Computers & Fluids Vol. 17, No. 3, pp. 497-508, 1989 0045-7930/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press pie

A COMPUTER EXTENDED SERIES EXAMINATION OF MIXED CONVECTION IN A BUOYANT PLUME

ROLAND HUNT ~ a n d GRAHAM WILKS 2

~Dcpartment of Mathematics, University of Strathclyde, Glasgow, Gl IXH, Scotland and 2Department of Mathematics, University of Keele, Keele, Staffordshire, ST5 5BG, England

(Received 23 March 1988; /n revised form 10 October 1988)

Almtrtct--An examination of the use of series extension techniques is presented in the context of the mixed convection flow of a uniform stream about a horizontal line source of heating. The implementation is monitored at various levels of arithmetic precision and detailed comparisons are made with full numerical solutions over a comprehensive range of Prandtl numbers. An attempt is made to assess the level of confidence which may be placed in the results obtained by these methods.

1. I N T R O D U C T I O N

In a recent article [I] numerical solutions covering a wide range of Prandtl numbers were presented for the vertical mixed convection flow of a uniform stream about a horizontal line source of heating. The solution algorithm exploited the acknowledge limiting similarity states, delineated by Afzal [2], of a weak plume near the source evolving to a pure convection buoyant plume far downstream. Using the method of continuous transformation [3] accurate evaluations of velocity and tem- perature distributions over the entire semi-infinite region downstream of the source were obtained. Comprehensive graphical representations of the physical characteristics of such flows, based on the detailed numerical solutions, were presented in Ref. [1].

The availability of such accurate information affords the opportunity to examine more closely the possible values of series extension techniques for predicting the details of the flow field. As pointed out by Afzal the velocity and temperature distributions near the source may be regarded as buoyancy induced perturbations about the weak plume similarity state. This appraisal of the flow in that vicinity may be used to formulate regular perturbation series solutions in a characteristic coordinate C which reflects the local relative importance of buoyancy, inertia and viscous forces. The series solutions are held to be valid for small ~. Direct use of the series confirms that satisfactory prediction of the flow field is only achieved for C "~ C0 where C0 is the radius of convergence. However Afzal [2], using a limited number of terms has indicated that by appropriate re-organisation of the series it is possible to obtain estimates of flow characteristics at all stations downstream of the source up to and including the ultimate pure free convection plume which is attained as C --" oo. In this paper this interesting work is developed and extended. A more detailed assessment of the numerical features of the implementation of the series extension method is presented. Particular attention is paid to examining the success of the method over the range of all Prandtl numbers. Against the background of available complete numerical solution an attempt is made to identify some of the elements of good practice if confidence is to be placed in the results which may be obtained for a variety of heat transfer problems using these methods.

2. G O V E R N I N G E Q U A T I O N S A N D SERIES S O L U T I O N S

The steady flow of a vertical uniform stream of velocity U~o and temperature T~ past a 2-D horizontal line source is to be examined. As pointed out by Wood [4] the flow has its origins in a diffusive zone in the immediate vicinity of the heat source. Out of the diffusive zone evolves a thin "wake" region in which diffusion and convection are comparable. On the assumption that the scale of the diffusive zone is small in relation to the semi-infinite region downstream of the source

497

498 ROLAND HUNT and GRAHAM WILKS

the governing boundary layer equations are

Cu Cv ?x + ~ = 0 .

au Cu O2u u ~ + r a y = V~y2 + gf l(T - T~),

gT CT O2T U - ~ - "+ " = K ox t ~ @2"

tl)

(2)

(3)

Gr~ (gflQ )2 x = ~ = (pc.)2vU~

where Grx and Rex are the local Grashof and Reynolds numbers given by:

Grx = gflQx-----~3" Rex U~ x p c p v 3 ' V

then the transformations appropriate to small values of ~ are:

=F d/ = (vV~.x)t;2f(¢, q); T -- T~c LU--~x..] p c p v 0(~ , 17); 1,1 L vx j

whereas away from the centre-line:

?.u ~T - = 0 on y = 0 , (4) ~y Cy

u ~ U ~ , T ~ T ~ . as y - * o c . (5)

An energy conservation constraint is obtained when eqn (3) is integrated across the mixed convection region at any downstream station, namely:

pc, u(T - T~)dy = Q, (6)

where cp is the specific heat of the fluid at constant pressure, p is the density of the fluid in ambient conditions and Q is the heat released per unit length of thermal source. Accordingly the energy transported across any plane is exactly that introduced into the flow by the source at x = 0.

If the local relative importance of buoyancy, inertia and viscous forces are characterised by the non-dimensional coordinate:

(7)

(8)

(9)

The introduction of the stream function qJ automatically satisfies the continuity eqn (1). The momentum and energy eqns (2) and (3) now read:

, ¢,,2 0 f , , , + if./',, + = ¢{fnf¢, - f c f , , } (10) ! - O.. + g O . f + O f . ) = ~ { f .O; - f eO , }, ( i 1) 17

where a = v/x is the Prandtl number.

Equations (10) and (11) are to be solved subject to the boundary conditions:

f(~, 0) = f , , ( ~ , 0) = 0,({, 0) = 0 (12)

f , (~,r t )-- , l and 0(~,q)--*0 as q - . ~ . (13)

Here (u,v) are velocity components associated with increasing values of coordinates (x, y) measured vertically from the heat source and normal to the axis of symmetry respectively. T is the temperature and g, fl, v and x are the acceleration due to gravity, the coefficient of thermal expansion, the kinematic viscosity and the thermometric conductivity. The Boussinesq approxi- mation is assumed to hold and the heat source is such as to generate positive buoyancy, relative to ambient conditions, in the fluid above the source.

The flow is assumed to be symmetric about the x-axis and consequently:

Mixed convection in a buoyant plume 499

The third term on the L.H.S. of eqn (10) indicates the growing influence of buoyancy on the developing flow field as ~ increases. The system of eqns (10) and (11) lend themselves readily to a regular perturbation series solution of the form:

f(~, r/) = ~. f.(r/),"; 0(¢, t/) = ~ 0.(r/)E", (14) n~O n--O

where the notation E = ~ J;2 has been introduced to coincide with that of the review of series extension techniques presented by Van Dyke [5]. Thus the leading terms of the series solutions are the closed form "weak plume" similarity solutions appropriate at ~ = 0 i.e.

f 0 ( ~ ) = ~ ; 00(~) = o exp - - i - " 05 )

The equations governing the higher order terms n >/1 can be conveniently written as:

- ~ f ;J 'o + ~n + l).f,f~;

1 . - I = -0 ._~+~ ~, { r f ; f ' _ , - ( r + l ) f , f Z _ , } = - F (16)

1 t t s 1 l o : + iO~fo - ½(n - 1)O,,fo + ½(n + 1)Oof. + ~Oof. O"

with boundary conditions:

f . (0) = f ~ (0) = 0~, (0) = 0;

l n - - I = ~ ~ {(r -- I) O,f',_, - (r + l)f,0:,_,} - G,

r--I

(17)

f~(r/),0.(r/)-,0 as r / ~ c o . (18)

3. S O L U T I O N S F O R f ~ ( n ) A N D 0.(t/)

A shooting-matching technique [6] has been used to evaluate f.(t/) and 0.(t/). With u. =f~, v. =f'~', w. = 0~ and supposing f , , u,, v,, 0,, w,, r = 1, 2 . . . . . n - 1 as known then ¢qns (16) and (17) may be written as the linear system:

where: I:] Un y_ = v . , K ( n ) =

O. Wn

.F '= K0/)Z + ~(r/), (19)

o , o o o ] I 0 0 0 0 0

--½(n + l)f~' ½nf~ --½fo 0 0 , f[(~)= . 0 0 0 0 I

(20)

If y ffi yp + sy_~ + ty_2 where ~ , yl and Y_2 are the solutions of:

.v~ -- K(r/)~ + g_;

y_; - K(r/) y_,;

y ~ = K(~/) Y_2;

D(0) = (0, 0, o, o, 0) r

y_l (0) = (0, 1 ,0 , 0, o) r

y,(0) = (0, 0, 0, 1,0) r, (21)

then ~ is a solution of system (19) satisfying the boundary conditions on ~ - 0 for all s and t. To satisfy the boundary conditions at ~ = co namely u.(co) = 0.(co) = 0 requires:

u~.(oo) + su,,(co) + tu~(oo) = 0

0~,(oo) + s0t.(co) + tOz.(co) = 0. (22)

500 ROLAND H U N T a n d GRAHAM W1LKS

Table I

Relative Distance o r/~ h accuracy ratio

I,'100 100 I,'18 7 × 10 ~ * 1.10 32 2.,225 3 x 1 0 ,0 6 × 10

I 10 1/180 2 x 1 0 '~ 7 x 1 0 6 10 10 1/180 5 × 10 -~ 4 x 10 '

I00 10 1/180 7 x lO "'7 *

Since f~(r / ) = uu(r/) = vz~(r/) - 0 for all t/

u..(oo). [0.~(oo) + s0~(oc)] s - uj~(oo)' t = - 0z~(oo) (23)

Equations (21) may be solved using a standard routine. However to facilitate analysis of the results it is required that the truncation error be regular and not randomised. This means that the step length h must not be calculated using tolerance criteria but specified beforehand. A Runge-Kutta-Fehlburg method was used which enables an estimate of the error to be made. The procedure is stable provided that the eigenvalues of K(r/) are greater than ~- - 3/h. Formally this can be shown to require:

In practice these are weak upper bounds and experience of the full numerical solutions indicates that ~ -- 10, ~ >/1 ; ~ -- 10/x/~, o < 1 are acceptable. Working in quadruple precision up to 30 profiles forfn(~) and 0,(~) were obtained for each a. By defining a distance ratio -- 12nd last grid value I/Imax value l and monitoring it for each profile it was possible to ensure that such ~ values were satisfactory. If the distance ratio is comparable with the tolerance within which the computations are aimed then ~ is acceptable. Table l contains specific details for various Prandtl number computations. Here the relative accuracy and distance ratios quoted refer to maximum values occurring for the final profile. The * in the final column indicates that for profiles n > 20 the distance ratio estimates are contaminated by "loss of significance", however for n ~< 20 the distance ratio is less than 10 -6 . Comparison of results using the series representation with the full numerical solution indicates that f~,(0), 0,(0), n = 1 , 2 , . . . , 30 are just as accurate as for other a.

4. RADIUS OF CONVERGENCE

The radii of convergence of the series solutions are determined by the position of the nearest singularity. A knowledge of the precise location and nature of the singularity plays a significant role in the subsequent reorganisation of the series with a view to extending its range of validity. If c~ is used to denote un(0) or 0n(0) then by D'Alembert's ratio test the radius of convergence of the series I:c~E n is given by

~;o = l i r a c n _ l . ( 2 5 ) n ~ o C n

A crude estimate of this D'Alembert fimit can be obtained using a Domb-Sykes plot (Domb-Sykes [7]) of inverse ratios cn/cn- ~ against 1/n and extrapolating to l /n = O. For functions of the form (~ + ~)~ the extrapolation is precisely linear and:

Cn : : [1 1 + ~ 1 (26, C n - I

Thus for most of the functions generally encountered the slope of the Domb-Sykes plot indicates the nature of the singularity and the inverse of the intercept its location.

In Fig. 1 the Domb-Sykes plots for various a are illustrated for both velocity and temperature series appropriate to the axis of symmetry. Each clearly has a common nearest singularity for a

Mixed convection in a buoyant plume 501

-0.25

-0.50

- 0 . 7 5

; IJ _,.oo

- 1.25

- 1 . 5 1 )

o -- f o ~ . ' ~

°" o. o

t

/ o

/o /o

/o/°°

/

/ / o

/o /o

/ o ° /o /

o

o

o

o

o

o

o o

o" = 0 . 0 1

o o

=" " 0 . I

- .~==~o-~ ~" o

o

I

/ /

/ o / o °

/ ~ °°

/ /

/ / o

/ o / o °

o /k / /

/ / d

/ o /

- / o

/ o °

/ o °

-/ /

Ibl I I

0 . 3 0 0 .1

o

o

o , = 1

o

o

o = 1 0

o

o

¢, = 1 0 0

(a)

-I.7~ I l l I o a~ 0.2 0.2 0.3

1

n

Fig. 1. Domb-Sykes plots of (a) velocity coe~cients, (b) temperature coefficients. The dashed lines are the asymptotes as predicted by Neville's algorithm.

given o and in all cases the slope is consistent with a singularity • = 1/2 for velocity and • = 3/2 for temperature. Provisional estimates of -E0 may be obtained from Fig. I. However more precise estimates can be obtained using Neville's algorithm for interpolating polynomials. Figure 2 illustrates the Neville tables using u,(0) for a = 1. The entries in the table p~,(0) are the polynomial interpolates of degree r passing through the data points 1/,, 1 / , - 1 . . . . . 1/n- r in Fig. 1 evaluated at zero. The value and accuracy of -~o is determined by observing the convergence in the right most columns. In Table 2 these results are compared with the exact values obtained from the full numerical integration of the adverse case to the point of breakdown from which ~0 is calculated.

5. S E R I E S R E O R G A N I S A T I O N - E U L E R I S A T I O N - C O M P L E T I O N

For all Prandtl numbers the nearest singularity has been found in Section 4 to be on the negative real axis. The singularity in each case may therefore be mapped to infinity by transforming to the new variable ~/(q + E). Further advantage however can be gained by extracting the dominant

CA.F . 17 !~ -F

502 ROLAND HUN1 and GRAHAM WILKS

0 1 2 3 4 5 6

3 2.61631061 4 2 . 1 7 3 2 9 9 3 6 0 . 8 4 4 2 6 5 6 3 5 1.97327752 1.17319015 1.66657694 6 1.85858414 1.28511723 1.50897139 1.35136585 7 1.783952;3 1.33616426 1.46378181 1.40382904 1 . 4 4 2 6 5 1 4 3 8 1.73140979 1.36360924 1.44594420 1.41621485 1.42890065 1.42065019 9 1.69236697 1.38002436 1.43747729 1.4205434B 1.42~9~427 1.42359717 1.42~507066

10 1.66219093 1.39060661 1.43293559 1.42233829 1.42503051 1.42410675 1.42444647 11 1.63815714 1.39781926 1.43027621 1.42318451 1.42466539 1.42422724 1.42432764 12 1.618556<57 1.40295148 1.42861259 1.42362173 1.42449616 1.424259'24 1.42429124 13 1 . 6 0 2 2 6 2 3 7 1.40673075 1.42751668 1.423B(:~.366 1.42440800 1.42426695 1.42427595 14 1.58850025 1.40959266 1.42676414 1.42400484 1.42435779 1.42426742 1.42426e0"3 15 1 . 5 7 6 7 2 0 9 6 1 . 4 1 1 8 1 0 9 0 1 . 4 2 6 2 2 9 4 7 1 . 4 2 4 0 9 0 7 9 1 . 4 2 4 3 2 7 1 4 1 . 4 2 4 2 6 5 8 2 1 . 4 2 4 2 6 3 4 3 16 1 . 5 6 6 5 2 3 6 7 1 . 4 1 3 5 6 4 3 7 1 . 4 2 5 8 3 8 6 2 1 . 4 2 4 1 4 4 9 3 1 . 4 2 4 3 0 7 3 5 1 . 4 2 4 2 6 3 8 3 1 . 4 2 4 2 6 0 ~ 2 17 1.55760898 1.41497396 1.42554592 1.42418001 1.42429401 1.424261C~ 1 . 4 2 4 2 ~ 1 18 1.54974869 1.41612376 1.42532215 1.42420327 1.424215469 1.42426044 1 . 4 2 4 2 4 ~ 19 1 . 5 4 2 7 6 5 8 0 1 . 4 1 7 0 7 3 6 8 1 . 4 2 5 1 4 7 9 7 1.42421111'00 1 . 4 2 4 2 7 7 9 7 1 . 4 2 4 2 5 q 1 8 1 . 4 2 4 2 ~ 6 4 7 20 1 . 5 3 6 5 2 0 8 7 1 . 4 1 7 8 6 7 3 3 1 . 4 2 5 0 1 0 2 4 1 . 4 2 4 2 2 9 8 0 1 . 4 2 4 2 7 3 0 3 1.424251120 1.4242 '~5Bq 21 1 . 5 3 0 9 0 2 6 0 1 . 4 1 8 5 3 7 0 9 1 . 4 2 4 8 9 9 8 3 1 . 4 2 4 2 3 7 3 3 1.424269"32 1 . 4 2 4 2 5 7 4 3 1.4242~d5~1 22 1 . 5 2 5 8 2 1 0 0 1 . 4 1 9 1 0 7 3 8 1 . 4 2 4 8 1 0 2 1 1 . 4 2 4 2 4 2 6 3 1 . 4 2 4 2 6 6 4 8 1 . 4 2 4 2 5 6 8 4 1 . 4 2 4 2 5 5 2 7 23 1.521202"d6 1 . 4 1 9 5 9 6 8 8 1 . 4 2 4 7 3 6 6 7 1 . 4 2 4 2 4 6 4 0 1 . 4 2 4 2 6 4 2 9 1 . 4 2 4 2 5 6 3 9 1 . 4 2 4 2 ~ 5 1 3 24 1 . 5 1 6 9 8 6 6 2 1 . 4 2 0 0 2 0 1 2 1 . 4 2 4 6 7 5 7 2 1 . 4 2 4 2 4 9 0 9 1 . 4 2 4 2 6 2 5 7 1 . 4 2 4 2 5 6 0 6 1 . 4 2 4 2 ~ 5 0 6 25 1 . 5 1 3 1 2 2 7 0 1 . 4 2 0 3 9 8 4 9 1 . 4 2 4 6 2 4 7 6 1 . 4 2 4 2 5 1 0 3 1 . 4 2 4 2 6 1 2 2 1 .42425.582 1 . 4 2 4 2 5 5 0 5 26 I.~0956840 1.42071105 1.42458180 1.42425244 1.42426015 1 .42425~4 1 . 4 2 4 2 ~ 7 27 1.50628791 1.42099507 1.42454531 1.42425345 1.42425929 1.42425,553 1 .4242~11 28 1 . ~ 0 3 2 5 0 7 1 1 . 4 2 1 2 4 6 4 3 1 . 4 2 4 5 1 4 1 2 1 . 4 2 4 ~ 4 1 9 1 . 4 2 4 2 ~ 8 6 1 1 . 4 2 4 2 ~ P 4 5 1 . 4 2 4 2 e ~ 1 8 29 1. 50043069 1 . 4 2 1 4 6 9 9 4 1. 42448729 1. 4 2 4 2 5 4 7 2 1. 42425806 1. 424251~1.1 1. 4 2 4 2 5 5 2 6 30 1.49780532 1.42166955 1.42446407 1 .424~811 1.42425761 1.42425540 1.4242~535

Fig. 2. Neville tables used to determine -~0 when ~ = I using velocity coefficients. Convergence in the columns indicates that ~, = 1.424255.

behaviour o f f ( l , r/) and 0(¢, r/) at large ~. When a balance between buoyancy, inertia and viscous forces is ultimately achieved all the terms on the left hand side o f e q n s (10) and (11) are comparable in magnitude. Pre-supposing dominant b e h a v i o u r s f ... ~ ", 0 .-. c ~, q ~ E" implies from eqn (10) that:

a - 3 c = 2 a - 2 c = I + b .

In terms o f f and 0 the energy conservation constraint reads:

f ~- f ~ O d q = 1 ,

and so also

(27)

(28)

a + b = 0. (29)

Equations (27) and (29) require a = - b = - c = 1/5. These values are entirely consistent with the known behaviour o f centre-line velocity and temperature for a plume as ¢ J:5 and ¢ -J/10 respectively.

Once again denoting u.(0) and 0,(0) in general by c. then the transformation to be invoked is:

A. C ~' c .c" = c; , (30) n=O n=O

where 7 is 0.4 for the u.(O) series and "/= - 0 . 2 for the 0.(0) series. Equating powers o f E gives:

Ca ~" E~ --~r-O (31)

Table 2

Numerical a Using u.(0) Using 0.(0) results

1/100 9.15250__. 1 9.158_+ I 9.15253_+3 1/10 3.29118:i:2 3.293_+ I 3.2912060+2

1 1.42455 _+ 1 1.425 _+ I 1.4242555 _+ I I0 0.82014 -+ 3 0.82013_+I 0.8201012_+3 I00 0.618-+I 0.6176-+ ] 0.617501_+4

Mixed convection in a buoyant plume 503

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- 1 , ~ 0 2 9 3 8 6 9 2 3 0 6 0 6 1 1 4 6 ` 5 4 E - 0 4 6 , 13916336317000166771[-0`5

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- 5 . 113103504~ )'~03041`513~76544930~-02 1. 7 4 9 6 8 7 1 8 2 ~ f 3 b O 1 9 4 9 4 9 2 7 3 0 7 B b 3 E ~ - 0 2

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- 4 . 1 7 3 5 8 2 1 6 6 1 ~ 94298935, .530= ] 2 7 3 8 1 9 1 [ - 0 4 2 . 2 2 0 3 6 3 4 7 0 4 . ~ eO 1 3 b 0 9 3 2 4 5 4 5 2 1 1 6 4 ~ E - 0 4

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2i 2 3 2 3 4 9 8 3 0 3 ~ . 3 2 2 8 4 7 0 2 1 8 4 4 9 3 7 9 0 3 1 E - 0 6 - 1. 2 9 8 1 0 1 3 4 7 9 ~ f 9 6 7 2 0 5 2 9 9 ~ 1 7 [ - 0 6

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2 . 6 1 ~1053004S~&66774742S3 l f f g f f641 ~ - 0 7 - 1. ` 5 4 & ~ b 3 8 6 8 3 ~ & 0 7 3 9 1 8 9 3 2 8 4 7 0 0 ~ 6 2 ~ - 0 7

9 . 16392~474~!~1S909626912F~26.531 lb1~-08 - 5 . 4 4 4 7 9 3 1 6 4 L 0 6 2 9 0 ~ 1 4 4 8 0 3 0 ~ B ~ ' ~ - 0 ~

3 . 2 4 3 3 7 2 6 2 4 7 ~ 7 7 b 0 4 5 9 6 1 ~ 1 0 0 0 8 1 1 ~ - 0 8 - 1 • 9 3 b b O 4 2 = 2 " J L ~ J 0 3 4 6 4 1 6 0 4 2 8 f f g ~ 8 7 6 3 E - 0 8

1 . 1 5 ~ 8 7 4 1 0 6 7 _ . ~ 7 9 5 ` 5 ~ 5 0 3 5 1 3 7 1 4 3 2 1 I E - 0 8 - 6 . 9 4 8 8 6 9 3 1 4 ~ ' 0 9 3 0 & 3 3 3 9 7 0 7 7 0 ~ 1 6 5 0 2 8 - 0 9

4 . 1 7 4 ` 5 8 1 2 3 7 2 1 5 4 5 : 5 3 1 2 2 3 6 2 7 2 b ~ 5 5 0 2 4 8 - 0 9 - 2 • 5 1 2 3 3 2 9 8 8 7 ~ 1 6 2 1 0 9 6 4 3 4 5 2 7 9 9 7 7 4 3 0 8 - 0 9

(b) S I N S L E

0 8 . 2 8 6 1 3 ` 5 1 9 6 7 8 - 0 1 1 l . 1 7 4 4 7 7 4 8 7 8 E - 0 2 2 - 1 • 8 4 3 2 4 5 0 0 1 5 8 - 0 3 3 - 2 • 3 3 9 8 4 4 3 9 1 9 1 [ - 0 3 4 - 2 . 0 4 3 1 4 8 7 3 , 5 5 t [ - 0 3 5 - 1 • 7 1 4 1 0 3 ~ 5 6 7 8 - 0 3 6 - 1 . 4 5 4 ~ - 1 3 2 8 7 9 8 - 0 3 7 - 1 • 2733 `57658~A [ -03 8 - 1 . 1 b 9 9 5 t ` 5 1 6 2 E ' - 0 3 9 - 1 • 1 4 8 3 2 6 9 0 4 5 8 - 0 3

10 - 1 . 2 1 5 5 9 7 6 1 4 & E - 0 3 11 - 1 . 3 7 3 ~ 9 9 5 b 1 [ - 0 3 12 - 1 • ` 5 9 8 6 0 3 4 4 1 4 E - 0 3 13 - 1 • 8 2 0 3 9 6 8 6 4 8 8 - 0 3 14 - 1 • 8 7 8 8 3 3 1 b ` 5 4 8 - 0 3 15 - 1 • 476": '09b 1 0 7 8 - 0 3 16 - 1 • &`5941 ~ 5 9 ~ E - 0 4 17 2 . 6 6 4 5 1 2 4 8 9 0 8 - 0 3 18 7 . 6 2 9 5 b 0 3 0 6 7 [ - 0 3 19 1 • 4 9 7 0 9 3 6 8 1 2 1 £ ' - 0 2 2 0 2 . 3 7 0 9 0 4 9 4 4 8 [ - 0 2 21 3 . O & 2 2 2 6 4 3 7 0 E ' - 0 2 2 2 2 . 5 2 7 9 " 3 1 0 7 b b E - 0 2 2 3 - 1 • 2 2 7 5 8 0 4 7 6 6 8 - 0 2 2 4 -1.2904`5094151[-01 2 5 - 4 . 1 2 1 0 5 2 0 2 6 7 8 - 0 1 2& - 1 . 0 4 1 3 0 4 7 0 7 5 8 ÷ 0 0 27 - 2 . 3 5 2 8 6 9 2 7 2 2 8 ÷ 0 0 28 -5 . 00441789638÷00 29 -1 • 02010&22021[+01 30 - I . 9992626190E÷01

DOUBLE

8 . 2 8 b 135~0433499bb977 b41[ - 0 1 w

1 . 1 ~ 0 4 7 4 3 7 5 0 7 S ` 5 3 1 2 1 4 1 1 8 - 0 2 - 1. I~B~270`52B6~b06 ' : '292`51K-03 - 2 • 3[77 ̀ 57 ̀51 "~9411291 ̀ 5 0 3 0 7 8 - 0 3 - 2 . 0 ~ 0 1 ` 5 4 9 0 0 4 8 7 9 1 8 8 1 6 f f T w - 0 3 - 1 . 7 1 2 3 1 9 0 3 8 9 2 7 1 1 8 9 6 2 3 3 7 1 r - - 0 3 -- 1 • 4 ~ 5 9 3 7 1 3 9 9 2 1 t 89575121 [ - -03 - 1 • ~ 9 8 4 2 1 1 0 5 6 4 1 0 1 5 8 3 6 8 8 - 0 3 - l j ~ 3 2 4 1 b ` s b 3 0 2 2 6 8 1 9 1 9 3 0 F - 0 3 - 8 . 9 1 8 1 b389`51 ̀ 5 5 3 6 1 4 5 8 3 7 8 - 0 4 --7 . 7 8 9 2 0 5 9 ` 5 0 4 ` 5 3 3 4 9 7 ` 5 9 0 4 8 - 0 4 - & . 8703`5197"~2~4972040`598- -04 - b . 1 1 2 8 3 5 ` 5 7 ~ O b 9 0 0 6 9 4 4 4 3 F - - 0 4 - ` 5 . 4 8 0 9 0 0 b 1 9 b b 2 4 9 b b 9 1 7 ! [ - 0 4 - 4 . 9 4 8 0 2 8 0 ` 5 9 ~ ' 3 1 1 3 3 7 0 2 ~ - 0 4 - 4 . 4 9 4 2 9 0 7 5 2 1 0 4 6 4 6 5 6 1 ` 5 0 ~ - 0 4 - 4 . 1 0 4 ` 5 1 5 0 5 1 7 9 4 5 3 8 4 2 5 3 8 ~ - 0 4 - 3 • 7 bb997919`53`5069639`53; [ -04 - 3 • 4 7 2 6 0 3 1 5 6 7 0 5 6 3 3 9 1 1 8 1 [ " 0 4 - 3 . 2 1 4 1 1 7 2 4 2 7 ~ 0 4 8 7 4 ` 5 4 3 , ` 5 w - 0 4 - 2 • 9 8 ~ 7 ` 5 4 4 7 1 6 7 9 9 ` 5 6 9 ~ 7 ~ 8 1 [ - 0 4 - 2 . 7 8 2 9 6 7 2 8 9 & 121~840634S~-04 - 2 . 6 0 1 8 9 5 1 0 2 2 b 12471 ~b I b l~ -04 - 2 . 4 3 9 4 7 & 3 5 9 5 ~ ; 7 5 ~ b 8 1 3 8 8 - 0 4 - 2 . 2 9 3 1 5 6 4 4 7 7 1 4 1 7 9 7 4 b ~ 0 t ~ - 0 4 - 2 . 1 6 0 8 0 9 0 8 2 4 b 1~03648231~--04 - 2 . 0 4 0 6 ` 5 2 6 2 2 4 0 2 3 8 3 7 6 & 141 [ -04 - 1 . 9 3 1 1 8 4 9 1 2 7 2 7 7 3 9 8 3 8 2 7 1 [ - 0 4 - 1 • 8 3 1 1 3 2 3 7 7 6 8 4 4 ~ 3 1 8 7 8 4 [ - 0 4 - 1 • 7 3 9 4 0 9 7 U 3 3 9 0 ~ 4 1 3 1 4 0 4 1 [ - 0 4 - 1 • 6 ` 5 ` 5 0 8 8 7 8 2 4 f f b 7 8 2 2 4 8 2 7 w - 0 4

QUADRUPLE

8 . 2 8 b I ; ] ' ~O4~k14~ . .~OO 1 1 7 1 4 ` 5 0 3 6 0 7 `52~-.O 1 1. 19047437~131~b`597 `5 , . . ~ . . 3~3916924E-02

- 1. ` 5 ~ 2 7 0 ` 5 2 1 ~ 1 ~ 14`5690`5.% 10922&, .~ 11 [ - 0 3 - 2 . 377.5751 ` 5 9 4 ~ 2 0 ~ 3 4 1 4 4 2 7 2 8 8 8 3 4 3 ~ 4 8 - 0 3 - 2 . 0701 ̀ 549004r~b 1 0 7 3 7 1 7 0 8 3 0 5 8 9 1 B f f 5 [ - 0 3 - 1 . 7 2 3 1 9 0 3 8 9 ~ 7 ` 5 2 3 0 5 ` 5 1 `57`5 J -44`50474bE-03 - 1 .43 `593713 'M~1877 ̀ 5 4 1 6 3 6 6 4 4 7 1 0 ~ 4 4 3 ~ - - 0 3 - 1 . 2 0 ~ 8 4 2 1 1 0 ~ 5 1 1 2 2 1 4 0 8 W 7 1 0 8 3 6 6 9 2 8 8 - 0 3 - 1. 032:41 b,562~ 9 2 1 3 4 6 5 9 2 5 0 6 ~ 0 1 5 6 4 2 2 [ - - 0 3 - 8 . 9 1 8 1 6 3 8 ~ t S b b 8 9 ` 5 7 9 7 2 5 7 3 6 9 3 0 1 b l S E - 0 4 - 7 . 7 8 9 2 0 5 9 ~ f 4 3 3 1 6 0 7 0 0 2 3 0 0 9 9 9 8 4 6 2 8 8 - 0 4 - 6 . 8703`519~ 3b 1 8 9 3 0 5 6 2 6 9 4 7 f f 2 6 8 8 9 7 0 E - - 0 4 - b . 11~83.5~, ~ b 2 7 8 8 3 7 7 9 4 6 9 4 ` 5 7 4 4 f f ~ - - 0 4 -`5. 4 8 0 9 0 0 b ] ~ 2 7 0 4 3 7 0 7 8 f f 3 6 0 4 8 0 7 3 4 1 2 [ - 0 4 - 4 . 9 4 8 0 2 8 0 2 $ 3 0 4 4 3 0 6 7 1 8 9 ` 5 3 0 1 0 7 4 ~ - 0 4 - 4 . 494290"~ T l b 2 3 3 4 7 b O ` 5 3 7 b O 4 8 4 3 8 6 9 [ - 0 4 - 4 . 1 0 4 5 1 5 0 r O 7 0 4 5 4 8 4 0 ~ l ~ i b ~ 3 9 1 [ - 0 4 - 3 . 7 6 6 9 9 7 9 ~ 7 0 7 9 9 6 5 0 8 7 1 1 2 3 9 4 2 6 4 f f T E - 0 4

- 2 • 6 0 1 8 9 ~ 8 9 7 1 4 4 4 0 8 1 1 9 b 4 5 2 2 2 8 9 b ~ - 0 4 - 2 . 4 3 9 4 7 ~ 1 b031 ~ S b 3 7 0 f f b b 6 0 4 2 b 2 0 1 $ 9 [ - 0 4 - 2 . 2931 `5~b&3927 b 104 f f92261227 '368041e -04 - 2 . l b O e O ~ 1 7 8 1 5 4 8 0 8 0 9 3 7 6 0 1 2 1 1 8 5 7 4 8 - 0 4 - 2 . 0 4 0 6 ` 5 2 ~ b 5 2 1 3 0 9 3 2 0 0 4 2 1 1 2 b k 2 2 1 5 2 1 [ - 0 4 - 1 • 9 3 1 1 ~ 1 0 b 8 5 9 4 0 ` s q 1 5 9 4 1 `5S 12~b�O�b�l['-04 - 1 • 8 3 1 1 ~ 9 0 9 3 2 1 8 2 6 0 6 6 2 9 8 6 3 6 2 ~ 1 ~ ¢ ) b [ - 0 4 -1 • 7 3 9 4 ~ ) 4 b t OO4`59225I~&`59367'e~,034E-04 - 1 • 65`50JSb 1 0 6 5 0 3 1 `5479011 ̀ 5187 b 7 ` 5 7 0 3 8 - 0 4

Fig. 3.(a) Velocity coefficients u,(0), n = 0, 1 . . . . . 30 for a = 0.72 at single, double and quadruple precision. (The jagged line indicates the last digit coincidence of single and double precision with quadruple precision.) (b) The Eulerised coefficients based on u,(0) of (a) at single, double and quadruple precision. (The jagged line indicates the last digit coincidence of single and double precision with

quadruple precision.)

In fact to examine the reorganised series the coefficients A, must be evaluated. The inversion leads to:

A, = r-~0 ( - 1)r 7 --r n E g - ' - ' cn_ ,. (32)

It is this process o f inversion o f coefficients which can lead to significant round-o f f errors and which therefore requires addit ional w o r k i n g precision in evaluat ing the initial coefficients c~. Figure 3(a) illustrates this point for the one case considered by Afzal namely a = 0.72. Profiles have been com p u t ed at single, double and quadruple precision respectively and the velocity coefficients at the axis listed. In Fig. 3(b) the transformed coefficients are listed and the significant figure agreement between single, double and quadruple precision indicated. Unfor tunate ly an error in Afzal 's second

504 ROLAND HUNT a n d GRAHAM WILKS

coefficient, for which an analytic representation is available, distorts any formal comparison. It is clear however that it is essential to be aware of the possible effects of round-off error in assessing the accuracy of results.

As the new series has unit radius of convergence, then:

limc-:" ~ c , c "= ~ A,. (33)

Consequently the sequence of partial sums of A, for each ~ and the associated coefficients should predict the known initial values for the pure free convection plume valid in the limit ~ --, cc. When only a finite number of terms are available an attempt to assess the contribution of the omitted terms is made by the process of series completion. In general any series of the form

.4

~.~ Cn (.n n = 0

can be completed by assuming that the remainder of the series behaves like K(~ + ~0)" where ~ is the nature of the singularity at ¢ = E0. the constant of proportionality is chosen so that the coefficients of cN coincide i.e.:

CN(0 ~'

The completed series is then:

In terms of the new expansion parameter ¢/, + *o the sum part of the right hand side of eqn (30) has unit radius of convergence and is singular of the order ~, say, at the point 1 on the positive axis. The general completion argument of eqn (35) holds and as a result:

lim e - ' c,~" = A. -,, ~ = 1 A, - ( - 1)N-"A, . (36)

Again note that c, refers to either u,(0) or 0,(0) of the series solutions. For the former ), = 0.4 and for the latter , / = - 0 . 2 and each case will generate its associated A,. However, since perturbations about the free stream velocity and temperature distributions at large ¢ are known to be 0(¢-us) = 0(E-2,5) = 0(1 -E/E + E0) 2/5 then a = 0.4 is a common value. Respective estimates of the initial values for the velocity and temperature distribution of the pure free convection plume may now be obtained. The results of these evaluations for various Prandtl number appear in Tables 3 and 4 where the completed series estimates are compared with the known exact values. Also included in the tables is an indication of the influence of the precision with which the nearest singularity is established. Approximate singularity locations 9, 3, 1, 1, 1 have been used in place of the accurate values in Table 2. These display various levels of discrepancy from the true values. The limited sensitivity to quite gross approximations is rather surprising but encouraging.

Further advantage may be taken of the knowledge of the asymptotic behaviour at large E. Since expansions at large E are based on powers of E-2,5 then

fRO

may be anticipated to behave as:

bt b2 . . . .

Mixed convection in a buoyant plume 505

Table 3. ]"(0)

Approximate singularity Exact singularity t~ Exact Comp. series Neville Comp. series Neville

1/100 0.52742 0.52668 0.5274 ± 0 0.52669 0.52740 _+ 2 1/10 0.71033 0.70938 0.7103 + 0 0.70942 0.71033 ± 1

1 0.81937 0.81755 0.81844-2 0.81781 0 . 8 1 9 6 ± 2 10 0.87516 0.87239 0.87513 ± 2 0.87199 0.87512 ± 1

100 0.91094 0.94613 0.908 ± 1 0.90667 0.91095 ± I

A Neville algorithm may once more be invoked to obtain S~ with data points l / [ ( n - i)2/5], i = 0, I . . . . . r. The results for this means of estimating velocity and temperature values also appear in Tables 3 and 4. They show significant improvements on the series completion estimates when used for both approximate and exact singularity location values.

The pure free convection plume similarity solutions are the solutions of

. r " + ff" - '-J": + g = o T(o) = r"(o) = 0"(o) = o

_I0. = + o ;r(oo) = 0(oo) = o. (7

The above tables compare exact solutions with series extension estimates based on (a) approximate singularity locations 9, 3, I, l, 1 and on (b) exact singularity locations 9.15250, 3.29118, 1.424255, 0.8013, 0.6176.

6. C L O S E R E X A M I N A T I O N

In previous sections a general description of the implementation of series extension techniques has been presented. It has been shown that with due care with respect to rounding errors impressive estimates of pure plume values at large ~ can be obtained from the coefficients of series ostensibly relevant only to small ~. In this section errors and accuracy of results are subjected to much closer scrutiny. For the particular case a = 2 there exists an exact analytic solution for the pure free convection plume and accordingly this provides the basis for such a detailed examination.

In a programme of tests for errors the following runs were made:

1. Standard run-using quadruple precision with step-length h = 1/180 and outer boundary at r / ~ = 10.

2. Double precision--as standard run except using double precision. 3. Single precision--as standard run except using single precision. 4. Truncation er ror- -as standard except h = 1/90. 5. Outer boundary er ror - -as standard run except r/oo = 12.5.

Comparing double and single precision results with the quadruple results gives maximum relative errors of 3 x l0 -:2 and 8 x l0 -3 for each precision respectively. The truncation error incurred in the standard run can be estimated as 1/15th the difference between its results and those when h = 1/90 (1/15th factor since R-K-Fehlburg is 4th order); the maximum relative truncation error is 2 x 10-12. Similarly comparing the results for r/~ = 10 and 12.5 gives the maximum relative error in the standard run due to the position of the outer boundary as 4 x l0 -12. Hence we conclude that the accuracy of quadruple and double precision results is l 1-12 significant figures and regard the single precision results as unacceptable.

Table 4. O'(0)

Approximate singularity Exact singularity o Exact Comp. series Neville Comp. series Neville

1/100 0.072083 0.071854 0.07208 + 0 0.071857 0.072085 ± 2 1/10 0.17840 0.17792 0.17840 ± 1 0.17795 0.17840 ± I

1 0.42754 0.42743 0.4273 ± 1 0.42739 0.4276 ± 0 10 I.I1742 I.I1951 1 .1175±1 1.11993 I . I I 7 3 ± 1

100 3.30179 3.30451 3.307 ± I 3.31112 3.3019 ± 3

506 ROLAND Ht;,~T a n d GRAHAM WILKS

Table 5. Double precision rounding and true errors. Q is either f,~ or 0".

Velocity Tempera ture

¢0 (Sth column of Neville table) t o (10th column of Neville table) Eulerised coel~cients ° Comple t ing the series Q ( ~ , 0) (3rd column of Neville table) Q ( x . 0) (6th column of Neville table) Q (7: , 0) (9th column of Neville table)

True error in % True error in Q (~c. 0) by complet ing the series True error in Q (.:~, 0) using Neville 's a lgori thm

4 × 1 0 t 2 , : 1 0 ~ 2 x 10 ~ 10 ' 6 x 10 " 6 x 10 ,0 6 x l 0 9 ~ 1 0 2 × 1 0 ~ 2 x 1 0 '

0.2 0.01 30 I

10 ~ 3 x 10 4 2 x 1 0 3 7 x 1 0 4 3 x 1 0 • 10 4

*These are absolute errors which are more appropr ia te in this case, the relative errors are 2 x 10- ~ and 10 -6 respectively.

These results are used to analyse and extend the region of validity of the series. Such transformations incur further errors which we will consider now. The techniques discussed in this paper so far are:

1. Determination of c o using Neville's algorithm with data set i/n. 2. Establishing the Eulerised coefficients. 3. Completing the series. 4. Estimating f ' (0) and 0(0) using Neville's algorithm with data set l /n 2;5.

Using these four techniques on the results from the five runs listed earlier and making suitable comparisons indicated the following conclusions. Changing the values of either h or ~/~ does not introduce errors significantly greater than those already present. Single precision results gave nonsense. Comparing double precision results with the quadruple results gave maximum relative errors due to double precision rounding as listed in Table 5. The correct values for ~0, f ' (0) and /7(0) are 1.167829, 0.837484 and 0561104 respectively and from these the true errors, which includes both truncation and rounding errors, are obtained and are given in Table 5. Comparing the true errors with the rounding errors we find that the determination of co using Neville's algorithm is satisfactory using double precision since convergence is usually obtained by the 5th or 6th column. Similarly completing the series is adequately performed in double precision. However the determination off ' (0) and 0"(0) using the l /n 2'5 data set and Neville's algorithm is unsatisfactory in double precision as can be seen by comparing the true error with, say, the rounding error in the 6th column. In fact using double precision resulted in an increase in the true error by about an order of magnitude.

In order to appreciate the magnitude of error propagation using Neville's algorithm we consider the worst case of rounding error distribution namely alternately E, - E , E, - E , . . . . i.e. ( - I )~E contaminating the ordinate values. Then using Neville's algorithm with the 1In data set and 30 profiles gives maximum possible error contamination of +4 x 10rE in the 5th column of the table and 13 × 101°E in the 10th column. This represents a loss of significance of 6 and 10 significant figures respectively. Using the l /n 25 data set the situation is worse, giving +5 x 10~E, +8 x 109E and +3 x 1013E in the 3rd, 6th and 9th columns respectively representing a possible loss of accuracy of 13 significant figures. Since double precision is of the order of 16 significant figures this represents a serious erosion of the results. In practice it was found that using the 1In data set is just adequate in double precision but the 1/n 2.'~ data is not and quadruple precision should be used.

7. R E S U L T S F O R 0 < ~ < oc

All previous discussions have been concerned with the prediction of flow characteristics valid in the limit ~ -~ oo. The question remains as to the level of accuracy which is achieved in estimates of centre line characteristics over the entire flow field. A comparison has therefore been made between predicted values from series re-organisation and exact numerical solutions. Table 6 shows the relative errors incurred in units of 10 -3 for various E using the Eulerised series [eqn (30)], completed series [eqn (35) with expansion parameter E/E + E0] and Neville's algorithm for which

M i x e d c o n v e c t i o n in a b u o y a n t p l u m e 507

Table 6. Relative errors

e/¢ o O.Ol

in units of 10 -3 using the Eulerised series (E), the completed series (C) and Neville's algorithm (N)

Velocity Temperature

0.1 l l0 ]00 0.01 0.1 I 10 100

E 0.0004 0 0 0.0001 0.0001 0.0001 0 0.0001 0 0 I C 0.0004 0 0 0.000! 0.0001 0.0001 0 0 0 0

N 0.0004 0 0 0.0001 0.0001 0.0001 0 0 0 0

E 0.002 0.000S 0 0.002 0.001 0.001 0.001 0.0004 0.002 0.001 3 C 0.007 0 0.0001 0.0002 0.0002 0.0005 0.0001 0 0 0.0001

N 0.001 0.0001 0.0001 0.0003 0.0003 0.006 0.0002 0 0.0002 0.0003

E 0.2 0.07 0.04 0.1 0.2 0.4 0.3 0.2 0.1 0.l 10 C 0.005 0.001 0.002 0.003 0.003 0.01 0.002 0.001 0.0002 0.001

N 0.01 0.002 0.004 0.005 0.006 0.02 0.01 0.006 0.004 0.004

E l0 5 5 l0 l0 30 20 20 l0 l0 100 C 0.2 0.1 0.2 0.3 0.4 0.5 0.3 0.1 0.2 0.3

N 0.1 0.1 0.1 0.2 0.5 0.3 0.6 0.2 0.3 0.3

E 50 30 20 50 60 100 100 70 40 40 oo C 1 1 2 4 5 3 3 0.4 2 3

N 0.3 0.001 0.2 0.04 I 0.3 0.3 0.4 0.3 0.2

the appropriate data set is [~/(~ + Eo)]" l/n 2/~. The full numerical results are accurate to about 6 significant figures and hence any error less than 10 -6 is not significant. For e ~< 5e0 the Eulerised series gives results as accurate as the full numerical solutions. For E > 5Eo the series becomes less accurate; however, the completed series improves these results by a factor 10-50 and we have 3-4 significant figures at ~ = 100 E0. Neville's algorithm only improves matters for very large ~, typically E > 103~0, again giving 3-4 significant figures of accuracy.

8. D I S C U S S I O N A N D C O N C L U D I N G R E M A R K S

In general the number of terms of the associated series solutions that are required for successful implementation of the series extension technique depends on two factors. The first may be an intrinsic difficulty of the problem. In such cases every attempt must be made to increase the number of terms available in order to clarify the subtle structure which is to be identified. The second however is more practical and simply relates to the facilities available to the investigator. Effectively a sensible balance between the desire to perform the analysis on as many terms as possible and the excessive demands this might make on storage and computer time must be found. The mixed convection plume has not revealed any inherent difficulties. On the contrary this problem has proved particularly amenable to series extension methods. Although for the purposes of detailed examination up to 30 terms of the series have been used here this number is perhaps excessive. For problems displaying the convenient structure of the mixed convection plume results sufficient for practical purposes may be obtained with significantly fewer terms, particularly as results are relatively insensitive to the precise location of the singularity. Nevertheless it is instructive to note the level of accuracy that can be achieved with large numbers of terms. This is true only so long as due regard is paid to accumulated round-off errors. By closely monitoring the discrepancies between quadruple and double precision it has been demonstrated that, for this class of problem, working in double precision, for the most part, adequately deals with this inherent pitfall in the method.

The work presented here has concentrated on internal comparisons between various levels of implementation of series extension techniques. After an exhaustive study of the application of these methods to the flow of a mixed convection plume, including extremes of Prandtl number, it is noteworthy that the methods have proven remarkably successful in estimating the physical characteristics of the flow. This is not to ascribe any specific advantage over finite difference methods but rather to point out what can be achieved by the present techniques. In view of the packages that exist for systems of ordinary differential equations the method perhaps constitutes an attractive, viable alternative to full numerical solution of the governing partial differential equations.

508 ROLAND HUNT and GRAHAM WILKS

R E F E R E N C E S

l, G. Wilks and R. Hunt, Vertical mixed convection flow about a horizontal line source of heating or cooling. Ira. J. Heat Mass Transfer 30, 1119-1131 (1987).

2. N. Afzal, Mixed convection in a two dimensional buoyant plume. J. Fluid Mech. 105, 347-368 (1981). 3. R. Hunt and G. Wilks, Continuous transformation computations of boundary layer equations between similarity

regimes. J. Comp. Phys. 40, 478-490 (1981 ). 4. W. W. Wood, Free and forced convection from fine hot wires. J. fluid Mech. 55, 419-438 (1972). 5. M. Van Dyke, Analysis and improvement of perturbation series. Q.J.M.A.M. 27, 423-450 (1974). 6. H. B. Keller, Numerical Method~for Two-Point Boundary Value Problems. Blaisdell, MA (1968). 7. C. Domb and M. F. Sykes, On the susceptibility of a ferromagnetic above the Curie point. Proc. R. Soc. 240A, 214-228

(1957).