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Transient natural convection around a horizontal circular cylinder
Fluid Dynamics Research 10 (1992) 25-37 North-Holland
FLUI D DI'NAMICS RESEARCH
Transient natural convection around a horizontal circular cylinder Takao Sano and Katuo Kuribayashi Department of Mechanical Engineering, UniL,ersityof Osaka Prefecture, Sakai, Osaka, Japan Received 8 October 1991 Abstract. The initial phase of transient natural convection around a suddenly heated circular cylinder is
investigated. The Grashoff number is assumed to be large but finite. It is shown that, due to the displacement effect, irrotational outer flow is caused in the outer region outside the boundary layer, which, together with the inner boundary-layer flow, forms vortices at both sides of the cylinder, and that the pressure drag D~ is equal to the skin-friction drag Df up to the present order of approximation.
1. Introduction
The problem of transient natural convection flow around a circular cylinder, whose temperature is suddenly increased to Tw from that of the surrounding fluid T~ at r ' (time) = 0, has been first investigated by Elliott (1970) and subsequently, by Gupta and Pop (1977) and by Katagiri and Pop (1979). Elliott obtained asymptotic solutions for small time of the boundary layer equations (Gr ~ oo). Gupta and Pop extended this work to a finite but large Grashoff number and showed the effect of the curvature term in the governing equations on skin friction and heat transfer. Katagiri and Pop, on the other hand, presented numerical solutions of boundary layer equations. Both Elliott and Gupta and Pop used the regular perturbation method with E = Ur%/r o as a perturbation parameter to obtain asymptotic solutions for small time, where u r is the characteristic velocity, % the characteristic time and r 0 the radius of the cylinder. According to their solutions, the tangential velocity is exponentially small outside the boundary layer, but the radial velocity, which is of smaller order than the tangential velocity in the boundary layer by Gr-1/4, has a finite value there and does not satisfy the boundary condition at infinity, u r (radial velocity)~ 0 as r (radial c o o r d i n a t e ) ~ oo. This suggests that, when Gr is finite, the flow which is not described by the solutions given by Elliott and Gupta and Pop exists in the outer region outside the boundary layer, that is, the present problem is a singular perturbation problem. This outer flow is the result of the displacement effect of the flow in the boundary layer, which Gupta and Pop neglected. The purpose of the present paper is to complete the previous works by Elliott and Gupta and Pop by taking the displacement effect into consideration. The analysis uses the method of matched asymptotic expansions, in which the velocity is expressed as separate, locally valid, expansions in term of E for two regions: the inner region (boundary layer region) and the outer region outside the boundary layer. Correspondence to: T. Sano, Department of Mechanical Engineering, University of Osaka Prefecture, 4-804 MozuUmemachi, Sakai, Osaka 591, Japan. 0169-5983/92/$04.25 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
26
7". Sano,
K. Kuribayashi / Natural convection around a circular cylinder
2. Governing equations Consider a horizontal circular cylinder immersed in a stationary fluid at temperature T=. Suppose that the temperature of the cylinder is suddenly increased from T= to Tw at r ' = 0. The non-dimensional equations governing the unsteady natural convection flow caused by this step change in wall temperature can be written under the Boussinesq approximation as div v = 0, --
Jr
(1) = - • Vp +
+ e(v'V)v
at
- - + •(V" V)/ =
1 (e)1/2
Vr
V 2 v - ti x,
V2I,
(2)
(3)
where V2 -
l O(rO ) 132 -- -r Or ~rr + r 2 30 2.
(4)
In the above equations, v ( = v ' / U r) is the velocity, r ( = r ' / r o) the time, p ( = p ' / p = U r 2) the pressure, t ( = ( t ' - T ~ ) / ( T w - T~)) the temperature, (r ( = r ' / r o ) , O) cylindrical coordinates with r = 0 at the center of the cylinder and 0 = 0 in the direction of the gravity, Pr the Prandtl number, i x the unit vector in the direction of gravity, Pr=v/a*,
• =Urro/ro,
Gr = g/3( T w - T ~ ) r 3 / v 2
(Grashoff number),
(5)
where prime denotes the dimensional quantity, U ~ = g / 3 r o ( T w - T~) is the characteristic velocity, ¢0 the characteristic time, p= the density at infinity, r 0 the radius of the cylinder, g the gravity constant, /3 the volumetric coefficient of expansion, v the kinematic viscosity and a* the thermal diffusivity. Introducing the stream function 0 defined by
1 00 b/r
r O0'
00 L/0 = - - , Or
(6)
where u r and u o are radial and tangential components of v, respectively, the continuity equation is automatically satisfied and eqs. (2) and (3) may be written as
a(V20)
1 a(0, V 2 0 ) { E.~_~1/2 Ot 10t +•-r ~.~ -~Gr] V40 +-sin0r 0 + - - - C30 O S r 0,
__3' +•1(--
00 0' + 30 0 ' ) ---- 1 ( • 11/2
Jr
~
r
Orr
-~r ~
P r I,GrrJ
(7)
(8) V2t"
The boundary and initial conditions are r<0
~0= 0, t = 0 ,
r>O
O=O, 00/Or=O,t=l
at r = l ,
0: finite, t ~ 0 as r --+ m.
(9)
For initial stage of motion, • is a small quantity and there is a thin boundary layer in which very large velocity and temperature gradients are formed near the surface (inner region). For this layer, the proper stretching of the radial coordinate axis is R = (r-
1)/8",
(10)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
27
where 6 " is a non-dimensional p a r a m e t e r having the same o r d e r of magnitude as the non-dimensional thickness of the inner region, that is,
~* = O(Vv~'r~ /ro) = O ( ( e / G r ) 1 / 4 ) .
(11)
By requiring that the non-dimensional tangential velocity in the inner region is of order unity, the inner-flow stream function ko'(i)(R, 0, T) is magnified as (12)
l/)'(i) = ~ / / ~ * .
As in the previous p a p e r ( G u p t a and Pop, 1977), we shall consider the most interesting case where 6* = e ,
(13)
that is, r e / G f G r = a e 2,
(13')
where a is a constant of order unity #~. Therefore, G r = O(e 3), suggesting that G r is very large. Now, we can expand the solutions in the inner region in term of • as (inner expansions) qr(i, = qt0O)(R ' O, r) + e ~ i ) ( R ,
0, ~') + " " ,
(14)
t°)=t~,i)(R, O, "r) + e t l i ) ( R , 0, v) + " - ' ,
(15)
where t°)(R, O, r ) = t(r, 0, ~-). T h e b o u n d a r y conditions on the surface are imposed on these inner solutions. Substitution of (10), (12), (13), (14) and (15) into (7) and (8) yields a set of equations for ~(~) and I,~i) - tl ll Ot oi,
0~- ~tR2
a ~
at~ i)
ce a2t~ i)
0~"
Pr 0R 2 '
82q,[i,
o4
0(~o0) , c32qt,~i)/8R2)
O-c ;JR I
O( R, O)
o3,/,2 '
or?
=a 0---~-+2a~+
0-r
30
--
OR
(16)
(17)
cO 0 ~ i) +----+
O-c OR 2
.
+ OR sin 0,
OR sin 0 +
+ - -
OR
--
00
Pr ~
00 cos 0,
+--~-]"
(18)
(19)
T h e first-order equations (16) and (17) have the same form as those for Gr---, ~. T h e second-order equations (18) and (19) have extra terms which take curvature effect into consideration. In the outer region, where r = O(1), the order of the tangential velocity is still unknown. D e n o t i n g this order by A(•), the outer-flow stream function is magnified as ~ ( ° ) ( r , 0, "r) = ~ / A ( • ) .
(20)
,~1 Only for this case, both the inertia (or convection) and the curvature terms, which are neglected in the first-order equations shown below, appear in the second-order equations.
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
28
The solutions in this region may be expressed as (outer expansions) ~(o) = ~0(O)(r, 0, ~-) + e ~ ° ) ( r , 0, r ) + . . . ,
(21)
t = t(o°)(r, O, r) + et~°)(r, O, r) + "..
(22)
The boundary conditions at infinity are imposed on these expansions. The energy equation (8) with the initial condition t = 0 yields t~°) = O,
(23)
for all n, suggesting that the buoyancy terms in the momentum equation play no role in the outer region. In view of (23), the matching condition for t °) may be written as t ° ) ( R , 0, r ) -~ 0 (exponentially)
as R -~ ~.
(24)
On the other hand, the momentum equation (7) with eq. (23) and the initial condition of irrotationality yields (25)
V 2 ~ b*(°) = 0
for all n. The outer flow is thus irrotational to all orders of E. The matching condition between ~ i ) and ~{o) may be written as E
lim ~ 0 ) ( R , 0, ~-) R ---~~
lira ~ O ) ( r , 0, 7). A(E-)
(26)
r--*l
3. Construction of the solution
The solution of (25) which satisfies the boundary condition at infinity and is symmetric with respect to the vertical plane 0 = 0 and 7r is oc
qt~,,, = ~" A , , k ( r ) r - k s i n ( k 0 ) ,
(27)
k~l
where the Ank are integral constants to be determined by matching with the inner solution. Applying the matching condition (26), we find that A(e) = E
(28)
and that the inner stream functions, q*0{i} and ~ i ) , should behave as ,m
~o i) ~ ~, Aok(r ) sin(kO) + o(1),
(29)
k~l
~i)~
_
~ Aok(r) k sin(k0) R k=l
+ ~ A l k ( r ) sin(k0) + o(1),
(30)
k-1
when R tends to infinity. Equation (28) suggests that both the tangential and radial velocities in the outer region, as well as the radial velocity in the inner region, are of the order e. The equation for t(0i) (17) is the heat conduction equation, and the required solution is t(0i)=-- e r f c ( f P r rt),
(31)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
29
where rl
R 2~Vc~-
r-1 2 ~r
(32)
2
Substituting (31) into (16), the equation for q,~i) may be written as C~ ~21~ (i)
~4 ltr(i)
Or OR2
a ~OR Pr)1/2
= -
~
e x p ( - P r r/z) sin 0.
r -'/2
(33)
The solution of (33) satisfying the boundary condition on the surface and the matching condition (29) is qt~i) = 8~-73/2F,,(r/)sin 0,
(34)
where Fo(rl) -
1
~(Pr
[i 3 erfc(~/Pr r/) - v/Pri 3 erfc(~7) + (~fp~ _ 1)/6~/~-],
1)
i n erfc(x) =f~ i n-1 erfc(y) dy
(n=l,2
.... ),
i ° erfc(x) = erfc(x).
(35)
It is easy to show by the limit process (Pr ~ 1) that, when Pr = 1, the expression (35), whose denominator is zero for Pr = 1, can be written as 1
F0(~7 ) = -~r/~ erfc(~7) + l ~ - ~ - ( 2 ~ 2 - 1) exp(-T/2) 1
+ 12~"
(35')
These solutions agree completely with those given by Gupta and Pop. Furthermore, the matching condition (29) determines unknown constants in qrl°) as Ao, = 3(a/'rr
Aok=0
Pr)l/2r3/2/Of~
+
1),
for k > 2 ,
(36)
and we have 4 qt[o)_ 3(,~P~ + 1)
(a)'/2 _
_
~ Pr
T3/2r-1
sin O.
(37)
In fig. 1, the streamlines calculated from the composite solution, which is obtained by adding (34) to (37) and then subtracting the common part, are shown for Pr = 0.7 and p ' r ' / r ( 2 = 1.5 X 10 - 3 . For comparison, the streamlines calculated from the inner solution (34) only are also shown. It is seen that there exist vortices in the flow, which do not appear in the previous solutions.
T. Sano, K. Kuribayashi / Natural conuection around a circular cylinder
30
I
/
/
it
j/ /
I /
;/
II
l//
I
~
,'
,,
//~/
/"
//k.. k(/,,5,,' ,'
/
,,'"
rsinO -
-
composite
eolutlon,
..............
inner
5olutlon
Fig. 1. F i r s t - o r d e r s t r e a m l i n e p a t t e r n s for P r = 0.7 a n d u ' c ' / r o = 1.5 × 10
~.
Now, we shall proceed to obtain solutions of second-order equations (18) and (19). Substituting (30) and (34) into (19), we have at] i)
a 02t] i)
Or
Pr
=
OR 2
/ - I
\~P~rl
'j2
7 -1/2 e x p ( - P r 7 2)
8 + ~/~ (Pr - 1) ~'[-i-~ e r f c ( ~ - ~ ) + f P r i 3 erfc(~) - (v/Pr - 1)/6~f~-~] × e x p ( - P r 72 ) cos 0.
(38)
The solution of (38) may be expressed as t]i)= ~.2f(~) cos 0 + af~g(~7).
(39)
The second term in (39) arises due to curvature effect. The equations for f and g are f"+2PrBf'-8Prf 32 Pr - ~ - v ( P r - 1) e x p ( - P r 72 ) × [ - i 3 erfc(~P~7) + fPr-i 3 erfc(~7) - ( ~ g"+2Pr
~g'-2Prg=4
Pr~exp(-Pr
- 1)/6~-~ ],
72).
(40) (41)
The solution for g has already been obtained by Gupta and Pop as g = -~7 erfc(~PT-q).
(42)
31
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
The solution for f, on the other hand, has been obtained only for Pr = 1 by Elliott. Here, we obtained the solution for f for arbitrary value of Pr. The result is 8 f = A ( P r ) i 4 erfc( P~/~-rl)- 15~w (V~7. + 1 ) e x p ( - P r .2) +
1
12(Pr
(4 pr2r/4 + 12 Pr 7"]2 + 3) erfc(~-~7) 2
1)
6 /v ( p r -
1)
(2 Pr rl ~ + 5"q) e x p ( - P r rl 2) erfe(v/-PTrl)
+
(pr
/'exp(
1)4 (4 pr2r/4 + 12 Pr r/2 + 48V~-~f ~ - ( P r - 1) + 3)jo
-
Pr x 2) erfc(x) dx
1
48-rrP ~ [2 Pr(Pr 2 + 4 Pr + 7)r/2 + 3 Pr 2 + 14 Pr + 3] e x p [ - ( P r + 1 ) . 2] 1
+ 48rr Px/~-rlPr__,- 1) [ 2 P r ( P r 4 + 4 P r 3 + 6 P r 2 + 4 P r - 7 ) @ + ( 5 pr4 + 20 pr3 + 30 Pr 2 - 12 P r - 3)r/] e x p ( - P r r/2) erfc(r/),
(43)
where 30 Pr 5/2 + 30 Pr 2 + 140 Pr 3/2 + 140 Pr + 286fPT + 30
A(Pr) =
15"rr,,/Pr (fP~ + 1) 2 + "rr Pr(Pr - l) [(Pr + 1) 4 arctg0/Pr ) - 4w Pr].
When Pr = 1, (43) can be written as
( 48)
(44)
4
i 4 erfc(r/) - - exp(-r/2) f = 8 + 15---~ 15"rr 2 112(47"/4+ 12~72 + 3) erfc(w 2) - ~ ( T / 2
q- 1) e x p ( 2 r / 2 )
1
+ -~'-~- (6r/3 + llz/) exp(__.2) erfc(r/).
(43')
Substituting (31), (35) and (39) into (18), we have the following equation for ~(~)" l
0 ~21Fli) Or
0R 2
I
•
041/tli) a ~ aR
16 -
pr - 1
erfc(n)
+2~(Pr-1)7
(5 Pr - 3) erfc(fPrrl)
e x p ( - P r ~02)] sin 0 +
8r 3/2 sin 0 cos 0 ~-(Pr-
1) 2
× { - [ i 2 erfc(rl) - i 2 erfc(fPr-rt)] [~/Pr i e r f c ( P ~ r l ) - i erfc(rt)] + [ ~ - / P r i 3 erfc(fPTrl) - i 3 erfc(~7) + (1 - pr l/2)/6~/ww ] × [erfc(~/) - Pr erfc(fPTrl)] } +
r 3/2 sin 0 cos 0 ~f(rl) 2v/~
art
(45)
32
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
The solution of (45) may be written as ~/-tli) = gf~-a'r7/2Fl(*/) sin 0 cos 0 + a r e [ G ( * / )
+ H ( * / ) ] sin 0,
(46)
where G and H arise from curvature and displacement effects, respectively. The equations f o r F 1,G and H a r e F~ v + 2*/F;" - 10F('
x [x/~-i e r f c ( x / P r , ) - i erfc(*/)] + [ l x / f ~ i 3 erfc( P ~ * / ) -
i 3 erfc(v) + ( 1 - Pr t/2)/6x/-w]
X [erfc(*/) - Pr erfc(fPr*/)]} - 8 d f ( * / ) / d * / . G TM +
2*/G'"-4G"-
16 Pr- 1
(47)
[(2 P r - 1) erfc(fPr*/) - erfc(*/)]
+ 8[erfc(fPT*/)- 2~*/exp(-
Pr */2)] ,
H TM + 2~/H'" - 4 H " = 0.
(48) (49)
Equations (47) and (48), for which solutions can be obtained without matching consideration, were already considered in the previous papers. Elliott obtained the solution of (47), but only for Pr = 1. Here we obtain it for arbitrary values of Pr. Since the detailed expression for F1 is too lengthy to reproduce here, it is omitted. The solution of (48) was obtained by Gupta and Pop for arbitrary values of Pr, which can be written as 4 i 2 erfc(*/) 8 ( 3 P ~ + 2) i 4 erfc(*/) + G = fP~(1 - Pr) Pr - 1
+
1 - P~Pr 4 p r ( g g 7 + 1)
2 + (Pr -
1) 2
+
16(2 P r - 1) )2 i4 e r f c ( f P r rl) pr(Pr - 1
(Pr(Pr-3)@ 6
1
6~fPT(Pr-
1) 2
Pr+l*/2 2
3Pr-1) 8 Pr
erfc(,~pT*/)
[2 er(Pr - 3)*/3 - (7 Pr + 3)*/] e x p ( - Pr */2).
(50)
When Pr = 1, (50) can be written as 2 G = (2./4 + ,/2) erfc(*/) - _ _ * / 3 exp(__*/2).
(50')
The solution of (49), which is required to satisfy the matching condition (30), that is, H~ -
8
3~/w ~/PT{~PT + 1) */+ "'"
as */
--+ m
,
as well as the boundary condition on the surface, can be obtained as 1
H = fPr(fPr
+ 1) [ - 16 i 4 erfc(*/) - ~-r//~/~w + 1].
(51)
T. Sano, K. Kuribayashi / Natural conL,ection around a circular cylinder
33
Finally, the matching condition (30) d e t e r m i n e s the integral constants A ~k a p p e a r i n g in the second-order stream function ~ o ) , and we have ~o)
o/
.1.2
4Pr
r
sin 0 + v ~ B ( P r ) ( r 7 / 2 / r 2) sin 0 cos 0,
(52)
where B(Pr) is a function of Pr. T h e expression of B(Pr) is too lengthy to reproduce here and is omitted.
4. D i s c u s s i o n s
We now introduce new non-dimensional time and stream functions as
T = aE 2 Grl/2"r = ~/gfl( Tw - T~l /r,, ~" = ( u r ' / r g ) G r 1/2, ~ * ~ ) = ( E / G r ' / 4 ) ( U r r o / u ) ~ ~x)= ~O'/v G r 1/4,
(53)
where x denotes i or o. In terms of these new variables, the solutions obtained in the previous section may be written as k0'*(i)= 8T3/2Fo(rl) sin 0 + TT/2FI(~) sin 0 cos 0
+(T2/Grl/4)[G(~7) + H ( r / ) ] 4T 3/2 ~,~o)=
(54)
1
3 ~r~-wrvvrtvvr r - ~m-" +l)-Sinr 1 T2 - - - -
sin 0 + " " ,
O+B(Pr)TT/2r -2sin 0 c o s 0
1
+ 4 Pr Gr 1/4 r
sin 0 + - - .
(55) TI/2
t = erfc(,,/-P-)-r/) + T 2 f ( r / ) cos 0
Grl/4r/erfc(vrPTrl) + "'-
(56)
From (54) and (55), the velocity c o m p o n e n t s can be written as ,
T3/2
bl r
Grl/Zv/r °
T7/2
8 G ~ i ~ F o ( ' r / ) cos 0
G r l / 4 F l ( r l ) ( c o s 2 0 - sin20)
T2 G r l / 2 [ G ( r / ) + H ( r / ) - 16r/F0(r/) ] cos 0 + ' " ,
(57)
!
/g0 G r 1/2 p / r 0
- 4TF~(r/) sin 0 + ~1 T 3 F ( ( r l ) sin 0 cos 0 T3/2
+ l/------~ 2 Gr [G'(r/) + H'(r/)]
sin 0 + - - - ,
(58)
in the inner region and as U'r Grl/2v/ro
4 -
T 3/2
1
3f~-wg~-( P~V/~+ 1) G r 1/4 r 2cOS 0 T 7/2
1
- B ( P r ) G r l / 4 r3 1
T2
( C O S 2 0 --
sin20)
1
4 Pr G r 1/2 r 2cOs 0 + "-- ,
(59)
T. Sano, K. Kuribayashi / Natural concection around a circular cylinder
34
(a) Presen
(b)
,
t
"
Present
GuptaS,Pop '
0.04
-
-
EUiott
0.4
Elliott
0.2
0.02
L
i
i
i
i
i
i
i
I
I
I
,
I
I .I
1.2
13
I
I
1.2
r
1.3
I
1./.,
r
F i g . 2. T a n g e n t i a l v e l o c i t y d i s t r i b u t i o n s f o r P r = 0.7, 0 = 90 ° a n d G r = 500. ( a ) T = 0.01, ( b ) T = 0.1.
u'o Gri/2u/ro -
4
T 3/2
1
T 7/2
1
3~-v ,fPT (fPr- + 1) Gr 1/4 r 2 sin 0 - 2 B ( P r ) Grl/4 r3 1
T2
sin 0 cos 0
1
4 Pr Gr ]/2 r 2
sin 0 + " . . ,
(60)
in the outer region. It is seen that the tangential velocity in the outer region is of the same order of magnitude as the two radial velocities. In fig. 2, the tangential velocity distributions calculated from the composite expansion are shown for Pr = 0.7, 0 = 90 ° and Gr = 500. For comparison, the previous results of Elliott and by Gupta and Pop are also shown in the figure. It is seen from the figure, as might have been expected from fig. 1, that the direction of outer flow is opposite to that of inner flow. The difference between the present results and those of Gupta and Pop is of course due to the displacement effect of the inner flow. Figure 3 shows examples of the streamlines for Pr = 0.7. The cores of the vortices move upwards slightly as time goes on, although it is difficult to see in the figure. The expression for the skin-friction on the surface of the cylinder may be written as I ~urO ) ]J'~ ~TFt r,=r 0
PU2 Gr3/4 {2T1/2F~'(O) F2
sin 0 +
1TS/2F('(O)
sin 0 cos 0
q- }(T/GF1/4)[G"(O) + / - / " ( 0 ) ] sin 0 + " " },
(61)
where F~'(O), G"(O) and H"(O) may be written, using the solution given in the previous section, as F(~'(O) = 1 / ~ / ~ ( ~ P r + 1), G"(0) = -H"(0)
= 4 / f ~ - ( P ~ + 1).
(62)
The expression for F('(0) is too lengthy to reproduce here and is omitted. It is seen from (62) that, as was already found by G u p t a and Pop, the effect of curvature is to increase the skin-friction, but the displacement thickness has the effect of decreasing it and cancels the increase of skin-friction due to the curvature effect; that is, up to O(e), the curvature and
T. Sano, K. Kuribayashi / Natural concection around a circular cylinder
35
(a)
(b) Fig. 3. Streamline patterns for Pr = 0.7. (a) Gr = 500, (b) Gr = 5000. displacement thickness dO not influence skin-friction as a whole. F r o m (62), the skin-friction drag per length L of the cylinder, Dr, can be obtained as Dr= r0
Gr 3/4 - - T P~PT+I
l/2+o(T 5/2)+o(Gr-l/4T)
.
(63)
The expression for the pressure on the surface of the cylinder is (P')~'=~(,-Po-
pv 2 Gr r~ { ( T 1 / 2 / G r ' / 4 ) [ ½ G ' " ( O ) + 2F(;'(0)] (cos 0 + ~(T1/2/Grl/4)H'(O)(cos
O - 1) + " " },
1) (64)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
36
where P0 is the pressure at the forward stagnation point, and G " ( 0 ) = - 16(2¢~- + 1 ) / ~ - ¢ ~ - ( v F P T + 1), H"(0) = 16/~
PfPT(~P~- + 1).
(65)
The first term in (64) is due to the curvature effect. Inserting the values of G"(0), Fd'(0) and
H"'(O) into (64), we have pu 2 Gr3/4 rX {[2T'/2/~(,,/~
(P')r'=~o--P0
+ 1)](cos 0 - 1 ) + " " }.
(66)
From this, the pressure drag per length L, Dp, can be obtained as Do=
/xuL
Gr3/4[2v/-~wTI/2/(g~PT + 1) + " " ].
(67)
r0
It is seen that Dp is equal to Df in the present order of approximation. Both coefficients increase with time as T t/2 and with increase in Gr as Gr 3/4. It should be noted here that, in the first-order boundary layer theory, the pressure distribution on the surface is uniform because of zero velocity in the outer region, and the pressure drag is zero. Therefore it is natural to expect that, when Gr is large, the pressure drag is negligibly small and tends to zero as Gr ---, ~. However, the present investigation shows that this expectation is not correct and that Dp has the same order of magnitude as Df. Finally, the local Nusselt number Nu, which is defined as Nu = h r o / k , h being the heat transfer coefficient,
h = - ( k / r o ) ( a t / a r ) r = ,, may be calculated from (31), (42) and (43) as Nu
fPT Gr '/4 V~w T 1/2
I G r l / 4 T 3 / 2 f ' ( O ) cos 0 + ~1 + . " ,
(68)
where f'(0) =~-
Pr 2 + 5 P r + l l 30 Pr s/2 + 30 Pr 2 + 140 Pr 3/2 + 140 Pr + 2 8 6 ~
+ 30
15"rrvrPT( PfPr + 1) 2 [(Pr + 1)4 arctg(v/~-) - 4"rr Pr]). rr P r ( P r - 1)
(69)
When Pr = 1, this equation becomes f'(0) = ~
1 (9_ 1-~w )'
(69')
The mean Nusselt number N----uaveraged over the surface is obtained from (69) as I --~ Nu=w~o--/ N u d 0 -
~ - Or 1/4 1 v~w T 1/---T + 2 + ' ' "
(70)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
References Elliott, L. (1970) Quart. J. Mech. Appl. Math. 23, 153. Gupta, A.S. and I. Pop (1977) Phys. Fluids 20, 162. Katagiri, M. and I. Pop (1979) Wiirme- und Stoffiibertragung 12, 73.
37
FLUI D DI'NAMICS RESEARCH
Transient natural convection around a horizontal circular cylinder Takao Sano and Katuo Kuribayashi Department of Mechanical Engineering, UniL,ersityof Osaka Prefecture, Sakai, Osaka, Japan Received 8 October 1991 Abstract. The initial phase of transient natural convection around a suddenly heated circular cylinder is
investigated. The Grashoff number is assumed to be large but finite. It is shown that, due to the displacement effect, irrotational outer flow is caused in the outer region outside the boundary layer, which, together with the inner boundary-layer flow, forms vortices at both sides of the cylinder, and that the pressure drag D~ is equal to the skin-friction drag Df up to the present order of approximation.
1. Introduction
The problem of transient natural convection flow around a circular cylinder, whose temperature is suddenly increased to Tw from that of the surrounding fluid T~ at r ' (time) = 0, has been first investigated by Elliott (1970) and subsequently, by Gupta and Pop (1977) and by Katagiri and Pop (1979). Elliott obtained asymptotic solutions for small time of the boundary layer equations (Gr ~ oo). Gupta and Pop extended this work to a finite but large Grashoff number and showed the effect of the curvature term in the governing equations on skin friction and heat transfer. Katagiri and Pop, on the other hand, presented numerical solutions of boundary layer equations. Both Elliott and Gupta and Pop used the regular perturbation method with E = Ur%/r o as a perturbation parameter to obtain asymptotic solutions for small time, where u r is the characteristic velocity, % the characteristic time and r 0 the radius of the cylinder. According to their solutions, the tangential velocity is exponentially small outside the boundary layer, but the radial velocity, which is of smaller order than the tangential velocity in the boundary layer by Gr-1/4, has a finite value there and does not satisfy the boundary condition at infinity, u r (radial velocity)~ 0 as r (radial c o o r d i n a t e ) ~ oo. This suggests that, when Gr is finite, the flow which is not described by the solutions given by Elliott and Gupta and Pop exists in the outer region outside the boundary layer, that is, the present problem is a singular perturbation problem. This outer flow is the result of the displacement effect of the flow in the boundary layer, which Gupta and Pop neglected. The purpose of the present paper is to complete the previous works by Elliott and Gupta and Pop by taking the displacement effect into consideration. The analysis uses the method of matched asymptotic expansions, in which the velocity is expressed as separate, locally valid, expansions in term of E for two regions: the inner region (boundary layer region) and the outer region outside the boundary layer. Correspondence to: T. Sano, Department of Mechanical Engineering, University of Osaka Prefecture, 4-804 MozuUmemachi, Sakai, Osaka 591, Japan. 0169-5983/92/$04.25 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
26
7". Sano,
K. Kuribayashi / Natural convection around a circular cylinder
2. Governing equations Consider a horizontal circular cylinder immersed in a stationary fluid at temperature T=. Suppose that the temperature of the cylinder is suddenly increased from T= to Tw at r ' = 0. The non-dimensional equations governing the unsteady natural convection flow caused by this step change in wall temperature can be written under the Boussinesq approximation as div v = 0, --
Jr
(1) = - • Vp +
+ e(v'V)v
at
- - + •(V" V)/ =
1 (e)1/2
Vr
V 2 v - ti x,
V2I,
(2)
(3)
where V2 -
l O(rO ) 132 -- -r Or ~rr + r 2 30 2.
(4)
In the above equations, v ( = v ' / U r) is the velocity, r ( = r ' / r o) the time, p ( = p ' / p = U r 2) the pressure, t ( = ( t ' - T ~ ) / ( T w - T~)) the temperature, (r ( = r ' / r o ) , O) cylindrical coordinates with r = 0 at the center of the cylinder and 0 = 0 in the direction of the gravity, Pr the Prandtl number, i x the unit vector in the direction of gravity, Pr=v/a*,
• =Urro/ro,
Gr = g/3( T w - T ~ ) r 3 / v 2
(Grashoff number),
(5)
where prime denotes the dimensional quantity, U ~ = g / 3 r o ( T w - T~) is the characteristic velocity, ¢0 the characteristic time, p= the density at infinity, r 0 the radius of the cylinder, g the gravity constant, /3 the volumetric coefficient of expansion, v the kinematic viscosity and a* the thermal diffusivity. Introducing the stream function 0 defined by
1 00 b/r
r O0'
00 L/0 = - - , Or
(6)
where u r and u o are radial and tangential components of v, respectively, the continuity equation is automatically satisfied and eqs. (2) and (3) may be written as
a(V20)
1 a(0, V 2 0 ) { E.~_~1/2 Ot 10t +•-r ~.~ -~Gr] V40 +-sin0r 0 + - - - C30 O S r 0,
__3' +•1(--
00 0' + 30 0 ' ) ---- 1 ( • 11/2
Jr
~
r
Orr
-~r ~
P r I,GrrJ
(7)
(8) V2t"
The boundary and initial conditions are r<0
~0= 0, t = 0 ,
r>O
O=O, 00/Or=O,t=l
at r = l ,
0: finite, t ~ 0 as r --+ m.
(9)
For initial stage of motion, • is a small quantity and there is a thin boundary layer in which very large velocity and temperature gradients are formed near the surface (inner region). For this layer, the proper stretching of the radial coordinate axis is R = (r-
1)/8",
(10)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
27
where 6 " is a non-dimensional p a r a m e t e r having the same o r d e r of magnitude as the non-dimensional thickness of the inner region, that is,
~* = O(Vv~'r~ /ro) = O ( ( e / G r ) 1 / 4 ) .
(11)
By requiring that the non-dimensional tangential velocity in the inner region is of order unity, the inner-flow stream function ko'(i)(R, 0, T) is magnified as (12)
l/)'(i) = ~ / / ~ * .
As in the previous p a p e r ( G u p t a and Pop, 1977), we shall consider the most interesting case where 6* = e ,
(13)
that is, r e / G f G r = a e 2,
(13')
where a is a constant of order unity #~. Therefore, G r = O(e 3), suggesting that G r is very large. Now, we can expand the solutions in the inner region in term of • as (inner expansions) qr(i, = qt0O)(R ' O, r) + e ~ i ) ( R ,
0, ~') + " " ,
(14)
t°)=t~,i)(R, O, "r) + e t l i ) ( R , 0, v) + " - ' ,
(15)
where t°)(R, O, r ) = t(r, 0, ~-). T h e b o u n d a r y conditions on the surface are imposed on these inner solutions. Substitution of (10), (12), (13), (14) and (15) into (7) and (8) yields a set of equations for ~(~) and I,~i) - tl ll Ot oi,
0~- ~tR2
a ~
at~ i)
ce a2t~ i)
0~"
Pr 0R 2 '
82q,[i,
o4
0(~o0) , c32qt,~i)/8R2)
O-c ;JR I
O( R, O)
o3,/,2 '
or?
=a 0---~-+2a~+
0-r
30
--
OR
(16)
(17)
cO 0 ~ i) +----+
O-c OR 2
.
+ OR sin 0,
OR sin 0 +
+ - -
OR
--
00
Pr ~
00 cos 0,
+--~-]"
(18)
(19)
T h e first-order equations (16) and (17) have the same form as those for Gr---, ~. T h e second-order equations (18) and (19) have extra terms which take curvature effect into consideration. In the outer region, where r = O(1), the order of the tangential velocity is still unknown. D e n o t i n g this order by A(•), the outer-flow stream function is magnified as ~ ( ° ) ( r , 0, "r) = ~ / A ( • ) .
(20)
,~1 Only for this case, both the inertia (or convection) and the curvature terms, which are neglected in the first-order equations shown below, appear in the second-order equations.
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
28
The solutions in this region may be expressed as (outer expansions) ~(o) = ~0(O)(r, 0, ~-) + e ~ ° ) ( r , 0, r ) + . . . ,
(21)
t = t(o°)(r, O, r) + et~°)(r, O, r) + "..
(22)
The boundary conditions at infinity are imposed on these expansions. The energy equation (8) with the initial condition t = 0 yields t~°) = O,
(23)
for all n, suggesting that the buoyancy terms in the momentum equation play no role in the outer region. In view of (23), the matching condition for t °) may be written as t ° ) ( R , 0, r ) -~ 0 (exponentially)
as R -~ ~.
(24)
On the other hand, the momentum equation (7) with eq. (23) and the initial condition of irrotationality yields (25)
V 2 ~ b*(°) = 0
for all n. The outer flow is thus irrotational to all orders of E. The matching condition between ~ i ) and ~{o) may be written as E
lim ~ 0 ) ( R , 0, ~-) R ---~~
lira ~ O ) ( r , 0, 7). A(E-)
(26)
r--*l
3. Construction of the solution
The solution of (25) which satisfies the boundary condition at infinity and is symmetric with respect to the vertical plane 0 = 0 and 7r is oc
qt~,,, = ~" A , , k ( r ) r - k s i n ( k 0 ) ,
(27)
k~l
where the Ank are integral constants to be determined by matching with the inner solution. Applying the matching condition (26), we find that A(e) = E
(28)
and that the inner stream functions, q*0{i} and ~ i ) , should behave as ,m
~o i) ~ ~, Aok(r ) sin(kO) + o(1),
(29)
k~l
~i)~
_
~ Aok(r) k sin(k0) R k=l
+ ~ A l k ( r ) sin(k0) + o(1),
(30)
k-1
when R tends to infinity. Equation (28) suggests that both the tangential and radial velocities in the outer region, as well as the radial velocity in the inner region, are of the order e. The equation for t(0i) (17) is the heat conduction equation, and the required solution is t(0i)=-- e r f c ( f P r rt),
(31)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
29
where rl
R 2~Vc~-
r-1 2 ~r
(32)
2
Substituting (31) into (16), the equation for q,~i) may be written as C~ ~21~ (i)
~4 ltr(i)
Or OR2
a ~OR Pr)1/2
= -
~
e x p ( - P r r/z) sin 0.
r -'/2
(33)
The solution of (33) satisfying the boundary condition on the surface and the matching condition (29) is qt~i) = 8~-73/2F,,(r/)sin 0,
(34)
where Fo(rl) -
1
~(Pr
[i 3 erfc(~/Pr r/) - v/Pri 3 erfc(~7) + (~fp~ _ 1)/6~/~-],
1)
i n erfc(x) =f~ i n-1 erfc(y) dy
(n=l,2
.... ),
i ° erfc(x) = erfc(x).
(35)
It is easy to show by the limit process (Pr ~ 1) that, when Pr = 1, the expression (35), whose denominator is zero for Pr = 1, can be written as 1
F0(~7 ) = -~r/~ erfc(~7) + l ~ - ~ - ( 2 ~ 2 - 1) exp(-T/2) 1
+ 12~"
(35')
These solutions agree completely with those given by Gupta and Pop. Furthermore, the matching condition (29) determines unknown constants in qrl°) as Ao, = 3(a/'rr
Aok=0
Pr)l/2r3/2/Of~
+
1),
for k > 2 ,
(36)
and we have 4 qt[o)_ 3(,~P~ + 1)
(a)'/2 _
_
~ Pr
T3/2r-1
sin O.
(37)
In fig. 1, the streamlines calculated from the composite solution, which is obtained by adding (34) to (37) and then subtracting the common part, are shown for Pr = 0.7 and p ' r ' / r ( 2 = 1.5 X 10 - 3 . For comparison, the streamlines calculated from the inner solution (34) only are also shown. It is seen that there exist vortices in the flow, which do not appear in the previous solutions.
T. Sano, K. Kuribayashi / Natural conuection around a circular cylinder
30
I
/
/
it
j/ /
I /
;/
II
l//
I
~
,'
,,
//~/
/"
//k.. k(/,,5,,' ,'
/
,,'"
rsinO -
-
composite
eolutlon,
..............
inner
5olutlon
Fig. 1. F i r s t - o r d e r s t r e a m l i n e p a t t e r n s for P r = 0.7 a n d u ' c ' / r o = 1.5 × 10
~.
Now, we shall proceed to obtain solutions of second-order equations (18) and (19). Substituting (30) and (34) into (19), we have at] i)
a 02t] i)
Or
Pr
=
OR 2
/ - I
\~P~rl
'j2
7 -1/2 e x p ( - P r 7 2)
8 + ~/~ (Pr - 1) ~'[-i-~ e r f c ( ~ - ~ ) + f P r i 3 erfc(~) - (v/Pr - 1)/6~f~-~] × e x p ( - P r 72 ) cos 0.
(38)
The solution of (38) may be expressed as t]i)= ~.2f(~) cos 0 + af~g(~7).
(39)
The second term in (39) arises due to curvature effect. The equations for f and g are f"+2PrBf'-8Prf 32 Pr - ~ - v ( P r - 1) e x p ( - P r 72 ) × [ - i 3 erfc(~P~7) + fPr-i 3 erfc(~7) - ( ~ g"+2Pr
~g'-2Prg=4
Pr~exp(-Pr
- 1)/6~-~ ],
72).
(40) (41)
The solution for g has already been obtained by Gupta and Pop as g = -~7 erfc(~PT-q).
(42)
31
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
The solution for f, on the other hand, has been obtained only for Pr = 1 by Elliott. Here, we obtained the solution for f for arbitrary value of Pr. The result is 8 f = A ( P r ) i 4 erfc( P~/~-rl)- 15~w (V~7. + 1 ) e x p ( - P r .2) +
1
12(Pr
(4 pr2r/4 + 12 Pr 7"]2 + 3) erfc(~-~7) 2
1)
6 /v ( p r -
1)
(2 Pr rl ~ + 5"q) e x p ( - P r rl 2) erfe(v/-PTrl)
+
(pr
/'exp(
1)4 (4 pr2r/4 + 12 Pr r/2 + 48V~-~f ~ - ( P r - 1) + 3)jo
-
Pr x 2) erfc(x) dx
1
48-rrP ~ [2 Pr(Pr 2 + 4 Pr + 7)r/2 + 3 Pr 2 + 14 Pr + 3] e x p [ - ( P r + 1 ) . 2] 1
+ 48rr Px/~-rlPr__,- 1) [ 2 P r ( P r 4 + 4 P r 3 + 6 P r 2 + 4 P r - 7 ) @ + ( 5 pr4 + 20 pr3 + 30 Pr 2 - 12 P r - 3)r/] e x p ( - P r r/2) erfc(r/),
(43)
where 30 Pr 5/2 + 30 Pr 2 + 140 Pr 3/2 + 140 Pr + 286fPT + 30
A(Pr) =
15"rr,,/Pr (fP~ + 1) 2 + "rr Pr(Pr - l) [(Pr + 1) 4 arctg0/Pr ) - 4w Pr].
When Pr = 1, (43) can be written as
( 48)
(44)
4
i 4 erfc(r/) - - exp(-r/2) f = 8 + 15---~ 15"rr 2 112(47"/4+ 12~72 + 3) erfc(w 2) - ~ ( T / 2
q- 1) e x p ( 2 r / 2 )
1
+ -~'-~- (6r/3 + llz/) exp(__.2) erfc(r/).
(43')
Substituting (31), (35) and (39) into (18), we have the following equation for ~(~)" l
0 ~21Fli) Or
0R 2
I
•
041/tli) a ~ aR
16 -
pr - 1
erfc(n)
+2~(Pr-1)7
(5 Pr - 3) erfc(fPrrl)
e x p ( - P r ~02)] sin 0 +
8r 3/2 sin 0 cos 0 ~-(Pr-
1) 2
× { - [ i 2 erfc(rl) - i 2 erfc(fPr-rt)] [~/Pr i e r f c ( P ~ r l ) - i erfc(rt)] + [ ~ - / P r i 3 erfc(fPTrl) - i 3 erfc(~7) + (1 - pr l/2)/6~/ww ] × [erfc(~/) - Pr erfc(fPTrl)] } +
r 3/2 sin 0 cos 0 ~f(rl) 2v/~
art
(45)
32
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
The solution of (45) may be written as ~/-tli) = gf~-a'r7/2Fl(*/) sin 0 cos 0 + a r e [ G ( * / )
+ H ( * / ) ] sin 0,
(46)
where G and H arise from curvature and displacement effects, respectively. The equations f o r F 1,G and H a r e F~ v + 2*/F;" - 10F('
x [x/~-i e r f c ( x / P r , ) - i erfc(*/)] + [ l x / f ~ i 3 erfc( P ~ * / ) -
i 3 erfc(v) + ( 1 - Pr t/2)/6x/-w]
X [erfc(*/) - Pr erfc(fPr*/)]} - 8 d f ( * / ) / d * / . G TM +
2*/G'"-4G"-
16 Pr- 1
(47)
[(2 P r - 1) erfc(fPr*/) - erfc(*/)]
+ 8[erfc(fPT*/)- 2~*/exp(-
Pr */2)] ,
H TM + 2~/H'" - 4 H " = 0.
(48) (49)
Equations (47) and (48), for which solutions can be obtained without matching consideration, were already considered in the previous papers. Elliott obtained the solution of (47), but only for Pr = 1. Here we obtain it for arbitrary values of Pr. Since the detailed expression for F1 is too lengthy to reproduce here, it is omitted. The solution of (48) was obtained by Gupta and Pop for arbitrary values of Pr, which can be written as 4 i 2 erfc(*/) 8 ( 3 P ~ + 2) i 4 erfc(*/) + G = fP~(1 - Pr) Pr - 1
+
1 - P~Pr 4 p r ( g g 7 + 1)
2 + (Pr -
1) 2
+
16(2 P r - 1) )2 i4 e r f c ( f P r rl) pr(Pr - 1
(Pr(Pr-3)@ 6
1
6~fPT(Pr-
1) 2
Pr+l*/2 2
3Pr-1) 8 Pr
erfc(,~pT*/)
[2 er(Pr - 3)*/3 - (7 Pr + 3)*/] e x p ( - Pr */2).
(50)
When Pr = 1, (50) can be written as 2 G = (2./4 + ,/2) erfc(*/) - _ _ * / 3 exp(__*/2).
(50')
The solution of (49), which is required to satisfy the matching condition (30), that is, H~ -
8
3~/w ~/PT{~PT + 1) */+ "'"
as */
--+ m
,
as well as the boundary condition on the surface, can be obtained as 1
H = fPr(fPr
+ 1) [ - 16 i 4 erfc(*/) - ~-r//~/~w + 1].
(51)
T. Sano, K. Kuribayashi / Natural conL,ection around a circular cylinder
33
Finally, the matching condition (30) d e t e r m i n e s the integral constants A ~k a p p e a r i n g in the second-order stream function ~ o ) , and we have ~o)
o/
.1.2
4Pr
r
sin 0 + v ~ B ( P r ) ( r 7 / 2 / r 2) sin 0 cos 0,
(52)
where B(Pr) is a function of Pr. T h e expression of B(Pr) is too lengthy to reproduce here and is omitted.
4. D i s c u s s i o n s
We now introduce new non-dimensional time and stream functions as
T = aE 2 Grl/2"r = ~/gfl( Tw - T~l /r,, ~" = ( u r ' / r g ) G r 1/2, ~ * ~ ) = ( E / G r ' / 4 ) ( U r r o / u ) ~ ~x)= ~O'/v G r 1/4,
(53)
where x denotes i or o. In terms of these new variables, the solutions obtained in the previous section may be written as k0'*(i)= 8T3/2Fo(rl) sin 0 + TT/2FI(~) sin 0 cos 0
+(T2/Grl/4)[G(~7) + H ( r / ) ] 4T 3/2 ~,~o)=
(54)
1
3 ~r~-wrvvrtvvr r - ~m-" +l)-Sinr 1 T2 - - - -
sin 0 + " " ,
O+B(Pr)TT/2r -2sin 0 c o s 0
1
+ 4 Pr Gr 1/4 r
sin 0 + - - .
(55) TI/2
t = erfc(,,/-P-)-r/) + T 2 f ( r / ) cos 0
Grl/4r/erfc(vrPTrl) + "'-
(56)
From (54) and (55), the velocity c o m p o n e n t s can be written as ,
T3/2
bl r
Grl/Zv/r °
T7/2
8 G ~ i ~ F o ( ' r / ) cos 0
G r l / 4 F l ( r l ) ( c o s 2 0 - sin20)
T2 G r l / 2 [ G ( r / ) + H ( r / ) - 16r/F0(r/) ] cos 0 + ' " ,
(57)
!
/g0 G r 1/2 p / r 0
- 4TF~(r/) sin 0 + ~1 T 3 F ( ( r l ) sin 0 cos 0 T3/2
+ l/------~ 2 Gr [G'(r/) + H'(r/)]
sin 0 + - - - ,
(58)
in the inner region and as U'r Grl/2v/ro
4 -
T 3/2
1
3f~-wg~-( P~V/~+ 1) G r 1/4 r 2cOS 0 T 7/2
1
- B ( P r ) G r l / 4 r3 1
T2
( C O S 2 0 --
sin20)
1
4 Pr G r 1/2 r 2cOs 0 + "-- ,
(59)
T. Sano, K. Kuribayashi / Natural concection around a circular cylinder
34
(a) Presen
(b)
,
t
"
Present
GuptaS,Pop '
0.04
-
-
EUiott
0.4
Elliott
0.2
0.02
L
i
i
i
i
i
i
i
I
I
I
,
I
I .I
1.2
13
I
I
1.2
r
1.3
I
1./.,
r
F i g . 2. T a n g e n t i a l v e l o c i t y d i s t r i b u t i o n s f o r P r = 0.7, 0 = 90 ° a n d G r = 500. ( a ) T = 0.01, ( b ) T = 0.1.
u'o Gri/2u/ro -
4
T 3/2
1
T 7/2
1
3~-v ,fPT (fPr- + 1) Gr 1/4 r 2 sin 0 - 2 B ( P r ) Grl/4 r3 1
T2
sin 0 cos 0
1
4 Pr Gr ]/2 r 2
sin 0 + " . . ,
(60)
in the outer region. It is seen that the tangential velocity in the outer region is of the same order of magnitude as the two radial velocities. In fig. 2, the tangential velocity distributions calculated from the composite expansion are shown for Pr = 0.7, 0 = 90 ° and Gr = 500. For comparison, the previous results of Elliott and by Gupta and Pop are also shown in the figure. It is seen from the figure, as might have been expected from fig. 1, that the direction of outer flow is opposite to that of inner flow. The difference between the present results and those of Gupta and Pop is of course due to the displacement effect of the inner flow. Figure 3 shows examples of the streamlines for Pr = 0.7. The cores of the vortices move upwards slightly as time goes on, although it is difficult to see in the figure. The expression for the skin-friction on the surface of the cylinder may be written as I ~urO ) ]J'~ ~TFt r,=r 0
PU2 Gr3/4 {2T1/2F~'(O) F2
sin 0 +
1TS/2F('(O)
sin 0 cos 0
q- }(T/GF1/4)[G"(O) + / - / " ( 0 ) ] sin 0 + " " },
(61)
where F~'(O), G"(O) and H"(O) may be written, using the solution given in the previous section, as F(~'(O) = 1 / ~ / ~ ( ~ P r + 1), G"(0) = -H"(0)
= 4 / f ~ - ( P ~ + 1).
(62)
The expression for F('(0) is too lengthy to reproduce here and is omitted. It is seen from (62) that, as was already found by G u p t a and Pop, the effect of curvature is to increase the skin-friction, but the displacement thickness has the effect of decreasing it and cancels the increase of skin-friction due to the curvature effect; that is, up to O(e), the curvature and
T. Sano, K. Kuribayashi / Natural concection around a circular cylinder
35
(a)
(b) Fig. 3. Streamline patterns for Pr = 0.7. (a) Gr = 500, (b) Gr = 5000. displacement thickness dO not influence skin-friction as a whole. F r o m (62), the skin-friction drag per length L of the cylinder, Dr, can be obtained as Dr= r0
Gr 3/4 - - T P~PT+I
l/2+o(T 5/2)+o(Gr-l/4T)
.
(63)
The expression for the pressure on the surface of the cylinder is (P')~'=~(,-Po-
pv 2 Gr r~ { ( T 1 / 2 / G r ' / 4 ) [ ½ G ' " ( O ) + 2F(;'(0)] (cos 0 + ~(T1/2/Grl/4)H'(O)(cos
O - 1) + " " },
1) (64)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
36
where P0 is the pressure at the forward stagnation point, and G " ( 0 ) = - 16(2¢~- + 1 ) / ~ - ¢ ~ - ( v F P T + 1), H"(0) = 16/~
PfPT(~P~- + 1).
(65)
The first term in (64) is due to the curvature effect. Inserting the values of G"(0), Fd'(0) and
H"'(O) into (64), we have pu 2 Gr3/4 rX {[2T'/2/~(,,/~
(P')r'=~o--P0
+ 1)](cos 0 - 1 ) + " " }.
(66)
From this, the pressure drag per length L, Dp, can be obtained as Do=
/xuL
Gr3/4[2v/-~wTI/2/(g~PT + 1) + " " ].
(67)
r0
It is seen that Dp is equal to Df in the present order of approximation. Both coefficients increase with time as T t/2 and with increase in Gr as Gr 3/4. It should be noted here that, in the first-order boundary layer theory, the pressure distribution on the surface is uniform because of zero velocity in the outer region, and the pressure drag is zero. Therefore it is natural to expect that, when Gr is large, the pressure drag is negligibly small and tends to zero as Gr ---, ~. However, the present investigation shows that this expectation is not correct and that Dp has the same order of magnitude as Df. Finally, the local Nusselt number Nu, which is defined as Nu = h r o / k , h being the heat transfer coefficient,
h = - ( k / r o ) ( a t / a r ) r = ,, may be calculated from (31), (42) and (43) as Nu
fPT Gr '/4 V~w T 1/2
I G r l / 4 T 3 / 2 f ' ( O ) cos 0 + ~1 + . " ,
(68)
where f'(0) =~-
Pr 2 + 5 P r + l l 30 Pr s/2 + 30 Pr 2 + 140 Pr 3/2 + 140 Pr + 2 8 6 ~
+ 30
15"rrvrPT( PfPr + 1) 2 [(Pr + 1)4 arctg(v/~-) - 4"rr Pr]). rr P r ( P r - 1)
(69)
When Pr = 1, this equation becomes f'(0) = ~
1 (9_ 1-~w )'
(69')
The mean Nusselt number N----uaveraged over the surface is obtained from (69) as I --~ Nu=w~o--/ N u d 0 -
~ - Or 1/4 1 v~w T 1/---T + 2 + ' ' "
(70)
T. Sano, K. Kuribayashi / Natural convection around a circular cylinder
References Elliott, L. (1970) Quart. J. Mech. Appl. Math. 23, 153. Gupta, A.S. and I. Pop (1977) Phys. Fluids 20, 162. Katagiri, M. and I. Pop (1979) Wiirme- und Stoffiibertragung 12, 73.
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