5.

6. 7. 8. 9. i0. Ii.

M. Ya. Ivanov and A. N. Kraiko, "The forward method for calculating two-dimensiOnal and three-dimensional supersonic flows," Zh~ Vychisl. Mat. Mat. Fiz., 12, No~ 3 (1972). V . M . Dvoretskii and M. Ya. Ivanov, "Computation of mixed flow in nozzles with an asymmetric subsonic part," Uch. Zap. Tsentr. A~ro-gidrodinam. Inst., ~, No. 5 (1974) o V . M . Dvoretskii, "Investigation of three-dimensional mixed flows in nozzles with asymmetric entrance secnions," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1975). A . A . Shishkov, Gasdynamics of Solid Rocket Motors [in Russian], Mashinostroenie (1974). W . C . Strahle, "A theory of the aerodynamics of the supersonic splitline gimbaled nozzle," J. Spacecr. Rockets, ~, No. 2 (1967). B . V . Orlov and G. Yu. Mazing, Basic Thermodynamics and Ballistics for Design of Solid Rocket Motors [in Russian], Mashinostroenie, Moscow (1968). G . I . Averenkova, E. A. Ashratov, G. G. Volkonskaya~ Yu~ N. D'yakonov, N. I. Egorova~ D. A. Mel'nikov, G. S. Roslyakov, and V. I. Uskov, Supersonic Jets of a Perfect Gas [in Russian], Izd. MGU, Moscow (1970).

METHOD OF SINGULAR PERTURBATIONS IN THE PROBLEM OF FREE CONVECTION WITH CONSTANT HEAT FLUX ON A VERTICAL SURFACE A. A. Berezovskii and Yu. A. Sokovishin

UDC 536.25

The process of laminar free convection for medium values of the Grashof number is examined. An asymptotic solution which accounts for the effect of the leading edge and interaction of the boundary layer with the external flow is constructed. A comparison with experimental data shows that the solution obtained is applicable for Ra > i0 =.

A description of the process of laminar free convection in a medium in contact with a heat-generating surface, in the framework of boundary-layer theory [1-3], leads to an appreciable and systematic discrepancy with experimental data in the region of medium Grashof numbers [4-6]. The explanation lies in the asymptotic nature of boundary-layer theory, whose equations are obtained from the complete Navier-Stokes equations and the equations of continuity and energy as one goes to the limit as Gr § =. Corrections to boundary-layer theory for medium Grashof number values must account for interaction of the boundary layer with the external flow [7] and also the leading edge effect [7, 8]. Attempts to account for interaction of the boundary layer and the outer flow, proposed in [9], lead to results differing considerably from the expected values, since the authors superimpose the effect of the trailing edge, simplified by the assumption that the streamlines in the wake are parallel to the flat plate. This paper establishes the effect of the leading edge and of interaction of the boundary layer with the external flow using the methods of singular perturbations. i. Statement of the Problem.

Basic Equations

We consider two-dimensional steady motion of a viscous incompressible fluid about a vertical semiinfinite flat plate with a given constant heat flux on the surface. The origin of the coordinates is located at the leading edge of the flat plate. The x axis is directed along the plate and the y axis, normal to it. Dimensionless variables are used: As unit length we choose the characteristic dimension L, the stream function is referenced to ~Gr*~2~ and the excess temperature is referenced to Lqw/%, where ~ is the kinematic viscosity, % is the thermal conductivity, and qw is the heat flux. In the Boussinesq approximation the Leningrad. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 129-136, March-April, 1977. Original article submitted July 26, 1976. This material is protected b y copyright registered in rite name o f Plenum Publishing Corporation, 227 West ] 7th Street, N e w York, iV. Y. 10011, NO part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any -form or by an), means, electronic, mechanical, photocopying, microfilming, recording or otherwise, w i t h o u t written permission o f the publisher. A copy o7" this article is available -from the publisher.for $7.50.

1 [

I

j

271

basic equations for the dimensionless stream function ~ and temperature 8, with no viscous energy dissipation, and the corresponding boundary conditions take the form

4u (V 24) x--4x ( V 24) ~=Gr *-'~ V '4+0u 4yO~-4~O~=Pr-'Gr*-'~V 20 4.=~u=O, O~=--l, y=0, x > 0 4~=~=0~=0, y=0, x < 0 4~=$~=0~0, r - ~ , ~ 0 r=(x~+y2) '~, ~ = a r c ~ y / x

(1.1) (1.2)

For l a r g e v a l u e s of t h e G r a s h o f number t h e probl em ( 1 . 1 ) , ( 1 . 2 ) , u s i n g t h e t e r m i n o l o g y of [10], is a problem in s i n g u l a r p e r t u r b a t i o n s . We choose E = Gr*-*P as t h e p e r t u r b a t i o n parameter. I n o r d e r to i n v e s t i g a t e t h e b e h a v i o r of t h e s o l u t i o n i n t h e b o u n d a r y l a y e r we i n t r o d u c e c o o r d i n a t e s which a r e of o r d e r u n i t y i n t h e b o u n d a r y l a y e r . Here t h e l o n g i t u d i n a l c o o r d i n a t e i s s l i g h t l y deformed i n o r d e r t o d e f i n e t h e t r u e p o s i t i o n of t h e b o u n d a r y l a y e r r e l a t i v e to the coordinate origin:

x=X+](X, Y)e,

y=Ys

(1.3)

The deformation function f is evaluated from the conditions that the similarity nature of the solution is preserved in the boundary layer and that the vorticity falls off exponentially at the outer edge of the boundary layer. The solution in the boundary layer can be represented as an asymptotic expansion for Gr § ~ with a fixed X and Y: 4(x, y; Or')=e~r Y)+8~ar Y)+... 0(x, y; Gr*)=e0o(X, Y) +e~O~(X, Y ) + . . .

"

(1.4)

Outside the boundary layer the solution has the form ~(x, y; G r ' ) = e ~ 0 ( X , y ) + e ~ ( X ,

y)+ . . . .

O(x, y; Gr*)~O

(1.5)

Here Gr + ~ f o r f i x e d v a l u e s X, y; x = X + f ( X , 0 ) e . The law of t r a n s i t i o n e x p a n s i o n s ( 1 . 4 ) and ( 1 . 5 ) .

t o t h e l i m i t d e t e r m i n e s t h e r e g i o n of a p p l i c a t i o n

of each of t h e

2. Zeroth Approximation Assuming the external flow to be irrotational, zeroth approximation to the outer flow:

from Eqs. (1.5) and (i.i) we obtain for (2 .i)

V~0=0 Since the boundary conditions are zero, without loss of generality we have (I)o(X, Y)~O

(2.2)

The matching procedure determines the boundary condition at the outer edge of the boundary layer: 40r (X, oo)=0 (2.3) The zeroth approximation for the boundary layer is written as a system of equations obtained by substituting Eq. (1.4) into Eq. (i.i): 4or0ox-4ox0or----Pr-'0orr

(2.4)

The system of equations (2.4) coincides with the well-known system of boundary-layer equations and has a similarity solution of the form [1-3]

4o(X, Y)----(5X)q'Fo(n), O0(X,Y)-=(5X)q'Ho(n),

7]=Y/(5X)'4

To determine Fo and Ho we have H0"+Pr (4FoHo'--HoFo')=0, Fo (0) =F0' (0) =Fo' (~) --H0 (~) =0,

Fo'"+4Fo"Fo--3Fo'Fo'+Ho=O,

(2.5) H0' (0) = - i

(2.6)

At the outer edge of the boundary layer F0(n) ~F0 (oo)+exp,

272

n-,-oo

(2.7)

Here the notation exp deno=es terms which are exponentially small for ~ + ~. 3. First Approximation For the firs= approximation to the outer flow we have V~=0

(3 ~

The boundary conditions are determined from the matching using Eq. (2.7):

gAt(X,O)-~(SX)'/~Fo(oo), X>O; @,(X, 0)=0, X < 0

(3.2)

The solution of Eqs. (3.1) and (3.2) is given by the Poisson integral 4 4

O, (X, y) = - F o (~) (Sr)'/'sin g (q~-~) / sin -~ ~

(3.3)

The matching procedure determines the bounda-y condition for the first approxi~ ation to the boundary layer:

,,:~ (X, oo)=e,v(X, O)+lr~ox(X , co)

(3.4)

The system of equations for the first approximation ~o the boundary layer has a similarity solution:

$,(X, Y)-~/*ox+F,(~I),

O,(X, Z)=fOox+(5X)-V,H,(N)

(3.5)

Here the exponential decay of the vorticity at the outer edge of the boundary layer is assured if

](X, Y)= rctg 4-~-_~ +i__ F, (oo) (5X) '4

(3.6)

To determine F, and Hz we have

Fi"+4F['Fo-2F(F~'+H~=O,

II['+Pr(4H(Fo+3H~F(-F,'H~)

F, (0) =F,' (0) =H,' (0) =H, (~o)=0, 4 F, (vl) ,~ -4F~ (~) ctg T ~ l + f ~ (r176+exp

=0

(3.7) ~t-+ oo

4. Second Approximation The second approximation for the ouner flow is given by the equation

V~F~=0, F~(X,v)=~(X,v)

F,(oo) (5X)'4~,x(X,Y) 4I Fo(~)

(4.1)

The boundary condition obtained from the ma~ching has the form

r

0) =2F,.(oo)

(4.2)

He nce

F~(X, y) =F,(~) (i-(p / ~)

(4.3)

~ (x, oo) = ~ (x, o) +l,-,,~(x, ~)-]~/~,o~(X, ~o) ,,,~(x, o~)=r 0) +21~,,~(X, ~)-21d~,o~r (X, ~ ) - I ~ , o ~ (x, oo)

(4.4)

The matching procedure gives

The system of equations for the second approximation for the boundary layer has ~he similarity solution of the form

fp~(X, Y)=~/~ff~oxx+jF, x+(5X)~4f,(~),O~(X, Y)=~/~fOoxx+f( (5X)-'4H~(,q))z+(SX)-V,H~(r~)

(4.5)

To determine Fa and Ha we have

=-~FinF( +48Fo'Fo+ ~2F(Fo%]--$Fo"FJ~=--f~F~n f o~ 2 + !2f on-/~F'~'' ~--2F~m ~]~ H2" +Pr (~f~'Fo+Tf-f~o'--F~,ff~.-4FJfJ) =4Ho+/d-7o'~l -H~#~]2.3p r F/H~ Fz(O) = F ; (0) =H~' (0) =H, (~)-----0, 2 l F2(n) ~8F0(=o)n - - - - F , (oo) ~]+F~(~) +ex p ~]-+=

(4.6)

273

TABLE i n

~

Pr

1

Po'(O)

H~(O)

Fo(~)

F,'(O)

H~(0)

F~(~)

1%"(0)

H2(O)

2.4850 0:9298 0,733 0.8089 t.0 0.7220 2.0 0:5592 0.0 0.3064

4.1977 1.6522 t.4798 t.3534 1.t300 0.7675

3,f149 0.67t4 0,5074 0.4347 0.332t 0.t810

-13.264 -0.7i69 -0.4t69 -0.25t5 -0.0522 0.0362

-23.785

-0.5246 -i.t667 -t.3506 -1.5285 -2.0368 -5.5372

-t.30~5 0.8t32 0,5671 .0.0261 0.5o44 0.t219

42,202 -5,9788 -5.5537 -4.2971 1.0502 2.9581

0.03

2

0,5

3 4 5 6

-2,442I -1.807 1.4248 -0.867t -0~3422

5. Eigen Solution In the solution process eigen solutions which satisfy the zero-order boundary conditions can be assumed. We therefore add to the expansion (1.4) terms of the type

~8%+~h(n) I (5X)~'(%-'), ~%+'gh(n) / (5X)~'(n-~i

(5.1)

Then to determine fk, gk we obtain

if' +4h"Fo+4( ~ - % ) ~ ' F o ' - 4 ( = ~ - i ) ~ F o " + ~ = 0 gk"+Pr (4~'F0+4 (a,--'h) g,Fo'-4 (=,-- t) ],Ho'-A'Ho) =0

(5.2)

The firs~ eigen solution, corresponding to the eigenvalue a, = 5/~, has the form /,=fiFo--Fo'~,

(5.3)

g,=Ho--Ho'n

The constant c~ is determined from the fact that the eigen solutions appear because of indeterminacy in the flow details at the leading edge [i0], while the deformation of the longitudinal coordinate has the objective of eliminating the indeterminacy

c,=--L (~) 14Fo(~) 6. Comparison with Experiments.

Discussion

Numerical calculations of the system of ordinary differential equations (2.6), (3.7), and (4.6) were performed for six values of the Prandtl number. The results, shown in Table i, indicate that the relative influence of the higher approximations increases with reduction in the Prandtl number. The largest number of experiments were conducted to determine the heat-transfer coefficients. In the general case the expressions for the local and average heat-transfer coefficients using the given theory have the form, respectively,

6

Nux

1.5 Z

~ I \3

f.#

#

!

g3

s Z

8.2

# Fig. 1

274

/

0.5

0.7:

,~

/

/

i

/ 10 g

Fig. 2

10 Z

- -

1

fO

t

/

i

t

1

1

I

q f

~0 o

fa-~

.

r o

103

fOg

t0

.

.

10i

t

.

~0 s

,

706

107

10#

#0 r I0 ~o

Fig. 3

l

o,,'~".

~ t

[ 5 (Gr')

N~=M~ --(a-t-b) l~a + a

25

a"

5's

i (3a+5b§

"'

a (or.. ''

y ~ :

b~+ c /

-',~

t

I

,Gr','/'

+y("+~

l)

a

.-7) - y -

a 9 Gr''-'l=

( Gr" "-'/' Gr . . . . /,_ l "2-0 (-a2"6abd-b2d'c) -,-'5-")

az

---a--,-t--~- (-.-~----)

(3a2+Sab+b~+c)

[Gr=.F',.+

5~', (--'5"-)

a=-Fl

+

,

Here the coefficients ~, b, and c, and also He(O), depend on the Frandtl number. Figure 1 compares the theoretical curve corresponding to Eq~ (6.2) with the approximate curve, interpreting the experimental data for both laminar and turbulent conditions [II]: 'h

NuL =0.825+0.387 Ra"/q [ t § (0.437/Pr)'/"]'/"

(6.3)

For curves 1-3, Pr = 0.03, and for curves 4-6, Pr = 0.7. Curves 2 and 5 relate to Eq. (6.2), curves 3 and 6 relate to Eq. (6.3), and curves 1 and 4 are for boundary-layer theory. The comparison shows tha~ ibis solution approximates to ~he empirical dependence for reduced Grashof numbers, but for increased Grashof numbers it approxima=es to boundary-layer theory. It should be noted that the asymptotic nature of the solution limits its region of application to the values Ra >_ I0 =. Nevertheless, the form of Eqs. (6.1) and (6.2) is of assistance in constructing the approximations

1

[/Grz'~'/'

1NUx ".(0) k ~ - " ~ ] Su~

--

H0(0)

~ \ 5 ]

t

t

G r z ' ~ - ' / - b I (Gr~,") -v, 2-5"t--~-

+--5-'--6 ~"('-~'--I q---In 4

q- 5'/,(5-v'--5 -`

.-S

]

(6.4) (6 5)

which g i v e a good r e p r e s e n t a t i o n of t h e e x p e r i m e n t a l d a t a r i g h t up t o t h e v a l u e Gr = 1, This i s i Z l u s t r a ~ e d • F i g . 2, w h i c h shows t h e e x p e r i m e n t s on t h e l o c a l h e a t - t r a n s f e r coefficient

f.O 0,8 0.6

- - _

\

.....

]

t I I

I # /

i

f

0.2

0.#

0.8

Z

1.8

2,

Fig. 4 275

r - f _

0.8

0.6

I

0.Z

0

Z

6

~

0

/0

IZ

I#

Fig. 5 for mercury. In addition to the theoretical curve [Eq. (6.4)], Fig. 2 shows also an empirical dependence for Pr = 0.028:

lg Nux=-0.551+0.i451g Grx'+0.00~(lg Grx*) ~

(6.6)

The d a t a a r e as f o l l o w s : 1) Eq. ( 6 . 4 ) ; 2) Eq. ( 6 . 6 ) ; 3) b o u n d a r y - l a y e r t h e o r y . A s i m i l a r picture is obtained also for the average heat-transfer coefficient, as shown in Fig. 3, where a comparison is made of the present theory with correlated experimental data [Ii]. The data are as follows: I) Eq. (6.3); 2) Eq. (6.5). Figures 4 and 5 show the temperature profiles in the boundary layer for water and mercury, computed from the present theory and determined experimentally [4-6]. The solid curves correspond to boundary-layer theory and the dashed lines, to the theory developed here for different values of the Grashof number; the shaded region is the experimental points. By comparing these profiles we can explain the large scatter of the experimental data, particularly for mercury, as attributable to the fact that the experiment was performed for various Grashof numbers. In Fig. 4, curve i corresponds to the values Gr = 108; curve 2, to Gr = lOS; curve 3, to Gr = 104; curve 4, to Pr = 5; curve 5, to Pr ffi i0. In Fig. 5, curve i corresponds to Pr = 0.024; curve 2, to Pr = 0.03; curve 3, to Gr = 108; curve 4, to Gr = 107 , and curve 5, to Gr = 106 . The deviation of the present results from boundary-layer theory becomes particularly understandable for reduced Grashof numbers, if we consider that the solution constructed describes the flow, beginning somewhat downstream of the leading edge. In fact, at the origin of coordinates because of Eqs. (1.3) and (3.6), we have

X=(-F,(~)

/ 4Fo(~))'/'(Gr'/5) -~

Hence it follows that the relative effect of the leading edge is increased, L and, therefore, the smaller the Grashof number.

the smaller

LITERATURE CITED I. 25 3% 4 5. 6.

276

E . M . Sparrow and J. L. Gregg, "Laminar free convection from a vertical plate with uniform surface heating," Trans. ASME, 78, No. 2 (1956). G. Wilks, "External natural convection about two-dimensional bodies with uniform surface heat flux," Intern. J. Heat Mass Transfer, 15, No. 2 (1972). N . N . Tran, "Sur la convection naturelle laminaire autour d'une plaque plane en incompressible," Compt. Rend. Acad. Sci., Ser. A, 275, No. 21 (1972). K . B . Chang and R. G. Akins, "An experimental investigation of natural convection in mercury at low Grashof numbers," Intern, J. Heat Mass Transfer, 15, No. 3 (1972). D . V . Julian and R. G. Akins, "Experimental investigation of natural convection h e a t transfer to mercury," Ind. Eng. Chem. Fund., 8, No. 4 (1969). R . G . Colwell and J. R. Welty, "An experimental investigation of natural convection with a low Prandtl number fluid in a vertical channel with uniform wall heat flux," Trans. ASME, Ser. C, 96, No. 4 (1974).

7. 8. 9. I0. ii.

K.-T. Yang and E. W. Jerger, "First-order perturbations of laminar free-convection boundary layers on a vertical plate," Trans. ASME, 86C, No. 1 (1964). K. Brodowicz, "An analysis of laminar free convection around an isothermal vertical plate," Intern. J. Heat Mass Transfer, ii, No. 2 (1968). K. S. Chang, R. G. Akins, and S. G. Bankoff, "Free convection of a liquid metal from a uniformly heated vertical plate," Ind. Eng. Chem. Fund., ~, No. 1 (1966). M. D. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York (1964). S . W . Churchill and H. H. S. Chu, "Correlating equations for laminar and turbulent free convection from a vertical plate," Intern. J. Heat Mass Transfer, 18, No. ii (1975) o

EQUATIONS OF VISCOUS SUPERSONIC JETS WITH A HIGH DEGREE OF OVEREXPANSION N. F. Borisov and V. A. Syrovoi

LrDC 621.43.011:533 ~

A complete system of equa=ions determining a viscous laminar, strongly overexpanded jet is obtained; the system is formed by shortened Navier--Stokes equations, equations for the metric of a coordinate system related with the form of the jet, and equations of transition from curvilinear coordinates to Cartesian. ~le problem of calculating the jet is formulated as a Cauchy problem for this system. Two- and three-dimensional flows are examined. Possible swirling of the jet is taken into account.

Despite the success in developing numerical methods (see, for example, [i, 2]), including automatic condensation of the mesh in regions with sharp gradients [i], the solution of Navier-Stokes equations presently represents a rather complex problem. Since the dimensionality of the arrays with which the computer must work is equal to the dimensionality of the problem, restrictions on the computer memory results in the calculation of two-dimensional flows being already at the limit of the capabilities of a B~SM-6 computer. We add to this that consideration of the multicomponent and multiphase character of the discharge products, nonequilibrium chemical reactions~ and phase transitions requires a considerable expansion of the computer memory and an increase of machine time. By virtue of this, shortened Navier-Stokes equations (and, particularly, an approximation of the boundary layer) will henceforth represent a constructive tool for investigating viscous flows with consideration of various dissipative processes. In this approximation the computer will operate only with two-dimensional arrays when solving three-dimensional problems. The purpose of the present work consists in supplementing the theory of viscous jets with equations for strongly overexpanded flows which, as far as the authors know, have not been obtained heretofore. In deriving thes~ equations it is necessary to introduce a coordinate system related with the form of the jet so that one of the axes is directed along the jet and the other is directed mainly across the mixing layer. The Navier-Stokes equations written in this system are simplified on the assumption of smallness of the longitudinal gradients in comparison with the transverse gradients. In other words, "shortening" of the Navier--Stokes equations consis=s in simplifying the viscous components adopted in boundary-layer theory and in preserving all inertial terms. Such equations, obviously, will describe both the inviscid core and the mixing layer and will automatically take into account their mutual effect. Continuous integration with respect to the cross section of the jet has an advantage over the iterative process beginning with build-up of the boundary layer along the boundary of the Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137-147, March-April, 1977. Original article submihted August 16, 1976. 10011. N o part o f ~his publication may be reproduced, stored in a retrieval system, or transmitted, in any form or b y any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the p~r A copy o f this article is available from the publisher f o r $7.50.

277