Perturbation techniques in free convection

^3>é.:is-Q..

PERTURBATION TECHNIQUES

IN

PERTURBATION TECHNIQUES

IN

FREE CONVECTION

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 1 NOVEMBER 1967 TE 14 UUR.

DOOR

HENDRIK KLAAS KU1KE^

WISKUNDIG INGENIEUR GEBOREN TE s-GRAVENHAGE

t f l

I

Dit proefschrift is goedgekeurd door de promotor Prof. Dr. R. Timman.

C O N T E N T S I INTRODUCTION 1 H i s t o r i c a l 1 2 D e r i v a t i o n of t h e b o u n d a r y l a y e r e q u a t i o n s of f r e e c o n v e c t i o n p a s t a v e r t i c a l f l a t p l a t e 3 2.1 analysis 6 2.2 a small 9 2.3 a large 11 2.4 boundary layer equations and boundary conditions . 14

3 S o m e i m p o r t a n t e x i s t i n g s o l u t i o n s . . . .17

3.1 isothermal surface 18 3.2 non-isothermal surface 22

4 H e a t t r a n s f e r 25 5 S k i n f r i c t i o n 28

II FREE CONVECTION AT LOW PRANDTL NUMBER

1 I n t r o d u c t o r y r e m a r k s 29

2 I s o t h e r m a l p l a t e 30 2.1 main t e r m i n n e r - a n d outer expansion 30

2.2 first perturbations 36 2.3 second perturbations 39 2.4 summary of the results 44 3 N o n - i s o t h e r m a l s u r f a c e 49

3.1 viscous boundary layer 49 3.2 inviscid boundary layer 50

III FREE CONVECTION PAST A NON-ISOTHERMAL VERTICAL FLAT PLATE

I R e v i e w 53 2 M e t h o d of s o l u t i o n 55

3 R e s u l t s a n d h e a t t r a n s f e r 58 4 Low P r a n d t l n u m b e r s 62

IV AXISYMMETRIC FREE CONVECTION PAST SLENDER BODIES

I S u r v e y 65 2 C y l i n d e r 67

2.1 analysis 67 2.2 heat transfer 71 2.3 applications: constant wall-heat flux 73

3 C o n e 76 3.1 solution 77 3.2 heat transfer and results 79

4 C o n c l u d i n g r e m a r k s 79

REFERENCES 81

NOMENCLATURE 84

SUMMARY 86

C H A P T E R I INTRODUCTION

1. H i s t o r i c a l

UNTIL recently the influence of body forces on the dynamics of a fluid has only been studied thoroughly in the well-known field of gravity waves. Laminar boundary layer flows past obstacles or in the free air which a r e due to gravity or to other, e. g. magneto-hydrodynamic, body forces did not receive much attention. As a matter of fact, a s late as 1950, only a few investigations had been presented concerning this type of flow. The bulk of these papers refers to a type of flow being present in non-isothermal fluids under the action of the force of gravity. The temperature differ-ences cause for a non-homogeneous fluid with respect to the density. According to the law of Archimedes those elements of fluid being lighter (having lower density) than those of the ambient fluid experience through gravity a force in upward direction. Naturally the r e v e r s e case holds too: on being suspended in lighter material a heavy element will move in downward direction. As this type of flow is present in the important fluid called the earth's atmosphere it originally drew the attention of such scien-tists as meteorologists, geophysicists etc. The mathematical description of free convection as embodied in the governing differential equations is a most difficult one and as a consequence the solution of very general problems did not prove to be obtained easily. The first major success therefore has been the d e s c r i p -tion of a simple problem. Pohlhausen [25] investigated mathemati-cally free convection in the vicinity of a vertical flat plate having a uniform temperature exceeding that of the semi-infinite fluid in which it is suspended. He proposes the boundary layer approxi-mations be applied to the basic equations and a solution be found by proceeding along exactly the same lines as was done in solving other problems in boundary layer flow, such as the problem of viscous forcedflow past a semi-infinite plate solved by Blasius [3]. Experiments performed by Beckmann and Schmidt [25] justify this method of solution.

Some other papers have appeared before the time mentioned above but the r e s e a r c h in this field was not abundant. At that time, however, other branches of fluid mechanics had come to consid-erable growth. A gamut of methods of solution had been presented

already and for example concerning the notoriously complicated boundary layer flow a large number of authors had shown a d m i r -able ingenuity to obtain useful results. This naturally had much to do with the practical importance of those studies in connection with the development of aircraft. Here the inexorable law of applied mathematics holds which has it that "problems of practical interest burgeon, while those of m e r e academic interest only receive incidental attention". Up to the period mentioned above free convection belonged practically to the class of problems having only academic interest.

In modern times, however, the action of body forces on the motion of a liquid has met with important practical applications. A noteworthy application seems to be the cooling in nuclear r e a c -t o r s -through free convec-tion. Moreover, -the applica-tion of o-ther body forces, such as the centrifugal force and forces giving r i s e to magnetohydrodynamic flows, worked as an incentive and gave impetus to the study of flows due to body forces. In the years following a great many papers dealing with free convection have appeared bringing this branch of science, partly through using experience gained in the other fields of fluid mechanics, with accelerated pace toward maturity.

Ostrach [23] has given a rigorous account of the derivation of the equations governing free convection, stating the conditions under which the equations may be simplified to the boundary layer type. In the next section we will consider this problem again for obtaining sharper conditions. Sparrow and Gregg [39] derived for free convection pas a vertical flat plate a class of surface t e m -p e r a t u r e s which allow for similarity solutions. This kind of solu-tion is most common in boundary layer theory and often they serve as the basis of the solution of more complicated problems e. g. with perturbation techniques. To date the application of p e r -turbation techniques to problems of free convection has been

rather s p a r s e . Among the exceptions we may note the study of flow past a vertical isothermal cylinder by Sparrow and Gregg [38] and of free convection at large Grashof number by Yang [44]. It is the purpose of this thesis to solve a number of problems of free convection with perturbation techniques. Mostly these problems a r e of a more complicated nature than those solved by similarity transformations, but in solving them, the knowledge of the t r a d i -tional problems is of great advantage. We will be strictly con-cerned with free convection along bodies suspended in a fluid extended to infinity, that is to say, free convection in a finite region, mostly referred to as natural convection, will not be considered here. Furthermore, the development of a boundary layer in the time is not considered; only the steady state will receive attention.

Apart from giving a rigorous account of the derivation of the boundary layer equations it is the purpose of the next sections of the INTRODUCTION to present the equations and to give a brief description of the most important methods of solution employed as yet. For further information with respect to this matter we may refer to the book "Theory of Laminar Flows" edited by F. K. Moore . Chapter F of this book "Laminar flows with body f o r c e s " written by S. Ostrach contains as far as the present author is aware of the most complete connected description of the work on free convection presented to date. The subsequent chap-t e r s of chap-this chap-thesis a r e devochap-ted each chap-to an original problem of free convection. So in chapter 11 the method of inner- and outer ex-pansions is used to investigate free convection at low Prandtl number. In chapter III solutions a r e found for free convection past a vertical plate having a nonisothermal temperature d i s t r i -bution of a very general t5^e. For some wall-temperatures it will prove to be possible to find over-all-valid solutions through applying a joined s e r i e s method. In chapter IV the influence of radial curvature on free convection past slender bodies of revolu-tion will be looked into.

It may be mentioned that a definition of the coefficient of heat transfer will be introduced different from that common in the literature on free convection. It is based upon the difference between the temperature of the body and the average (bulk) t e m perature of the fluid rather than upon that between the t e m p e r a -ture of the surface and the tempera-ture of the ambient fluid. F u r t h e r m o r e , the condition be imposed throughout this thesis that the temperature of the body exceeds that of the ambient fluid. The r e v e r s e case naturally is the same from a mathematical point of view as has been stated by several authors [23, 38]. In order to have a unified treatment and to avoid ambiguity it is necessary that this condition be mentioned here. It is constantly assumed that the fluid properties be constant with the exception of the one (density) causing for the effect of buoyancy.

2. D e r i v a t i o n of t h e b o u n d a r y l a y e r e q u a t i o n s of f r e e c o n v e c t i o n p a s t a v e r t i c a l f l a t p l a t e

As is well-known to those familiar with the complexity of the Navier-Stokes equations the boundary layer approximations a r e a powerful means of reducing these equations to proportions which allow them to be solved. One should, however, always be aware of the fact that an approximation will he accompanied by r e s t r i c t -ing conditions. In apply-ing an approximation one has to check if these conditions a r e satisfied. Therefore it has been judged n e c -e s s a r y h-er-e to d-eriv-e th-es-e r-estricting conditions car-efully. Th-e

mathematical model to be used has been given in figure (1.1). \ \ \ \ \ \ \ \ \ \ \ \ gravity

y

FIG. 1.1. F r e e convection past flat plate: physical model. ÏÏ we r e s t r i c t ourselves in the analysis to ordinary liquids the equation of state giving the relation between the density and the temperature will be

P=Pc={l - ^ ( T - T „ ) } . (1.1) This naturally is the linear approximation of a more intricate equation of state. It may be used if

e =p{T^ - T ^ ) « 1. (1.2) For simplicity we assume T-^^^ constant in the present analysis. Apart from the equation of state there a r e four other equations governing the phenomenon of free convection in steady state, v i z . , the equation of continuity

4(pu)+^(pv) = 0,

(1.3)

the momentum equations

au au U + V = u ax ay By Bv u — + V — = u , a^u a^u ax'^ ay'^ la^v a^v —=- + ax ay lax ay' 1 dp „ ~&> p dx 1 dp > (1.4) (1.5) p dy

Here the last t e r m of equation (1.6) is the dissipation function. The task which now lies before us is to determine the condi-tions under which the Navier-Stokes equacondi-tions and the energy equation may be reduced to their boundary layer approximations. Especially Ostrach [23] has performed a great deal of work to achieve this aim. The derivation to be given here deviates in many respects from that of Ostrach who states that for the Grashof number

g^(T^

-

Tji^

Gr = (1.7)

large enough the second order derivatives with respect to x may be neglected in comparison with those taken with respect to y. This would mean that for an inviscid fluid (i^=0) there would a l -ways be a boundary layer, which is certainly not t r u e . We anti-cipate that for small viscosities Ostrach's condition Gra » 1 has to be replaced by a condition not involving the viscosity. In d e r i v -ing the restrict-ing conditions one probably has been led too much by the achievements of forced laminar boundary layer flow. A d e s i r e for parallelism made earlier investigators decide to i m -pose the condition that the viscous t e r m s and the inertia t e r m s had to be of the same order of magnitude in the free convection boundary layer. This in close analogy with the earlier findings of forced laminar boundary layer flow. As a consequence the

Prandtl number

a = — ^ (1.8) representing the connection between the viscousand the t e m p e r

-ature boundary layer appeared in the energy equation. It i s , how-ever, a known fact that the answer to the question as to which t e r m s have to be of the same order of magnitude in the equations of free convection depends highly on the value of the Prandtl number. So rather than trying to find one single expression decid-ing about the applicability of the boundary layer approximations

we have to expect different c r i t e r i a for different Prandtl numbers. Although the boundary layer equations ultimately found by Ostrach a r e correct the derivation leads in some cases to dubious con-clusions about the conditions which, on being satisfied, permit the boundary layer approximations. As an example we may give, that instead of Gra » 1 we have to give CT Grz » 1 as the condition justifying the boundary layer approximations at low Prandtl number (nearly inviscid fluid). This will be proved later on.

This last condition is found through considering that for CT< 1 the temperature boundary layer is always somewhat thicker than the velocity boundary layer. Moreover, the viscous effects d e -c r e a s e with a. That is why for a < 1 the boundary layer equations will be derived through stating that for small Prandtl numbers the convection- and conduction t e r m s of the energy equation have to be of the same order of magnitude. In the equations obtained the Prandtl number will be the coefficient of the viscous t e r m thus asserting that for a ^ 0 the viscous s t r e s s e s a r e of minor impor-tance.

It may be clear through these statements that there is still need for an investigation, having the character of a scrutiny, concerning the conditions permitting the boundary layer approxi-mations.

2.1 analysis

Let us introduce non-dimensional variables - indicated with b a r s - through

x =£x, y = öly, u = UÜ, p = p p,

„ _^ (1.9) P = P ^ U ^ P , T = T ^ + ( T ^ - T ^ ) T .

Apparently the equation of state (1.1) will become

p = 1 - ef. (1.10)

Bearing in mind that 6 is the thickness of the boundary layer an integration of equation (1.3) through the boundary layer gives

v

=

- U ö j - f — (pü)dyL (1.11)

' p •' O öx )

Obviously v = 0(u6) so that in addition to (1.9)we may introduce

Substituting (1.9), (1.10) and (1.12) in (1.3), (1.4), (1.5) and (1.6) gives 3u Sv (1.13) au _ a u V 1 1 (S'^u Ü — + V _ :. ^

ax ay U i 6^ 1-cT lay

, a^u

9- = - 1 - ^ 1 7 - + 6'^ - ^ 1 9 - ^ + a x '

1 ap g i

l - e T ax " U2 ' (1.14) a2v

_ av au V

ax ay u / 6^ l-eT ( ay

2 1 , r r ) ::i-T72 a ^ ax2 1 ap i - e T 52 ay ' (1.15) _ aT _ aT U + V ax" ay X 1 1 p^Cp^U 62 1-eT |a2T ^ ^2 a^T Iay2 ax^

u 1 (_ ap _ ap

Cn(Tw-T„) l - e T T ü ""^ ay

-p

t' u

W OCJ> 1 C p A T ^ - T „ ) "6^ 1-cT av >2 'lay a x / 3 1 ^x a y ' (1.16)

Let us now first direct our attention to equation (1.15). Since 6 is supposed to be small (62 « 1) it is obvious that the p r e s s u r e t e r m is much larger than the inertia t e r m s (left hand side). It i s , however, also essential that the viscous t e r m s a r e much smaller than the p r e s s u r e t e r m . If not, the equations a r e p r o -hibitively difficult to solve. Hence we impose

We consequently a r r i v e at the equation 3p

^ ^ = 0. (1.18)

ay

Switching on to equation (1.14) we see that this equation reduces to

i E = . l i (1.19)

as y-»oo^. This follows directly from the boundary conditions ÏÏ -> 0, T -> 0 a s y-» » . On account of equation (1.18) the ejqjression (1.19) i s also valid in the boundary layer. Insertion of (1.19) in (1.14) yields dU a i T c i / 1 3 % e gi U +T" =' ÖX ay f u ' 1 3 ^ e e i I

JT2^*^^\^' -0(.)}. (1.20)

Here we have already made use of 6 « 1. Under the same con-dition equation (1.16) will become

_ 3p _ Sp u -Z + v — 1+

ax ay

while for the equation of continuity we find

_ + __ = 0 ( e ) . (1.22) Sx ay

We have now arrived at the important question a s to which conditions should determine 6. If we would decide that after dropping the 0 ( c ) t e r m s every t e r m of equation (1.20) has to be of the same order of magnitude we would obtain the traditional r e -sults. It is felt, however, that in this way too much emphasis is put on the momentum equation. In free convection the energy equation should also receive adequate attention a s it is the t e m -perature-differences which a r e lying at the very root of this phenomenon. The known results of free convection suggest that

our investigation split up in two different studies. One should be concerned with small Prandtl numbers ( a < 1), the other should refer to large Prandtl numbers ( a > !)•

2.2 or small

F i r s t we investigate low Prandtl numbers, i. e. nearly in-viscid fluids. We impose the condition that the convection and the conduction t e r m s of equation (1.21) a r e of the same order of magnitude. Hence

6^ = . (1.23) P„Cpi U

Since in free convection the sole driving force is represented by the buoyancy t e r m this t e r m must be of the same order of magni-tude as the largest term(s) of equation (1.20). In the present case these obviously a r e the inertia t e r m s . We consequently find

u 2 = e g / = ^ g / ( T ^ - T ^ ) . (1.24) The combined knowledge of (1.23) and (1.24) yields

6^ = CT"^ G r - ^ (1.25) Obviously for small Prandtl numbers the condition

( T G r ^ » l (1.26) has to be satisfied for the boundary layer approximations to be

valid. Using (1.24) condition (1.17) imposes

G r 2 » l . (1.27) Apparently (1.27) is satisfied if (1.26) is, since a is small. Since

Cp usually is very large the t e r m s

U ^ _ g;8£ _ V \J 1 _ g|8i Cp(Tw^T„) " " ^ ' C p / ( T ^ - T „ ) 72- - ^ —

a r e small. Hence the last two t e r m s of equation (1.21) may be neglected. On finally imposing the condition e « 1 (1.2) we find for small Prandtl numbers the set of governing boundary layer equations

au av — + — a^x ay aü aü" U + V = ax ay aT aT U + V — ax ay = 0 ,

a^ü

ay^

a^T ay2 (1.28) + T , (1. 29) (1.30)

It may be noted that the Prandtl number occupies a realistic position where it determines, about the influence of the viscous s t r e s s e s . As in the analysis of Ostrach the viscous s t r e s s e s a r e judged to be of the same order of magnitude as the inertia t e r m s it evidently applies to the viscous part of the free convection boundary layer which is known to dwindle beyond any limit as cr - 0. So this analysis can give interesting information about the ratio of the thickness of the viscous layer öy and the thickness of the full free convection boundary layer. On making all t e r m s in the momentum equation of equal order of magnitude we readily derive

6v = G r - 4 . (1.31) With (1.25) this clearly gives

6 v / 6 = a 2 . (1.32) The boundary layer momentum- and energy equation now

obvi-ously a r e given by

. 2rr

(1.33)

(1.34)

which is, as far as the position CT occupies is concerned, the traditional way of presenting the boundary layer equations of free convection. Chapter 11 of this thesis is completely devoted to free convection at low Prandtl number. Hence we see fit to refrain from further discussion of this matter here.

_ aü u —

ax

a\ u + V aT ax

au

ay

+T

a^u 9y2

a T 1

ay t

+ T , a^T By2 '

2.3 o large

As we have remarked e a r l i e r the t h e r m a l - and the velocity boundary layer a r e of about the same thickness for a < 1, the former being somewhat thicker than the latter. Concerning this we may refer to chapter IT or to the work of Sparrow and Gregg on low Prandtl numbers [40]. This contrary to the situation in forced flows where the velocity boundary layer is drowned in the thermal boundary layer as a - 0. In free convection a growing part (as cr -< 0) of the velocity layer is inviscid so that the Prandtl number cannot supply information about the relation between the t h e r m a l - and the complete velocity boundary layer. For cr > 1, however, the physical pattern reflects the same features as in forced convection, i. e. the thermal boundary layer is thin in comparison with the velocity boundary layer. The velocity bound-ary layer is totally viscous. If we fix our attention now to very large values of CT(CT»1) the following picture emerges (see

Ostrach [24] figure F3b and F3c or in present thesis figures (1.3) and (1.5)). Let us consider a fluid of large viscosity and small thermal conductivity. For such a fluid the Prandtl number is large. Now, obviously, the temperature boundary layer will be very thin thus only admitting buoyancy forces in this very thin layer. In this layer the fluid will be dragged in upward direction. Due to the large viscosity the fluid will also move upwards in an adjacent layer of considerable thickness where no buoyance forces exist. We obviously have to use the following model in deriving the boundary layer equations and in stating the conditions of their applicability. In the thin thermal boundary layer the convection t e r m s and the conduction t e r m s a r e of the same order of magni-tude. In the momentum equation the buoyancy t e r m and the viscous t e r m have to be of the same order of magnitude. Making use of these considerations we may find the thickness of the thermal boundary layer and, what is very important, a c h a r a c -teristic velocity. Since in the layer where no buoyancy effects a r e present the flow can only be retarded this velocity must also be the characteristic velocity of the complete viscous layer. Using this velocity and the condition that in the viscous layer t h e i n e r t i a -and the viscous t e r m s a r e of the same order of magnitude we can derive an e g r e s s i o n for the thickness of the free convection boundary layer at large Prandtl number. Here the outer fringes of the viscous layer determine this thickness.

Now fixing our attention first to the thermal layer the condi-tion of the conduccondi-tion- and conveccondi-tion t e r m s being of comparable magnitude leads to (see equ. (1.21)).

The suffix T naturally refers to the fact that of is not the thick-.

ness of the complete free convection boundary layer but only of that part of it where tangible temperature differences with the ambient fluid exist. Our condition about the buoyancy t e r m match-ing up with the viscous t e r m leads to (equ. (1.20))

gpi{T^~Tj 2

^ = Orr. Substitution of (1.35) in (1.36) renders

^2 _ ëPlJT^-Tj

a and 6^ = CT-2 G r - 2 .

The fact, as expressed by (1.37), that the velocity decreases as cr increases is in complete agreement with e a r l i e r findings (see figure (1.3)). As has been remarked above the character of a free convection boundary layer at large Prandtl number is one of a viscous layer in forced flow the force being exerted through buoyancy in a very thin layer adjacent to the wall. As a c o n s e -quence we can only impose one condition for determining the thickness of the layer. This condition naturally is the same as the one used by Prandtl [26] in discussing a viscous boundary layer of forced flow: it expresses that in the layer the i n e r t i a - and the viscous t e r m s are of the same order of magnitude. Using equation (1.20) we find

6^ =JL . (1.39)

For reasons e a r l i e r advanced we may use equation (1.37) as ex-p r e s s i n g adequately the characteristic velocity in equation (1.39). This leads to

6^ = a 2 G r - i . (1.40) As the condition 6 « 1 coincides with (1.17) we have to impose

in case a is large

-CT-2 Gr2 » 1 (1.41) for the boundary layer approximations to be valid. Another i n t e r

-(1.36)

(1.37)

esting outcome of the present analysis is that it supplies evidence about both the viscous- and the thermal layer. Using (1.38) and (1.40) we find

6 ^ / 6 = a-4 . (1.42) The figures found by Ostrach [23] about free convection at large

Prandtl numbers confirm qualitatively esq^ression (1.42). As one is left with a certain amount of uncertainty in choosing the outer edge of a boundary layer the = sign could be replaced best by a ~ sign. Bearing in mind that for cr < 1 the thermal boundary layer is predominant while for a > 1 this is the case with the viscous layer both formulas (1.32) and (1.42) lead to

6 ^ / 6 ^ = a'2 . (1.43) After having thrown light upon the different aspects of large

Prandtl number free convection boundary layer flow it may have become clear that the only way to solve it realistically is using the method of matched inner- and outer expansions. The inner problem can be studied through creeping into the exiguous dimen-sions of the thermal boundary layer. Substitution of (1.37) and (1.38) in (1.20) and (1.21) then leads to the following momentum -and energy equation

(1.44) d U u — 3 x ü + V óT ax" au = Oi ay aT + V — öy Ö ^ U TÏT — T + T ' ay2 ( a2T " ay2 • (1.45)

In solving these equations one has to use the inner boundary con-ditions (concon-ditions at the wall). The remaining concon-ditions for y -•<» have to be found through matching with the solution of the outer problem.

Inserting the expressions (1.37) and (1.39) in the equations (1.20) and (1.21) we scale up to the larger dimensions of the complete viscous layer. The equations become

— _ 2—

au au a u ^

ü" - 3 + V -— = - ^ + a T , (1.46)

a ] u — + v — ' [ - —9 . (1.47) < 3x öy ) 9y^

At first sight it seems rather contradictory that the buoyancy t e r m in the momentum equation contains the larger parameter o. We should, however, remember that for the main t e r m of the outer ejqjansion the temperature is exactly equal to zero. As a consequence the buoyancy t e r m plays, as expected, no part in the main t e r m of the outer problem. This main t e r m is a solution of the differential equation

- _

2-_ Su 2-_ au 3 u ,. ^o\

u ^ + V — = ^ 2 (1.48)

ax ay ay^

and has to satisfy the outer boundary condition u - 0 as y -co. The remaining inner boundary conditions have to be found through matching with the inner problem according to the well-known matching rule (see [47]). In this thesis we will not deal with large Prandtl number free convection in more detail as has been done as yet. Low Prandtl number free convection has been chosen in this thesis as the topic to be dealt with the sophisticated method of matcned inner- and outer expansions (chapter II).

2.4 boundary layer equations and boundary conditions

Although the separate analyses have been performed for ex-t r e m e values of ex-the Prandex-tl number iex-t may be expecex-ted ex-thaex-t ex-the results drawn therefrom a r e qualitatively consistent for a larger Prandtl number range as long as in this range the basic a s s u m p -tions remain the same qualitatively. Consequently the results obtained for small Prandtl numbers a r e expected to give informa-tion for CT < 1, while those found for large Prandtl numbers a r e believed to be valuable for CT > 1. Hence for CT < 1, we have that the boundary layer approximations a r e valid provided

a G r ^ > M . (1.49) Here we have refrased equation (1.26) through introduction of a

very large number M so as to give it a more definite character. For CT > 1 we find through (1.41) the analogous condition

1 er

FIG. 1.2. Conditions for boundary layer approximations, doubly shaded: present work; singly shaded: Ostrach [23]. Although near a = 1 the graph of figure (1.2) may have to be changed somewhat it clearly exhibits the result of the present analysis. While Ostrach's analysis merely gives Gra > M the present investigations reveal that s t r i c t e r rules have to be i m -posed upon the Grashof number if the boundary layer approxima-tions a r e to be valid. The singly shaded region applies to the Ostrach condition while the doubly shaded region is a result of the present investigations.

We finally give the boundary layer equations in the original coordinates. So if e « 1 and the Grashof-Prandtl combination refers to the doubly shaded region we have

1° the equation of continuity

Su 9v

— + — = 0 , (1.51)

ÖX By

2° the momentum equation

2

u — + v — = y — 5 + g ) 8 ( T - T „ ) , (1.52)

3 the energy equation

u — + V — = —K. (1.53) ax ay pCp öy'^

Here and in the following p is understood to have the value of p^. The boundary conditions to be imposed concern both the velocity and the temperature. Starting with those of the velocity the in-cluding of viscous forces requires a no-slip condition at the wall. Thus we have

u = 0 at y = 0. (1. 54) At the wall we also have

V = 0 at y = 0, (1. 55) since the wall is impermeable. At infinity (y-.<x=) we have to put

u -. 0 a s y - "> (!• 56) since the surrounding fluid is quiescent. With respect to the temperature two conditions have to be imposed. F i r s t , there is a prescribed surface - temperature.

T = T ^ a s y = 0. (1.57) Second, the temperature of the ambient fluid is constant

T - T ^ as y-.a=. (1.58) It must be remarked that all boundary conditions a r e given

for special values of y. One would expect that at x = 0 (leading edge) some conditions have to be given. However, these conditions a r e usually discarded, not in the last place for the mathematical difficulties they imply. Generally one takes for these conditions whatever a convenient, e.g. a similarity solution, gives. On a c -count of the fact that the equations a r e of a parabolic type this somewhat "loose" attitude does not seriously affect the solution somewhat downstream. It is a known fact, verified by many exact solutions, that parabolic equations have solutions which a r e built up by information immediately present. These solutions tend to forget about remote information. Now, since a boundary layer is a thin layer of fluid attached to a wall the conditions at this wall a r e much more important than those present at some isolated

place. Due to the dimensional properties of a boundary layer a special value of x (such as the leading edge) indeed represents an isolated place.

3. S o m e i m p o r t a n t e x i s t i n g s o l u t i o n s

As an introduction to the more intricate problems receiving attention in later chapters some simple problems of free convec-tion, well-known to the specialists of this field, will be treated here. These problems not only serve to usher those not familiar with the subject into the rudiments of it, they also constitute the indispensable tool in comprehending those intricate problems. To achieve this goal it seems to go without reasoning that the body to be investigated first will be the flat plate. In fact the problem of free convection past an isothermal flat plate has been most t h o r -oughly investigated and a large body of results is available. The most apparent reason for the successful studies of this problem is the fact that through a similarity transformation the partial differential equations can be reduced to ordinary ones. Though being non-linear these equations a r e more tractable and the numerical integration, especially with the aid of modern computers, is a simple thing to do. The great success of the s i m i l a r -ity transformations caused Sparrow and Gregg [39] to search for other transformations allowing for such solutions. They proved the existence of a class of wall-temperatures giving r i s e to a similarity boundary layer.

Also some work has been done concerning similarity flows about bodies of a different shape. As to this we may mention the work of

Merk and P r i n s [20] on axisymmetric bodies : cone, Millsaps and Pohlhausen [21] on vertical cylinders,

Yang [43] on vertical cylinders, Hering and Grosh [13] on cones,

Braun, Ostrach and Heighway [5] on twodimensional and a x i -symmetric bodies with closed lower ends. These investigations a r e of a more or less special nature, how-ever, and we consequently will not elaborate on them here.

Up to now we only have mentioned those problems which can be solved with exact methods. Here we call the result of a numerical integration of an ordinary differential equation exact. However, approximate solutions can be obtained through applying the integral method of Karman-Pohlhausen. Although the application

of this method has proved to give satisfactory results in many c a s e s we will not consider it here. For those interested we may refer to the work of Squire [41], Sparrow [35],Hama a n d C h r i s -tiaens [12], Fujii [8], Braun and Heighway [4], Scherberg [29] etc.

3.1 isothermal surface

For the isothermal flat plate the equations (1.51) through (1.58) have to be considered. Naturally T ^ in equation (1.57) is a constant for which T ^ > T^,. F i r s t , the equation of continuity (1.51) is solved by introduction of the streamfunction i/j with

u = — , V = - — . (1.59)

9y ax

Substitution of (1.59) in (1.52) and (1.53) gives

- - 2 = ' ' — 3 " + g-3(T-T„), (1.60)

ay axay ax ay^ ay' a0 aT a0 aT x a^T

ay ax ax ay pcp ay^ (1.61)

The boundary conditions of the velocity have to be changed into

0 at y = 0 , (L 62) 30 d0

ay ax a0

ay - 0 as y - » . (1.63)

Now completely in accordance with the solution of a linear time-space dependent heat conduction problem (e. g. heat con-duction in a semi-infinite rod) one wonders whether one single dependent variable, containing x and y, can be found for the d e -scription of the problem. This indeed turns out to be possible. The independent variable, which is called a similarity variable, to be chosen is

Upon introduction of two non-dimensional functions i{rj) and Qirj)

ij) = Avcx' i{r]) , (1.65) T = T^ + {T^-TJ eiv), (1.66) the momentum- and energy equation a r e transformed into

0 , (1.67)

^ + Serf — = 0 . (1.68) d??^ dt]

On account of the fact that u and v in the new coordinates a r e ex-p r e s s e d as

u = Auc^x^ — , (1.69) drj .

df

i/cx~4 |77 3f ) , (1.70)

the boundary conditions the functions f and 6 have to satisfy a r e

f = _ = 0 , 9 = 1 at 17= 0, (1.71) d77

df

dï7 -> 0 , 6 -> 0 as 77 -.CO . (1.72) The system of equations (1.67) and (1.68) with the boundary

conditions (1.71) and (1.72) have been solved by numerous authors. We may mention Pohlhausen [25], Schuh [31], Ostrach [23], Sparrow and Gregg [40]. In the figures (1.3) through (1.6) graphs have been given covering an extended Prandtl number range. These graphs show that for a -> 0 the coordinate r) becomes l e s s and less suitable for the description of the problem. One has to integrate up to very large values of 77 to satisfy the asymptotic boundary conditions. To cope with this deficiency a new analysis

exposing clearly the salient features of the different p a r t s of the boundary layer will be presented in chapter II. The reason for this awkward behaviour becomes quite clear if we rewrite TJ as

i l 0.6 0.5 0.4 0.3 0.2 0.1 / / / '

r

/ . / Pr 0.01 072 ^ 10 ^ TOOÖ^ ^ \ , \ \ ^ \ s. ^ ^ N , S w ^ f ^ ^ \

h > .

^

X

^ N N \ ^ _^ 1 2 3 k 5 6 .y. 7

FIG. 1.3. Velocity function for isothermal flatplate (Ostrach[23]).

^ 10 dT) 0.9 0.8 0.7 0.6 05 0 4 0.3 0 2 01 0 5 10 15 20 25 30 35 /.O 45 50 55 60 65 70 X

FIG. 1.4. Velocity function for isothermal flat plate (Sparrow and Gregg [40]).

\

k^

^ \ \ \

k

\

N

^ 0.003 ^,0008 ^ \ ^ -1 • ^

1 d

t — 1 1 1

1.0

e

0.8 0.6 0.4 0.2 \ \ \ lOOff^ • - > ^

1

AN

\ \ ^ \

y\

> Pr 001 A07

K ^

\ _\

k

^ ^ I \ \

\ \ '

_{^ .} ^ _{^ : : ^} ^ ^==3 ^ ^ ^ ^ "o 1 2 3 U 5 6

FIG. 1.5. Temperature function for isothermal flat plat (Ostrach [23]).

e

0.9 0.8 0.7 0.6 0.5 0.4 0 3 02

°:

\ \ \\ \ \

A

\ ^

A

\

A

\

k

\

V

\ ^ oDr-0.003 \ 0.008 \ s 5 ^ \ k V . \

^ - - .

^ . _ _ - - — . . "O 5 10 15 20 25 30 35 40 45 50 55 60 65

FIG. 1.6. Temperature function for isothermal flat plat (Sparrow and Gregg [40]).

z

e

70

7j = - | - r ^ | . (1.73)

with Gr^ as the local Grashof number

Grx = ^ . (1. 74)

In view of (1.7), (1.9) and (1.31), with i replaced by x, it is clear that 77 is associated with the viscous part of the free convection boundary layer. According to (1.32) the complete boundary layer

is 0(a~2) thicker than the viscous layer, whence we may expect that 77 has to reach values of this same order for arriving at the outer fringes of the boundary layer.

Through (1.40) we conclude that for large Prandtl numbers the same inefficient character of TJ exists. Here the stretching is of the order O {ai).

3.2 non-isothermal surface

Sparrow and Gregg [39] and Finston [7] have proved the exist-ence of a class of wall-temperatures giving r i s e to similarity solutions. Most important a r e the wall-temperatures

T ^ = T^ + Nx^^. (1.75) For k=0 their analysis naturally has to reduce to the isothermal

case treated in section (3.1). Again the equation of continuity is solved by the introduction of the streamfunction i/i as in equation (1.59). The reduction of the partial differential equations (1.60) and (1.61) to ordinary ones will be done through introduction of the similarity variable

k-1 ,

r? = c y x , c = ^ — 1 (1.76)

and the non-dimensional streamfunction f and the temperature 9 k43

^ = 4 i ; c / x ^ f(T7), (1.77)

1

V

/ / / /

r

-^ ^ \ '^

N

^ 0

p^3

1—01 l \ \ \ \ \ \ \ 0 'VJ N \ ^ ^

s^

-Ti !N ^ 130 e 0 90 OSD 0 60 / / /

i\N

\\

y

_\ ^ ^ r\ \ \ \ \ ^ \ \ \ \

V

\ \ \

V

v^

W O ^ \ ^ \ s \ \ \ -DB \ \ V*" \ ^^ \ ^ ^ — f ^ =— DS 10 IS 20 25 30 35 i 0 iS

FIG. 1.7. Velocity function for FIG. 1.8. Temperature function non-isothermal flat plate for non-isothermal flat plate (Sparrow and Gregg [39]). (Sparrow and Gregg [39]).

The result of the substitution of (1.76), (1.77) and (1.78) into (1.60) and (1.61) is the set of ordinary differential equations

d^f dT?3 + (k+3) _{f - i - - 2(k+l) - + 6 = 0 , (1.} d^f /df \2 dt)' \ d77 79) d^e (, , de df , — I + a (kw<3)f — - 4 k — 6 [ = 0 . drj ( drj dry (1. 80)

The boundary conditions to be satisfied by f and 9 follow from (1.58), (1.62), (1.63) and (1.75) df f = — = 0 6 = 1 at 7? = 0 , (1.81) df • 0 dri e - 0 _as 77 ^ c (1. 82)

The expressions for the velocity components u and v in the s i m i -larity coordinate a r e

k+1 . 2/7- 2 df

k-1 _ 4

V = - vex (k-l)rj — + (kH^)f df

drj (1. 84) These formulae may facilitate the derivation of the boundary con-ditions for f.

It is easily verified that for k=0 the equations a r e in complete agreement with those of section I (3.1) as has to be expected. For a = 0.7 (air) Sparrow and Gregg [39] integrated the equations for some special values of k (figures (1.7)and (1.8)). The most striking result is that for k=-0.8 the temperature of the part of the fluid immediately adjacent to the wall is higher than the temperature of the wall itself. As for k=-0.5 this behaviour is not present it can be concluded that for some value of k lying between -0.5 and -0.8 the slope of the temperature distribution must become zero on approaching the wall. From numerical ejqjeriments for different Prandtl numbers Sparrow and Gregg deduced that this value of k has to be equal to -0.6 independently of a. This can be proved directly through integration of equation (1.80) for k=-3/5

de de

dr7 d?7

+ —aie = 0 (1. 85)

T? = 0

Since lim f(T7) is bounded and 9 and dB/dri tend to zero as 77 ^

Tj -* CO

we find for k=-3/5

de

drj at

T) = 0 (1. 86)

independently of a. Although lim \lim f(T7)|~a~ 2 is not bounded cr- 0 JTj -°> (

(chapter II) the result (1.86) holds for cr -» 0 a s an inspection of equation (1.85) shows.

An important application follows from the e^qjression of the heat flux through the wall

3y y = 0 i 5k-l N l M X 4 ^ 4 i / ^ l dr] (1. 87) T? = 0

In fact for k = l / 5 the heat flux through the wall is constant. Upon solution of N from (1.87) and substitution in (1.75) it is seen that

the wall-temperature induced on a flat plate by a constant heat flux q is 1 , > 4 '5 l-\y — A» + 4 t . 2 q 4 ^ \ 5 g^k4 de

9(0)

I - — .(1.88) T? - 0

In chapter IV we will prove that the t e r m

9(0) d9 dTj _4 5 (1. 89) 0

is a constant independent of the boundary conditions imposed upon the temperature at 77=0. So one may find it by integrating the equations with 8(0)=1. In the first column of table (4.1) values of this constant may be found for different Prandtl numbers. The problem of constant heat flux has been treated in detail by Sparrow and Gregg [37].

An inspection of the similarity variable (1.76) shows that for k > 1 the boundary layer is of a different character than for k < 1. As for some special large value of r], e.g. TJ=10, the ambient condi-tions a r e satisfied up to a very high degree of accuracy we con-clude that for k > 1 the thickness of the boundary layer becomes smaller a s x -.<». This means that the l a r g e r the temperature differences the thinner the boundary layer. This also follows from (1.64) where a growing of T ^ - T» makes TJ larger. Thus for T;v - Toe large the outer edge of the boundary layer is reached for a small value of y. For k < 1 we have a r e v e r s e c a s e . The bound-ary layer grows with x. From this we may infer that the boundbound-ary layer always tends to grow in the x-direction, but that a growing temperature difference will curb this tendency. For k < 1 the former effect wins while for k > 1 this is the case with the latter. For k=l there is a boundary layer of constant thickness.

4. H e a t t r a n s f e r

An important feature of problems of the present type is the amount of neat being transferred through the wall to the fluid. In equation (1.87) this quantity has been derived already for the Sparrow-Gregg similarity solution. More generally, however, use is made of the local Nusselt number

Nu, ax (1.90) where a is the coefficient of heat transfer defined by

ST - \ —

ay _{y = o} = C^(Tw-Tj

Using (1.75) and (1.87) it is easily derived that

Nu, G r x l " de

drj

T? = 0

(1.91)

(1.92)

Here is Gr^^ the local Grashof number defined by (1.74). In many textbooks on heat transfer, however, the definition of the coef-ficient of heat transfer is different from that of equation (1.91). In fact a definition widely used in engineering applications is [2, 11]

9T - X — By

= a (T^-Tav) * (1.93)

y - 0

where T^y is the average (bulk) temperature defined by

JJ

u T d A

• a v (1.94)

u d A

Here the integration is performed over the whole of a plane x = constant. In forced flows where the fluid outside the boundary layer is non-quiescent the definition of T^y as given in (1.94) yields Tav = T;„5 so that the definitions (1.91) and (1.93)coincide. This can be seen directly for the flat plate case through writing

'av T„ + N x

J"

edy

ƒ"

(1.95) dy

If for y -•» the velocity u tends to some positive constant the integral of the denominator will be divergent. The integral of the numerator is convergent since 6 tends to zero in an ejqjonential way as y -><=. In those problems where only the boundary layer is

n o n - q u i e s c e n t , which i s the c a s e in f r e e convection, t h e a v e r a g e t e m p e r a t u r e T^y will g e n e r a l l y differ f r o m T^,. In d e a l i n g with t h e b o u n d a r y l a y e r flow a l o n g a continuous moving s u r f a c e Kuiken [17] g i v e s e x p r e s s i o n s for t h e N u s s e l t n u m b e r b a s e d upon t h e d e f i n i tion (1.93). Applying (1.94) w e find for Tay in c a s e t h e w a l l t e m -p e r a t u r e i s given a s in (1.75) Nx^ r'- df 1'av _{f(cc) J} I n t e g r a t i o n of equation (1.80) gives 0 e — d r , , ( 1 . 9 6 ) dr? de dr? /•co d f (5k+3) (3 d — dr], ( 1 . 9 7 ) J„ dn w h e n c e ri = 0 O N 3 ^ de T „ „ = T a v " ( 5 k + 3 ) a f ( - ) dr? ^ ^ Q We now e a s i l y d e r i v e t h e modified N u s s e l t n u m b e r ( 1 . 9 8 ) de d77 — a x / Gr.. \ T7 = 0 (5k+3)CTf(-) drjI^^Q It i s i n t e r e s t i n g to c o m p a r e t h e p r e s e n t definition of t h e N u s s e l t n u m b e r (1.99) with the f o r m e r one (1.90) for a w e l l - k n o w n p r o b l e m of t h e l i t e r a t u r e of f r e e c o n v e c t i o n . To that end let u s c o n s i d e r t h e i s o t h e r m a l flat p l a t e . In t a b l e (1.1) i n f o r m a t i o n c o n -c e r n i n g t h i s h a s b e e n a s s e m b l e d . U s e h a s b e e n m a d e of r e p o r t s of O s t r a c h [23] and of S p a r r o w a n d G r e g g [40]. It i s s e e n t h a t t h e d i f f e r e n c e b e t w e e n t h e two definitions a s e x p r e s s e d t h r o u g h t h e i r r a t i o d i s a p p e a r s a s cr - «. F o r ff- 0 t h e r a t i o s e e m s to a p p r o a c h a c o n s t a n t v a l u e . T h e a n a l y s i s of c h a p t e r n will give m o r e definite i n f o r m a t i o n about t h i s . It suffices to m e n t i o n h e r e t h a t t h e r a t i o a p p r o a c h e s t h e v a l u e 2.3736 in t h e l i m i t CT->0. The b e h a v i o u r for (;->» c a n b e explained a s follows. T h e r e s u l t s of O s t r a c h [23] show that for o -> <^ t h e t e m p e r a t u r e b o u n d a r y l a y e r b e c o m e s t h i n n e r w h i l e t h a t of the velocity g r o w s (figures (1.3) and (1.5)).

Consequently the value of the integral of the numerator of T^y (1.94) becomes smaller as compared with that of the denominator. In the limit <j = « the ratio will be zero, thus giving T^y =

Ta-xable 1.1 cr 0.003 0.003 0.010 0.02 0.03 0.72 1 2 10 100 1000 N u ^ / ( G r ^ ) ' 0.0319 0.0513 0.0574 0.0789 0.0952 0.3568 0.4010 0.5066 0.8269 1.549 2.804 NUx/(Grx)* 0.0751 0.1163 0.1663 0.1737 0.2053 0.5868 0.6304 0.7178 0.9803 1.639 2.851 Nux/Nu^ 2.356 2.269 2.896 2.200 2.157 1.644 1.572 1.417 1.186 1.053 1.016 ref. [40] [40] [23] [40] [40] [23] [23] [23] [23] [23] [23] 5. S k i n - f r i c t i o n

An important quantity in forced flows is the skin-friction. Although this quantity is of less importance in free convection we will give it h e r e for reasons of completeness

(1.100) = 0 'w pv 3u ay Au'^pc^x d ^ drj2

C H A P T E R I I

FREE CONVECTION AT LOW PRANDTL NUMBERS

1. I n t r o d u c t o r y r e m a r k s

THE study of free convection under low Prandtl number conditions has received the attention of a considerable number of authors. The reason for this is quite obvious a s low Prandtl numbers a r e realized by liquid metals, which a r e known to have a large c o -efficient of heat conduction. The free convection cooling with liquid metals has important applications e. g. in nuclear r e a c t o r s .

The methods of solution employed as yet maybe divided in two c l a s s e s . The first class of methods is concerned with the integra-tion of the original partial differential equaintegra-tions (1.51), (1.52) and (1.53) by means of the integral method of Karman-Pohlhausen [6]. The advantageous side of this method is that it gives results which display the Prandtl number explicitly. Disadvantages a r e its in-accuracy and its inherent systematic e r r o r s . In the other class of methods the transformed ordinary differential equations (1.79) and (1.80) a r e integrated by means of an electronic computer [40]. This may yield exact results. For each Prandtl number, however, a separate integration of the differential equations has to be p e r -formed. The results instead of being represented in one single formula have to be tabulated. Another disadvantage of the methods of the second class is embodied in the fact that for low Prandtl numbers the equations a r e of a somewhat "lopsided" nature and so a r e the solutions. This naturally is very inconvenient from a numerical point of view.

From this it may be clear that there is still need for a method having the favourable characteristics of both methods. It should be accurate and it should yield results containing the Prandtl number explicitly. In this chapter such a method will be supplied using a singular perturbation technique with matched asymptotic expansions [47]. Apart from having the required characteristics just mentioned the analysis will reveal much of the anatomy of a boundary layer of free convection. This is because the method of matched asymptotic expansions exposes the predominant factors in different p a r t s of the boundary layer. It goes without saying that the knowledge of these factors at low Prandtl numbers is of great theoretical interest.

F i r s t we will be concerned with the study of low Prandtl num-ber free convection for the isothermal flat plate. The knowledge acquired through this study will cause us to investigate the more complex Sparrow-Gregg equations (1.79) and (1.80) which govern some special cases of non-isothermal walls. Naturally the ana-lysis and its results will be compared with previous investigations, especially with those presenting exact solutions for a p a r -ticular small Prandtl number. It is of great advantage that a report of Sparrow and Gregg [40] gives extended account of results fit for comparison.

2. I s o t h e r m a l f l a t p l a t e

2.1 main t e r m inner- and outer expansion

In the first chapter we derived the set of ordinary differential equations governing free convection past an isothermal flat plate

d^f d2f / d t \ 2 ^ + 3f —5 - 2 — + 6 = 0 , (2.1) dT]3 dr,^ \dTjy d^e de — r +3CTf — = 0 , (2.2) dr)^ dT)

with the boundary conditions df f = _ = 0, 9 - 1 at r] = 0 , (2.3) dtj df . 0, 6 - 0 as Tj - « . (2.4) d-q

It is obvious that the first step of an investigation to flow at low Prandtl numbers will be to put C7=0 in equation (2.2). The general solution of the resulting equation for e then is

e = a + brj. (2.5) It follows that this solution cannot satisfy the two boundary

condi-tions (2.3) and (2.4) imposed on 6. So, according to the method of inner- and outer expansions the solution for 9 found in this way is valid, either near the flat plate where it can satisfy the inner boundary condition at r]=0 (2.3) or far from the plate (r?-")whereit

can satisfy the outer boundary condition (2.4). We have to select the first possibility since the equation depicting the dynamical part of the problem (2.1) contains the viscous t e r m d^f/dr?^. As viscous effects a r e only of importance near a solid wall the r e t e n -tion of this term is only realistic in a region adjacent to such a wall. In order to support these statements it is necessary to gain deeper insight into the physical implications. We may for example consider a superconductive (X =<»^CT = 0)fluid having finite v i s -cosity, density and heat capacity. For such a fluid the Prandtl number is zero. It now follows that the temperature of the surface of the plate is maintained in the fluid. For real fluids having large but finite conductivity this means that there exists a h o r i -zontal temperature distribution near the plate. Due to the ambient conditions (2.4) the whole (for CT=0) of the fluid experiences a

uniform force in upward direction. Consequently at some distance from the plate, where the viscous effects may be neglected, the fluid has a uniform acceleration in upward direction. This means that there exists a potential flow U which is proportional to X2. This follows directly from an elementary law of mechanics. So, for CT=0 the viscous p a r t of the boundary layer of free convection past an isothermal plate is related to the Falkner-Skan wedge-flow with a potential wedge-flow U=px2. It is well-known that the ordinary differential equation governing the viscous boundary layer wedge-flow with a potential wedge-flow

U=pxt (2.6) I S d^G d^G 2 t / d G \ 2 + G - ^ + ^ 1 - } = 0 , (2.7)

UMJ

dn^ dy2 i + t ) \dn/

where G is affiliated with the streamfunction like f is [30].Now upon substitution of 9=1 (uniform temperature) and

tin) = 18-^ G(M) , iu = 3(18)-577 (2.8) equation (2.1) becomes indeed identical with (2.7) for t =^ . In

dealing with the more complicated Sparrow-Gregg equations (1.79) and (1.80) at the end of this chapter we will endeavour to obtain systematically insight into this matter.

The considerations brought forward a s yet make us decide that there is an inner expansion

which is valid m the vicinity of the plate. The boundary conditions at the surface (2.3) apply to this expansion. The main term of the inner expansion ÏQ naturally has to satisfy the differential equation

d^fo d2fQ / dfo\2

•^ + 3fo — ^ - 2 - ^ ) + 1 = 0 (2.10) ?3 " dr?2 \dr]/

dr?

for reasons already mentioned. For the zeroth perturbation of the inner expansion Sinner °^ ^ ^ temperature we take for reasons explained before

:.o = 1- (2.11) Consequently there will be an inner expansion for the temperature

dinner = ^ + ^l^^^^l + ^2 (^) ^2 + (2. 12) For the expansion p a r a m e t e r s (^{0) and ï^{a) the following condi-tions hold

lim ^ = 0, lim ^+^ = 0 , e^ = TQ = 1. (2. 13) CTiO «i ffiO "^i

For the integration of equation (2.10) we lack up to now one bound-ary condition since the inner boundbound-ary conditions (2.3) only supply fQ=dfQ/dT7=0 at 77=0. The remaining third boundary condition will be found later through matching with an outer expansion to be derived next.

It is quite obvious that after having discussed the part of the boundary layer experiencing viscous forces the inviscid layer has to receive attention. F r e e convection of an inviscid fluid has been studied already by Lefevre [18]. So we may anticipate that the first t e r m of our outer expansion will be a solution of Lefevre's equations. Introducing the new variables

£ = 3(18)-4 T?ai, (2.14) W e r = 18-i CT-* F ( | ) , (2. 15)

W e r = ^^^' (2.16)

o , (2. 17)

d^* d$

—K + F — = O . (2.18) d?^ df

Indeed on taking 0^=0 the viscous t e r m d^F/d^^ is left out of the equation as should be. The forces of inertia and those due to buoyancy balance each other. The flow is not hampered by any viscous s t r e s s e s . Consequently the equations (2.17) and (2.18) a r e feasible for the description of the outer part of the free convection boundary layer. The boundary conditions to be satisfied by these equations naturally a r e those depicting the ambient conditions (2.4). So we have

dF

' 0 , f(f) - 0 as 4 - CO. (2.19) d4

The remaining boundary conditions have to be found through matching with the inner expansion. The outer expansions to be analysed will be given by

W e r = 1 8 - ? a - H F o + 6 i ( a ) F i + 62(cr)F2 + }, (2.20)

V t e r = % +"6i(or)*i + ' 0 2 ^ * 2 + ' ( 2 - 2 1 ) with

lim i l ? = 0, lim ^+1 = 0, 6Q ="00 = 1. (2.22) CTiO '^i ffiO ÖJ

Upon insertion of the outer expansions (2.20) and (2.21) in the equations (2.17) and (2.18) FQ and «ï^ indeed turn out to satisfy the equations of Lefevre

d^$0

dèf

2-„

Now according to the matching principle [47] we have to impose the conditions

lim f(T)) = lim f = lim 18" i <j-hF{^) , (2.25) n-^ inner ^^^ outer ^ , Q

lim e(Tj) = lim e = lim <ï>(£) , (2.26) ^^^ inner ^IQ outer ^ ^ Q

for (jiO. In the present analysis these conditions can be applied best [47] by writing (2.20) and (2.21) in inner variables (i. e. in T)) and expanding the expression for small values of cy. Comparison with the asymptotic representations (17-=°) of fjjjjjg^ and 9 inner then supplies the remaining boundary conditions. Now, upon a s -suming that FQ and ^Q can be expanded in a s e r i e s of increasing powers of Ë, as

F Q = F Q ( 0 ) + positive powers of Ë,, <jQ=<i'Q(0) -(-positive powers of È,,

an expansion of FQ and <E>Q for small values of CT will be (use is

made of equation (2.14))

fouter = 18-5 o'h Fo(0) + O(a^), r > - J (2.27) V t e r = *o(0) + 0 ( ' ^ ) . s > 0 . (2.28) From the fact that the inner expansion does not contain any

t e r m s with a negative power of a we may directly infer, using (2.25) that F Q ( 0 ) = 0 . Comparison of the inner expansion for 9 (2.12)

with (2.28) gives according to (2.26) that $ Q ( 0 ) = 1 . In fact these boundary conditions were also used by Lefevre. Hence the zeroth perturbation of the outer expansion has to satisfy the system of

equations (2.23) and (2.24) with the boundary conditions.

FQ = 0, *o = ^ ^* f = 0> (2.29) dFQ

> 0, $n - 0 a s 4 - » . (2.30) d^ "

A s e r i e s solution of FQ and $ Q valid near Ë, -0 is

7 2 , 1 0 o T ^0 Q ^0°0 s n 2, 13 H

7 , 2 3 ^ 0 4 %% 3 3

*0 = 1 + ao 4 - J i | 3 _ _ 0 . |4 _ _ L ao bo €3 + 0 ( 4 3 ), 6 12 130 (2.32) with ag =-0,582983, (2.33) bo = 0.13744 . (2.34) Another quantity of importance is

FQ(C) = 1.007366 . (2.35) Inspection of the first term of the inner expansion shows that

for CT iO the outer expansion should supply a term not involving CT . Indeed using the first t e r m of expansion (2.31) the matching p r i n -ciple (2.25) yields

lim 10(77) = -^ 77V2 + t e r m s of order lower than rj. 77-» TO

This is equivalent with dfo 1

- — - r ^^2 a s 7 7 - - . (2.36) dr? ^

This condition is exactly the one we had to expect since on using (2.8) it can be shown that (2.36) is in agreement with the asymp-totic boundary condition of the traditional Falkner-Skan problem

dG

> 1 as /i - °°.

d/i

Integration of equation (2.10) with the inner boundary conditions and (2.36) renders

d2fo

— — = 1 . 0 6 9 9 4 9 6 at rj = 0, (2.37) d77'^

fo(")~ 0.7071068 7?-0.359348+exp - as 77-". (2.38)

Here exp- means terms of exponentially small order.

Graphs of the main terms of the inner- and the outer expan-sions may be found in the figures (2.1) through (2.3).

expansions depicting two important regions of the boundary layer of free convection. Before proceeding to the t e r m s of higher order it is interesting to compare the results derived up to now with results of the literature. In the figures (2.4) and (2.5) the continu-ous curves a r e the results of a numerical integration of Sparrow and Gregg [40] for (7=0.003 and a =0.03. It follows quite clearly that the inner expansion covers that part of the boundary layer where large velocity gradients exist; the outer expansions refer to the small velocity gradients. The predominant part of the free convection boundary layer is non-viscous and tends to create a uniformly accelerated flow at the surface. This follows from (1.69), (2.15) and the fact that dFQ/d4 = l at 4 = 0 (see (2.31)). How-ever, on coming within the range of the viscous s t r e s s e s a viscous boundary layer of forced flow type is created, which in the limiting case of cr = 0 is exactly the same as the Falkner-Skan boundary layer originated by a potential flow which is the same as the velocity distribution of Lefevre at the flat plate. It is worth to compare the numerical figures of the present analysis with those of Sparrow and Gregg (table (2.1)). F i r s t it is seen that the values of d^i/dq^ at TJ =0 quite naturally tend to the limiting value given in this thesis d2fQ/d77^ = 1.0699496 at T? = 0(2.37). Anticipating already the findings of the next section we may deduce from the figures of table (2.1) that the differences a r e of the order 0(CT2). Whereas the figures of the first column refer to an outcome of the inner expansion, the figures of the other two columns have to do with the outer expansion.

Table 2.1 d^f/dTj^ _{f(«>) 18"4ff~2Fo(")} ?7=0 0.003 1.0223 8.7060 8.9290 0.008 0.9955 5.4018 5.4678 0.020 0.9590 3.4093 3.4582 0.030 0.9384 2.7878 2.8236 2.2 first perturbations

In view of the fact that the first t e r m s of the expansions (2.31) and (2.32) a r e integer powers of 4 we infer that the expansion variables ej(cT), ëj(a), öi{o) and öj(cr) a r e all equal to a2. This is because f is proportional to a s . insertion of the inner- and outer expansions (2.10), (2.12), (2.20) and (2.21) in their respective

systems of differential equations (2.1), (2.2) and (2,17), (2.18) two systems pertaining to the first perturbations follow. For the inner problem we have

0, (2.39)

(2. 40)

while for the outer problem the differential equations a r e

F j + * j = 0, (2.41)

0 . (2.42) d^fj d^fj

+ 3 fo

dr?^ dT}2

dfo dfi d2fo - 4 +3 —-t^+e^ dr] dr] dr] d^ei d77^ 3 2

Fo

d^Fj d | 2 - 2 d 2 * j ...2 dFo d | + F d F i + d€ d # j 0 .,. 3 2 + d^Fo d^^

d*o

F

d r d4 d4

If we first pay attention to the temperature it is seen through (2.40) that

9 J = cc^n- (2.43) Here we have used the inner boundary condition 9j(0)=0. The two

term inner expansion for the temperature now is

Sinner = 1 + ^iVo'^- (2.44) Obviously the asymptotic behaviour of öjnner ^^^ rj-rois described

by (2.44) too. If for a moment we assume that $2(0)^0 a two t e r m outer expansion written in inner variables for a iO gives (see (2. 32))

*outer = 1 + ( 3 a o l 8 - i r j + * i ( 0 ) ) a è . (2.45) Comparison of (2.44) and (2.45) gives according to the matching principle (2.26)

ttj = 3 ao 1 8 - Ï = - 0 . 8 4 9 1 0 , (2.46)

and

The general solution of (2.41) and (2.42) which satisfies the condition (2.47) can be represented near 4 = 0 as

3 7 3 3 2 ^ 1 "" ^1 + "2 ^0 ^1 ^ + -3 ^1 ^0 ^ + (2 ^0 ^1 + '^l) ^ + _7 _8_ + c j 4^ + ^ a i b o 4^ + 0 ( 4 ^ ) , (2.48) 10 ^ 1 = ^ 1 ^ - i ^ h ^^ - ( i ^ o ^ + i v ^ ^ + 0 ( 4 ^ ) . (2.49) Writing down a two term inner expansion and a two term outer expansion for f as required by the matching principle (2.25)

lim finner ~ \ V'^^ - 0.359348 + CTifi(rj-.-), (2.50)

li'" W e r ~ i ^ ' ^ 2 + 18-^ ai +

4 i 0

+ a2{9(18)"^ aQTj^ + t e r m s of order lower T?^} . (2.51) it follows that

a^ = -(0.359348)(18)'* = - 0 . 7 4 0 1 7 4 (2.52)

and

. 18 aQ = - 1 . 2 0 0 8 1 a s r ) - ™ . (2.53) drj^

Among the t e r m s of order lower rj^ in equation (2.51) occurs a t e r m which originates from the expansion of F j (2.48). This is the t e r m

| - a o a i rj ^ 2 = 0.45768 77. (2.54) The asymptotic behaviour of fi(Tj) turns out to be after integrating

(2.39) with the inner boundary conditions (2.3) and the one found through matching (2.53)

Apparently once the condition (2.53) is imposed the term (2.54) appears automatically in the asymptotic behaviour (2.55). The fact that both coefficients of TJ in the expressions (2.54) and (2.55) a r e results of completely different integrations gives a valuable means of determining the accuracy of the numerical work. Apparently at least five decimal places a r e significant. Yet the integration of the equations of the outer problem is rather difficult as the coef-ficient of the highest derivative in one of the equations is zero at one of the boundaries (Fo(0)=0). For the skin friction an important figure is

d \

= - 1.Ó01023 at 7] = 0. (2.56)

dTj2

The system of equations (2.41) and (2.42) has to be integrated with the outer boundary conditions (2.19) and with F2(0)=aj((2.48) and (2.52)) and $j(0)=0 (2.47). This integration naturally supplies the numerical values of the remaining constants b j and c j of the expansions (2.48) and (2.49). For subsequent use the value of b j will be given here

b i = 0.31445 . (2.57) Another quantity of importance is

Fi(<») = - 0.10867 . (2.58) Graphs of the first perturbations can be found in the figures (2.1)

through (2.3).

2.3 second perturbations

Inspection of the expansions (2.31) and (2.48) shows that the second perturbation is influencedby t e r m s like | V 3 and ^^/^ etc. A three term outer expansion for f and 6 gives for the t e r m following CT2

^

5

2

2 I_ 5 4

, 3 r „ 6 „ 3 , 3 „ 1 2 „ 6 „ , 3 W e r = o {2 3 borj + 2 3 7 a^ bQ T, + 4 o

r

1. 0.8 0.6 O.A 0.2 d F \ d^ "o 1 FIG. 2.] 1. 08 0.6 OM 0.2 0 ] 1 FIG. 2.2. dFo ^ \

x:\

^ ^ ^ 2 3 ^ 5 6 7 ^

. Universal functions of outer expansion (velocity).

\ & \

\ - ^

2 : } i, 5

S 7 1

1

0.5 1. 1.5 2. 2.5 T) 3

FIG. 2.3. Universal functions of inner expansion (velocity).

FIG. 2.4. Comparison of Sparrow-Gregg [40] solution (a = 0.003)

(continuous curves) with one-term inner- and outer expansions

(dotted lines).

As a consequence the expansion variable e2('^) °^ ^ ^ inner ex-pansion should be CT^/^. Consideration of an expression for ÖQytpr

analogous to (2.59) suggests that e2(a)be merely g a s the fractional powers enter the outer problem for 6 in a much later stage. So for the second perturbation of the inner expansion we merely have a differential equation for f2(T?)

( i \ d.\ dfo dfo d^fQ

+ 3 fo ^ - 4 + 3 ^ f2 = 0 . (2.60) drj-^ drj^ dri drj dr]^

On account of the matching principle this equation must have an asymptotic behaviour coinciding with the expression between brackets of equation (2.59). Using the asymptotic expansion (2.38) of fg the solution (2.60) for large values of 77 has to satisfy the equation

d^fg d \ dfg

+ 3(èTj^2 - p) 2 ^ 2 = 0 . (2.61)

drj^ d?}^ d77

Here t e r m s of exponentially small order have been omitted and p is the constant 0.359348. Now it is easy to show through substitu-tion of •? 1 1

77 = 2 3 i M + 2 p ' that K = df2/dii satisfies the equation

- 2AJ + -I K = 0 . (2.62)

2 3 d/i d;i

This equation is known to have the general solution [22]

^ 2 2 1 9 1 3 9

— = A H [ - | , i - ; M^] + BMH[--i, - | ; M^ ],(2.63)

where H[a, b; x] is the confluent hypergeometric function

00

Tir V. 1 - 1 ^ ^ a(a+l) x^ _ V^ W n ^ ^ H[a, D, xj - 1 + ^ jT- + ^ ^ ^ ^ 2T + • • • • - ^ (b)n n ! '

Application of the asymptotic expansion for the confluent hyper-geometric function (2.64) (see [33])

[L]

H[a, b; - x] ~ x-a -fM^ J ] ^^^n^^+^-^^n x " " + n=0 " *

+ 0 ( x - ^ - [ L ] - l ) as x - c (2.65) makes it easy to prove that the highest order t e r m s of df2/dr7 for

77-- a r e 4 ^ 2 iv - p^/2)^ (r; - pV2)^ (rj - p^2) ^ etc.

It is thus proved that matching with the outer e^qjansion (2.59) is possible. We will confine ourselves to these qualitative r e m a r k s about the second perturbation of the velocity since it is in the first place the heat transfer results which deserve the attention and for which numerical results a r e important. As to this it is possible to draw an important result from the numerical figures obtained as yet. Upon proceeding along exactly the same lines as was done up to now it is easy to show that the expansion variables to be used now a r e

€3(0) = T2(CT) = 62(cf) = 02(a) = CT.

The inner expansion for the temperature then yields a solution

62(77) - « 2 ^ (2-66) giving the three t e r m inner expansion

dinner = 1 " 0.84910T7CT2 + a 2 ^ ' ^ - (2.67) As a three t e r m outer expansion for the temperature is

^outer = 1 - 0-84910rja-2 + [3(18)-ïbjT7 +

+ t e r m s of order lower than TJ ]a (2. 68) the constant is easily proved to be

02 = 3(18)-5 b j = 0.45799. (2.69) With respect to this it is interesting to r e m a r k that the t e r m with