) = 0, C,(O) + D(0) = 0, E(0) + c1 = -f,
(3.1 lc)
and a is a constant to be determined. Following Benton[9] and Chawla[5], we assume solution of the above equations as power series in the form
eecni, D =
Go = c2~an
c2cd,,
eecnz, F =
c3Cfn e-cn(,
(3.12b)
where c = 0.88447 and a, and b,, are tabulated in Benton[9], and d,,, fn, c,and e,, are given by the recursion relations n-l n’(n - 1)dn + C {(n’ - 3nj + 3j2)d,b,_j + 4fja,-j} = 0, (3.13a) j=l n-1 n(t~
-
lyn
+
C
(2j - n)(ajd,-j
+f,b,-j)
= 0,
(3.13b)
j=l n-1
~{(r~~-3nj+3j~)c,b~_~+4e~a~_~J=n(3n-2)d~--~
n’(n--l)c,+
j=l n-1 +
j(24b,-j
C
(3.14a)
-bjd,-j),
j=l n-l t~(n
-
l)e,, + 1 (2j - n)(ajc,_j
+ ejb,_j) = (2n - 1% + $bn
j=l n-l +
C
V;b,-j
(3.14b)
-Q,-,).
j=l (n
=
1, 2, 3, . . . .)
A. K. VERMA
168
The coefficients c,, e, in (3.14) are expressible in terms of the leading coefficients c,, e,. Setting n = 1 in (3.14) we obtain d, andf; and rest of the coefficients d,,ji can be generated from (3.13). c, and e, are so chosen as to satisfy the boundary conditions Cc, = 0 and Cnc,, - Cd, = 0. The remaining boundary condition Cc’e, + a = -i is then used to evaluate the constant c(, introduced in (3.9). A sixty-term power series expansion yields c, = -4.41770, e, = 0.56731. a = 1.27797.
(3.15)
The solution of the second set of eqns (3.8) can be written in the form Hi = K + [L + [‘A4 + $I0
+ @Z;), Gf = P + [Q + [*R + ;(2G” + CGF). (3.16)
Adopting the same procedure as explained above, we write cm,, pn, cy,, c*r,
and obtain the leading terms and the constant /I introduced
>
eecni
(3.17)
in (3.16) as
k, =.4.05402,p, = -2.74082, /I = 0.24417.
(3.18)
The low frequency first harmonic solution is not carried further from this stage. We now solve the differential set (3.4) governing the timemean induced flow. Following the above method of analysis, the zeroth order solution in (Tmay be written in the form H;(i) = S + CT + C*U + ;(HO + [Hi), GX) = X + [Y + 1*Z + $(2G” + CG;),
(3.19)
in which y is an unknown constant and is to be evaluated from the corresponding boundary condition. Now, writing (X T, U, X K Z) = f (cs,, c*t,, c3u,, c*x,, c3y,, c4z,) epcni, n=I
(3.20)
s, = 2.92469, x, = -2.43868, y = - 12.44341.
(3.21)
we find that
It can be easily shown that the quasi-steady terms of the second harmonic solution of the set (3.5) are given by %,(i) = $X(i),
G:(i) = fG;(i).
(3.22)
For sixty-term power series, the above solution leads to the following expressions for the radial and azimuthal components of the skin friction on the disk - H,,(O, t) = 1.02045 + t (0.21879 cos at + 0.95834~ sin at) -t2(0.79582
+ 0.79582 cos 2ot),
(3.23a)
-G&O, t) = 0.61591 - ~(O.O5164cos at + 1.057230 sin at) - ~~(0.52204 + 0.52204 cos 2ot).
(3.23b)
Also the inflow velocity at infinity is -H(co,
t) =
0.8844711 +~(1,27797cosot
- 0.244170 sin at)
-~‘(6.22171 + 6.22171 cos 2ot)).
(3.23~)
Symmetric flow about a rotating disk
169
6.0-
5.0 -
0.0
20
6.0
80
100
120
14.o
Fig. I. Variation of the superposed (coefficient of cz) time-mean flow components with 4.
We notice that .the imposed torsional oscillations in the far field reduce the components of the mean tangential shear at the rotating disk. The time-mean axial inflow in the far region is also decreased. The components of the superposed mean flow Hf,, G; and H& are shown in Fig. 1 which clearly exhibit that the boundary layer corresponding to the induced flow are much thicker than that of the basic von-k&man flow (see Benton[9]). The imposed periodic fluid motion not only increases the mean angular momentum within the boundary layer but also introduces steady injection out of the boundary layer which results in the decrease of the net mean axial inflow in the far field. The superposed pumping of the fluid out of the boundary layer is a direct consequence of the thickening of the boundary layer, in which the fluid particle just outside the basic K&man layer are undergoing angular momentum, albeit small. The terms of orders c4 are likely to introduce spatial oscillations in the superposed mean flow. From the expressions (3.23) we can derive formulae for the skin friction corresponding to the steady fluid flow, when the disk is rotating with angular velocity CJ and the fluid at infinity is slowly rotating with angular velocity ~$2in the same or in the opposite sense. We put u = 0 in (3.23) and get - H,[(O, t) = 1.02045 +0.21879e - 1.59164~’ + O(t’),
(3.24a)
-G,(O, t) = 0.61591 - 0.05164~ - 1.04407~‘+ 0(c3).
(3.24b)
The above results are compared with the results of Evans[3] (for e < 0) and with those of Rogers and Lance[l] (for c > 0), in Table 1. The agreement seems to be fairly good.
4. HIGH
FREQUENCY
SOLUTIONS
At the highest frequencies, the skin-effect results in the establishment of the Stokes layer over the rotating disk. As such the fluid flow varies on two distinct length scales. In order to expose the effect of high frequency oscillations at large distances over the whole of the fluid regime, we employ the method multiple scales. This method is effective, though
170
A. K. VERMA Table 1. L
-____
f&,(0,1)
G,(O, 1)
Relation (3.24a) E/R and L Relation (3.24b) -0.16 -0.15 -0.1 0.1 0.2
-0.98257 -0.98613 - 1.00155 - 1.02343 - 1.01049
- 0.94666 -0.95255 - 0.98261 - 1.02679 - 1.00374
-0.60667 -0.60804 -0.61327 -0.60811 -0.58987
E/R and L
-0.57767 -0.58761 -0.60825 -0.60155 -0.57208
complicated, largely because the induced fluctuating flow is rendered devoid of secular terms. Moreover, the solution thus obtained is valid for 6 of O(1). The inner Stokes layer is of O(o - l/*) times as thick as the outer von-K&man layer. We, therefore, consider velocity components as functions of the three independent variable I, c and 5 = a”*c; the 5 -dependence is so chosen as to suppress all secular terms in the perturbation process. The problem is now to find out H([, {, t) and G(&‘,e, t) where H,, - H,,, = 6 { 3H 551- H,, - HH,, + $H;} + 6*(3H,,, - 2HHt, + H,H,) + 63{Hcic - HH,, + ;H; - 2G2 - 4cG cos T},
G, - G,, = 6(2G,, - HG, + H,G + cH, cos T> f 6*{Gci - HG, + H,G + cH, cos T},
(4. la)
(4.lb)
H(O, 0, T) = 0, H,(O, 0, T) + 6H,(O, 0, T) = 0, G(0, 0, T) = 1 - ;(ewir + eir>, (4. lc) He(co, 00, T) + GH,(oo, co, T) = 0, G(co, 03, T) = 0,
(4.ld)
where 6 = o-~/*, T = ot and L N 0( 1). In order to obtain a uniformly valid solution of the set (4.1) we follow closely Benney’s[lO] method of analysis and write H = f
PH(“@‘, {, T) and G = f
PI=0
P’G@)(~,5, T).
?I=0
(4.2)
The solution of (4.1) valid upto second order in 6, is Ho = ag(i),
Go = b&l)
+ gj”)(i, 5) eiT + grco)(5, [) ewiT,
g{O)([, 0 = b(y)(c) e=(’ +l~~IJz,
(4.3a) (4.3b) (4.4a) (4.4b) (4Sa) (4Sb) (4.5c)
The arbitrary functions a$)(i), bf)([) occuring in the above solution, are given by the following system of boundary value problems (for details of the method of derivation of
171
Symmetric flow about a rotating disk
this system see Benney [ lo] and Chawla[ 1l] a&
- a$)a$‘& + iagf - 2bg2 = 0, b& - aRb& + a&b8 a&‘(O)= 0, a&(O) = 0, b#(O) = 1, a&(m)
= 0,
= 0 = b$$(oo),
2b\q’,- a#b\:) = 0, b\;)(O)= -c/2,
(1 + i)(26#
- a$$b$\))= fi(b
b&c - agb&
+ a&b8
b\!)(O)= 0,
+ b#a& = 0,
(4.8)
(4.9b)
a&O) = 0, a&(O) = -~*/2, b@(O) = 0, a&(m) = 0 = b&)(a),
(4.9c)
a$:\ = 2kb# , a\:)(O)= 0 7
(4.10)
2a& - a#a$q = 4ib$fb\:)‘, a@(O) = 0,
(4.11)
(1 + i)(2b& - a$$b.$) - a@b$) + bga$t)) = $(b\$,
- agb#
+ a&b\‘&
b\:‘(O)= 0, @(2a\-‘,:
(4.6b) (4.7)
#Cc- a$jb& + a&b@,
- b&a8
(4.6a)
(4.12a) (4.12b)
- aga\?) = (1 + i)(3a$& - 2a$ja\:\ + a&a\‘,‘) + 4b$$bt’,), a\:)(O)= (1 + i)E /Jz
(4.13a) (4.13b)
2a& - a#a$? = 0, a!&)(O)= 7.
(4.14)
The flow field characterized by a&$and 68, governed by the eqns (4.6), is the classical von-K&ran flow due to a rotating disk. The time harmonic terms in (4.3~(4.5) correspond to the shear wave solution whereas (4.7), (4.8), (4.10)-(4.14) give minor modifications in the thickness of the oscillatory Stokes layer due. to their interaction with the steady flow. The system (4.9) yields the exact solution in the form (4.15a)
(4.15b)
The system of eqns (4.6) as a whole represents the solution in the outer layer; proper matching with the inner solution is provided by the boundary conditions. In addition, it gives the complete interaction between the steady and unsteady parts of the field functions. The high frequency solution obtained above leads to the following radial and azimuthal components of the coefficients of shearing stress at the disk -H&O, T) = 1.02045 - c-“*{0.7071 lo’+ 1.41421c(cos T + sin T) + 0.37868c2(cos
2T + sin 29)
+ a-‘(0.51808~~
+ 1.84773~ sin T) + 0(aw3/*), -Gr(O, T) = a1’*{0.70711~(sin T -cos
(4.16a)
T)) + 0.61591
+ a-‘(0.27881~~ + 0.63777~ sin T) + O(O-~/~),
(4.16b)
172
A. K. VERMA 0.6
/ 0.5
I ‘1 f
/-\
1 \ \
\ \
\ ’
Fig. 2. Variation of the superposed (coefficient of D - ‘~3 time-mean flow components H”(axial), G” (azimuthal) and HF (radial) [for (a) CT= 3, (b) u = 5, (c) u = 91, with [.
whereas the axial inflow at infinity is given by -H(co,
T) = 0.88447 + a-‘(0.13346~~ + 6.5011~ sin T) + O(O-~‘~).
(4.17)
The non-dimensional composite components of the mean velocity, uniformly valid for the entire flow regime, are
-a&
G’“‘(~) = b&c) + c
-‘c2
>
(4.18a)
.
(4.18~)
+
The last two terms in the expressions for HP)([) are contributed from the solution of O(a-3’2). The time mean flow (proportional to a-‘~~), represented by H”(C), H’;(i) and Cm(c) is shown in Fig. 2. The broken curves, representing the radial flow, evidently exhibit the spatial oscillations in the superposed flow and vary with the frequency parameter. The spatial oscillations are more pronounced as g is increased. The spatial oscillations which are confined to the inner layer result from the Ekman effect of the superposed mean
Symmetric flow about a rotating disk
113
rotation just outside the Stokes layer. The skin effect is responsible for additional mean torque at the disk, whereas the radial component of the skin friction is decreased. We notice that to O(o-‘E’) the boundary layer sucks in additional fluid at the edge of the outer layer. The contribution to the axial flow from the last two terms of (4.1Xb) (which is in the opposite direction to that shown in Fig. 2) is O(o -3’2t:2). Acknowledgement-me author wishes to thank Dr. S. S. Chawla for the ~ntinuing many valuable discussions throughout the investigation. REFERENCES [l] M. H. ROGERS and G. N. LANCE, J. Fluid Mech. 7, 617-631 (1960). [2] H. OCKENDON, Q. J. A4ec!r. Appl. Math. XXV, 291-301 (1972). [3] D. S. EVANS, Q. J. Me&. Appl. Mul. XXII, 467-485 (1969). (41 R. J. BODONYI, J. Fluid Mech. 67, 657-666 (1975). [S] S. S. CHAWLA, Acta Me~hunica 25, 207-219 (1977). [6] Th. VON KARMAN, Z.A.M.M. 1, 233-252 (1921). 171W. G. COCHRAN, Proc. Cumb. Phil. Sot. 30, 365-375 (1934). i8i H. E. FETTIS, 4th Midwest Conf. Fluid Mech. Purdue, pp. 93-114 (1955). 191 E. R. BENTON. J. Fluid Mech. 24. 781-800 (1966). ;iOj D. J. BENNEY,‘J. Fluid Mech. 18, 385-391 (1964). .I l] S. S. CHAWLA, Q. J. Mech. Appl. Muth. XXVI, 355-370 (1973). (Received 26 January 1983).
guidance and for the
1984 0
002s7225/84 1984 Pergamon
$3.00 + 03 Press Ltd.
ROTATIONALLY SYMMETRIC FLUCTUATING FLOW ABOUT A ROTATING DISK ARUN KUMAR VERMA Department of Mathematics, Indian Institute of Technology, Kharagpur, India Abstract-Superposed
mean flow due to fluid performing torsional oscillations at large from a rotating disk is analysed. For low frequency of oscillations an analytical-numerical is obtained for small values of the amplitude parameter, which agrees fairly well with the numerical results in the limit of zero frequency. Method of multiple scales is employed a solution for high frequency range and for finite values of the amplitude parameter.
distances solution available to derive
1. INTRODUCTION
motion induced by a rotating disk has been considered by a number of authors and is by now well understood. Rogers and Lance [ l] have considered the physical situation in which the fluid away from the disk is also rotating with an angular velocity which differs from that of the rotating disk but is in the same sense. A slightly different case, in which the fluid in the far field is rotating in the sense opposite to that of the disk, has been discussed by Ockendon[2], Evans[3] and more recently by Bodonyi[4]. The specific purpose of present paper is to analyse the fluid motion superposed by imposed torsional oscillations in the far field over that induced by a rotating disk. The resulting fluid motion fluctuates with a non-zero mean. The flow characteristics of such a fluid motion are studied separately for low and high frequencies of the imposed periodic motion. The Fettis-Benton method as employed by Chawla[5] enables us to obtain the low frequency solution as power series expansion. The high frequency solution, on the other hand, is analysed by introducing an additional length scale. We derive specific expressions for the radial and azimuthal components of the skin friction which are in good agreement with the available numerical results of Rogers and Lance[ l] and Evans[3] in the limit of zero angular frequency (corresponding to slow steady rotation in the far region).
THE FLUID
2. MATHEMATICAL
FORMULATION
We take the disk to be of infinite radius and coincident with the plane z = 0. The region z > 0 is occupied by a viscous incompressible fluid of semi-infinite extent. The disk is rotating with angular velocity R, in contact with the fluid which is performing torsional oscillations of angular frequency o and amplitude ER in the far off region. We take Cartesian coordinates (x, y, z) in a non-rotating frame of reference, with z as the time. In consistency with the continuity equation, it is appropriate to assume the velocity components (u, V, w) and the pressure p in the form u=R
, x(G+mm)--g 2x
-;f$-y(G+acosot)
w = (vQ$H(~, t),p = p@22(i(x2+ y2)c2 cos2 at - xyc sin at} - pvRP([, t),
(2.la) (2.lb)
with 1’2.z,t =Rz,a
=ojR,
(2.2)
where cr is a non-dimensional frequency parameter. In (2.1), flow components H and G are functions of the dimensionless variables [ and t. Introducing the above expressions in the Navier-Stokes’ equations and equating the terms proportional to x and the terms proportional to y separately to zero, we arrive at the following set of differential equations H,, = Hrir - HH,, + ;Ht2 - 2G2 - 4G cos at, G, = G,, - HG, + H(G -I- df, cm ot, 165
(2.3a)
166
A. K. VERMA
where a suffix denotes a partial derivative. The appropriate situation under consideration are
boundary conditions for the
H(O, t) = 0, H,(O,t) = 0, G(0,t) = 1 -E cos at, H,(cq t) = 0, G(co, t) = 0. (2.3b) We develop two separate solutions of the governing eqns (2.3) valid for small and large values of the frequency parameter cr. 3. LOW
FREQUENCY
SOLUTIONS
For 1~1< 1, we may develop the low frequency solution of the differential system (2.3) in the form H([, t) = Ho(<) + L{I?([) eio’+ H’*(l) e-‘“‘} + c2{H”([) + H2(c) e2ior+ H2*(c) e-2iuf}+ 0(c3), G(L t) =
(3.la)
Go(l) + c{G’(() eiat+ G'*(c) eCor) + t’{G”(() + G2(l) eziar+ G2*(t') e-2io’} + 0(c3),
(3.lb)
where an asterisk denotes the complex conjugate of the function under it. Substituting (3.1) in (2.3) and equating same order of 6, we obtain the following set of equations for (HO, GO), (H’, G’), (H”, G”), (H2, G’): H& - H”H$ + +H;’ - 2G02 = 0, G;[ - HOG; + H;G” = 0, Ho(O)= 0 =
H;(O), GO(O) = 1, H@)
Hi,, - jaHi - H’H:,
= 0 = Go(~),
+ HFH[ - HRH’ - 4G”G1 = 2G”,
(3.2a) (3.2b) (3.3a)
Gk - iaG’ - HoGi + HfG’ - GFH’ + GoHi = -iHF,
(3.3b)
H’(0) = 0 = Hi(O), G’(0) = -;, H;(m)
(3.3c)
H;,, - HOG”,,+ H;H;
- HRH” - 4GOG” = H’H:; + H’*H$
Gf, - HOG; + H;G” - G;H” + G”Hf = H’Gt* + H’*G;
- HiHi’+
- G’H;*
H”(O) = 0 = H;(O), G”(O) = 0, H;(m)
= 0 = G’(m),
4G’G’* + 2(G’ f G’*),
- G’*H;
- f(H;* _t Hi),
= 0 = GS(cn),
H& - 2iaHf: - H”Hfc + HFHt - H&H2 - 4G”G2 = H’H!, - iHi + 2Gi2 + 2G1,
(3.4a)
(3.4b) (3.4c) (3Sa)
Gtc - 2iaG’ - HOG: + HFG2 - G;H2 + GOH; = H’G; _ H;G I_ #I,
(3Sb)
H2(0) = 0 = H;(O), G2(0) = 0, H;(co) = 0 = G2(~).
(3.5c)
The steady state problem characterized by the set (3.2) has been solved by von K&r-man[6], Cochran [7], Fettis [8] and Benton [9]. The superimposed oscillatory and mean flow given by (3.3~(3.5) evidently depends on Ho and Go. For the subsequent analysis of (H’, G’), etc., we shall make use of the steady solution obtained by Benton[9]. For small values of a, we look for a solution of (3.3) in the form H’ = ‘f (ia)“HA, G’ = 2 (ia)“GA. n=O
IT=0
(3.6)
167
Symmetric flow about a rotating disk
Substituting (3.6) in (3.3), we obtain HA,,, - H’H;,,
+ H;H& - HicH; - 4G’G; = 2G”,
GA,,- HOG& + H;G; - G;H:, . + G”H& = -fH;, H;(O) = 0 = H&(O), G;(O) = -;, - H’H;,,
Hh,
+ H;H;,
H&(m) = 0 = G;(m),
- HicH; - 4G’G; = H&,
G;,, - HOG!, + H;G; - G;H; + G”H;[ = GA, H;(O) = 0 = H;,(O), G;(O) = 0, Hf,(co) = 0 = G:(m),
(3.7a) (3.7b) (3.7c)
(3.8a) (3.8b) (3.8~)
and so on. In order to solve (3.7) to (3.8) which gives the quasi-steady flow corresponding to (r = 0, we write the solution in the form H:, = C + CD + ;(H” + CH;), G; = E +
(3.9)
where C, D, E and F are given by D,, - HOD,, + H;D, - H;[D - 4G°F = 0, F,, - H’F, + HfF - GFD + G”Dc = 0, C,,, - H°CcI + H;C, - H$
- 4G”E = 2G” - 3D,, + 2H”D, - HFD,
EcI - HOE, + H;E - G;C + GOC, = -;H;
- 2F, + H°F - GOD,
(3.10)
(3.1 la) (3.11b)
with C(0) = 0, C,(O) + D(0) = 0, E(0) + c1 = -f,
(3.1 lc)
and a is a constant to be determined. Following Benton[9] and Chawla[5], we assume solution of the above equations as power series in the form
eecni, D =
Go = c2~an
c2cd,,
eecnz, F =
c3Cfn e-cn(,
(3.12b)
where c = 0.88447 and a, and b,, are tabulated in Benton[9], and d,,, fn, c,and e,, are given by the recursion relations n-l n’(n - 1)dn + C {(n’ - 3nj + 3j2)d,b,_j + 4fja,-j} = 0, (3.13a) j=l n-1 n(t~
-
lyn
+
C
(2j - n)(ajd,-j
+f,b,-j)
= 0,
(3.13b)
j=l n-1
~{(r~~-3nj+3j~)c,b~_~+4e~a~_~J=n(3n-2)d~--~
n’(n--l)c,+
j=l n-1 +
j(24b,-j
C
(3.14a)
-bjd,-j),
j=l n-l t~(n
-
l)e,, + 1 (2j - n)(ajc,_j
+ ejb,_j) = (2n - 1% + $bn
j=l n-l +
C
V;b,-j
(3.14b)
-Q,-,).
j=l (n
=
1, 2, 3, . . . .)
A. K. VERMA
168
The coefficients c,, e, in (3.14) are expressible in terms of the leading coefficients c,, e,. Setting n = 1 in (3.14) we obtain d, andf; and rest of the coefficients d,,ji can be generated from (3.13). c, and e, are so chosen as to satisfy the boundary conditions Cc, = 0 and Cnc,, - Cd, = 0. The remaining boundary condition Cc’e, + a = -i is then used to evaluate the constant c(, introduced in (3.9). A sixty-term power series expansion yields c, = -4.41770, e, = 0.56731. a = 1.27797.
(3.15)
The solution of the second set of eqns (3.8) can be written in the form Hi = K + [L + [‘A4 + $I0
+ @Z;), Gf = P + [Q + [*R + ;(2G” + CGF). (3.16)
Adopting the same procedure as explained above, we write cm,, pn, cy,, c*r,
and obtain the leading terms and the constant /I introduced
>
eecni
(3.17)
in (3.16) as
k, =.4.05402,p, = -2.74082, /I = 0.24417.
(3.18)
The low frequency first harmonic solution is not carried further from this stage. We now solve the differential set (3.4) governing the timemean induced flow. Following the above method of analysis, the zeroth order solution in (Tmay be written in the form H;(i) = S + CT + C*U + ;(HO + [Hi), GX) = X + [Y + 1*Z + $(2G” + CG;),
(3.19)
in which y is an unknown constant and is to be evaluated from the corresponding boundary condition. Now, writing (X T, U, X K Z) = f (cs,, c*t,, c3u,, c*x,, c3y,, c4z,) epcni, n=I
(3.20)
s, = 2.92469, x, = -2.43868, y = - 12.44341.
(3.21)
we find that
It can be easily shown that the quasi-steady terms of the second harmonic solution of the set (3.5) are given by %,(i) = $X(i),
G:(i) = fG;(i).
(3.22)
For sixty-term power series, the above solution leads to the following expressions for the radial and azimuthal components of the skin friction on the disk - H,,(O, t) = 1.02045 + t (0.21879 cos at + 0.95834~ sin at) -t2(0.79582
+ 0.79582 cos 2ot),
(3.23a)
-G&O, t) = 0.61591 - ~(O.O5164cos at + 1.057230 sin at) - ~~(0.52204 + 0.52204 cos 2ot).
(3.23b)
Also the inflow velocity at infinity is -H(co,
t) =
0.8844711 +~(1,27797cosot
- 0.244170 sin at)
-~‘(6.22171 + 6.22171 cos 2ot)).
(3.23~)
Symmetric flow about a rotating disk
169
6.0-
5.0 -
0.0
20
6.0
80
100
120
14.o
Fig. I. Variation of the superposed (coefficient of cz) time-mean flow components with 4.
We notice that .the imposed torsional oscillations in the far field reduce the components of the mean tangential shear at the rotating disk. The time-mean axial inflow in the far region is also decreased. The components of the superposed mean flow Hf,, G; and H& are shown in Fig. 1 which clearly exhibit that the boundary layer corresponding to the induced flow are much thicker than that of the basic von-k&man flow (see Benton[9]). The imposed periodic fluid motion not only increases the mean angular momentum within the boundary layer but also introduces steady injection out of the boundary layer which results in the decrease of the net mean axial inflow in the far field. The superposed pumping of the fluid out of the boundary layer is a direct consequence of the thickening of the boundary layer, in which the fluid particle just outside the basic K&man layer are undergoing angular momentum, albeit small. The terms of orders c4 are likely to introduce spatial oscillations in the superposed mean flow. From the expressions (3.23) we can derive formulae for the skin friction corresponding to the steady fluid flow, when the disk is rotating with angular velocity CJ and the fluid at infinity is slowly rotating with angular velocity ~$2in the same or in the opposite sense. We put u = 0 in (3.23) and get - H,[(O, t) = 1.02045 +0.21879e - 1.59164~’ + O(t’),
(3.24a)
-G,(O, t) = 0.61591 - 0.05164~ - 1.04407~‘+ 0(c3).
(3.24b)
The above results are compared with the results of Evans[3] (for e < 0) and with those of Rogers and Lance[l] (for c > 0), in Table 1. The agreement seems to be fairly good.
4. HIGH
FREQUENCY
SOLUTIONS
At the highest frequencies, the skin-effect results in the establishment of the Stokes layer over the rotating disk. As such the fluid flow varies on two distinct length scales. In order to expose the effect of high frequency oscillations at large distances over the whole of the fluid regime, we employ the method multiple scales. This method is effective, though
170
A. K. VERMA Table 1. L
-____
f&,(0,1)
G,(O, 1)
Relation (3.24a) E/R and L Relation (3.24b) -0.16 -0.15 -0.1 0.1 0.2
-0.98257 -0.98613 - 1.00155 - 1.02343 - 1.01049
- 0.94666 -0.95255 - 0.98261 - 1.02679 - 1.00374
-0.60667 -0.60804 -0.61327 -0.60811 -0.58987
E/R and L
-0.57767 -0.58761 -0.60825 -0.60155 -0.57208
complicated, largely because the induced fluctuating flow is rendered devoid of secular terms. Moreover, the solution thus obtained is valid for 6 of O(1). The inner Stokes layer is of O(o - l/*) times as thick as the outer von-K&man layer. We, therefore, consider velocity components as functions of the three independent variable I, c and 5 = a”*c; the 5 -dependence is so chosen as to suppress all secular terms in the perturbation process. The problem is now to find out H([, {, t) and G(&‘,e, t) where H,, - H,,, = 6 { 3H 551- H,, - HH,, + $H;} + 6*(3H,,, - 2HHt, + H,H,) + 63{Hcic - HH,, + ;H; - 2G2 - 4cG cos T},
G, - G,, = 6(2G,, - HG, + H,G + cH, cos T> f 6*{Gci - HG, + H,G + cH, cos T},
(4. la)
(4.lb)
H(O, 0, T) = 0, H,(O, 0, T) + 6H,(O, 0, T) = 0, G(0, 0, T) = 1 - ;(ewir + eir>, (4. lc) He(co, 00, T) + GH,(oo, co, T) = 0, G(co, 03, T) = 0,
(4.ld)
where 6 = o-~/*, T = ot and L N 0( 1). In order to obtain a uniformly valid solution of the set (4.1) we follow closely Benney’s[lO] method of analysis and write H = f
PH(“@‘, {, T) and G = f
PI=0
P’G@)(~,5, T).
?I=0
(4.2)
The solution of (4.1) valid upto second order in 6, is Ho = ag(i),
Go = b&l)
+ gj”)(i, 5) eiT + grco)(5, [) ewiT,
g{O)([, 0 = b(y)(c) e=(’ +l~~IJz,
(4.3a) (4.3b) (4.4a) (4.4b) (4Sa) (4Sb) (4.5c)
The arbitrary functions a$)(i), bf)([) occuring in the above solution, are given by the following system of boundary value problems (for details of the method of derivation of
171
Symmetric flow about a rotating disk
this system see Benney [ lo] and Chawla[ 1l] a&
- a$)a$‘& + iagf - 2bg2 = 0, b& - aRb& + a&b8 a&‘(O)= 0, a&(O) = 0, b#(O) = 1, a&(m)
= 0,
= 0 = b$$(oo),
2b\q’,- a#b\:) = 0, b\;)(O)= -c/2,
(1 + i)(26#
- a$$b$\))= fi(b
b&c - agb&
+ a&b8
b\!)(O)= 0,
+ b#a& = 0,
(4.8)
(4.9b)
a&O) = 0, a&(O) = -~*/2, b@(O) = 0, a&(m) = 0 = b&)(a),
(4.9c)
a$:\ = 2kb# , a\:)(O)= 0 7
(4.10)
2a& - a#a$q = 4ib$fb\:)‘, a@(O) = 0,
(4.11)
(1 + i)(2b& - a$$b.$) - a@b$) + bga$t)) = $(b\$,
- agb#
+ a&b\‘&
b\:‘(O)= 0, @(2a\-‘,:
(4.6b) (4.7)
#Cc- a$jb& + a&b@,
- b&a8
(4.6a)
(4.12a) (4.12b)
- aga\?) = (1 + i)(3a$& - 2a$ja\:\ + a&a\‘,‘) + 4b$$bt’,), a\:)(O)= (1 + i)E /Jz
(4.13a) (4.13b)
2a& - a#a$? = 0, a!&)(O)= 7.
(4.14)
The flow field characterized by a&$and 68, governed by the eqns (4.6), is the classical von-K&ran flow due to a rotating disk. The time harmonic terms in (4.3~(4.5) correspond to the shear wave solution whereas (4.7), (4.8), (4.10)-(4.14) give minor modifications in the thickness of the oscillatory Stokes layer due. to their interaction with the steady flow. The system (4.9) yields the exact solution in the form (4.15a)
(4.15b)
The system of eqns (4.6) as a whole represents the solution in the outer layer; proper matching with the inner solution is provided by the boundary conditions. In addition, it gives the complete interaction between the steady and unsteady parts of the field functions. The high frequency solution obtained above leads to the following radial and azimuthal components of the coefficients of shearing stress at the disk -H&O, T) = 1.02045 - c-“*{0.7071 lo’+ 1.41421c(cos T + sin T) + 0.37868c2(cos
2T + sin 29)
+ a-‘(0.51808~~
+ 1.84773~ sin T) + 0(aw3/*), -Gr(O, T) = a1’*{0.70711~(sin T -cos
(4.16a)
T)) + 0.61591
+ a-‘(0.27881~~ + 0.63777~ sin T) + O(O-~/~),
(4.16b)
172
A. K. VERMA 0.6
/ 0.5
I ‘1 f
/-\
1 \ \
\ \
\ ’
Fig. 2. Variation of the superposed (coefficient of D - ‘~3 time-mean flow components H”(axial), G” (azimuthal) and HF (radial) [for (a) CT= 3, (b) u = 5, (c) u = 91, with [.
whereas the axial inflow at infinity is given by -H(co,
T) = 0.88447 + a-‘(0.13346~~ + 6.5011~ sin T) + O(O-~‘~).
(4.17)
The non-dimensional composite components of the mean velocity, uniformly valid for the entire flow regime, are
-a&
G’“‘(~) = b&c) + c
-‘c2
>
(4.18a)
.
(4.18~)
+
The last two terms in the expressions for HP)([) are contributed from the solution of O(a-3’2). The time mean flow (proportional to a-‘~~), represented by H”(C), H’;(i) and Cm(c) is shown in Fig. 2. The broken curves, representing the radial flow, evidently exhibit the spatial oscillations in the superposed flow and vary with the frequency parameter. The spatial oscillations are more pronounced as g is increased. The spatial oscillations which are confined to the inner layer result from the Ekman effect of the superposed mean
Symmetric flow about a rotating disk
113
rotation just outside the Stokes layer. The skin effect is responsible for additional mean torque at the disk, whereas the radial component of the skin friction is decreased. We notice that to O(o-‘E’) the boundary layer sucks in additional fluid at the edge of the outer layer. The contribution to the axial flow from the last two terms of (4.1Xb) (which is in the opposite direction to that shown in Fig. 2) is O(o -3’2t:2). Acknowledgement-me author wishes to thank Dr. S. S. Chawla for the ~ntinuing many valuable discussions throughout the investigation. REFERENCES [l] M. H. ROGERS and G. N. LANCE, J. Fluid Mech. 7, 617-631 (1960). [2] H. OCKENDON, Q. J. A4ec!r. Appl. Math. XXV, 291-301 (1972). [3] D. S. EVANS, Q. J. Me&. Appl. Mul. XXII, 467-485 (1969). (41 R. J. BODONYI, J. Fluid Mech. 67, 657-666 (1975). [S] S. S. CHAWLA, Acta Me~hunica 25, 207-219 (1977). [6] Th. VON KARMAN, Z.A.M.M. 1, 233-252 (1921). 171W. G. COCHRAN, Proc. Cumb. Phil. Sot. 30, 365-375 (1934). i8i H. E. FETTIS, 4th Midwest Conf. Fluid Mech. Purdue, pp. 93-114 (1955). 191 E. R. BENTON. J. Fluid Mech. 24. 781-800 (1966). ;iOj D. J. BENNEY,‘J. Fluid Mech. 18, 385-391 (1964). .I l] S. S. CHAWLA, Q. J. Mech. Appl. Muth. XXVI, 355-370 (1973). (Received 26 January 1983).
guidance and for the