Partial differential equation for heat transfer in plane incompressible laminar jet is reduced by similarity transformation to ordinary differential equatio

Millsaps, K.

Soong, N. L.

English

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HEAT TRANSFER IN A PLANE 
~ INCOMPRESSIBLE LAMINAR JET 
https://ntrs.nasa.gov/search.jsp?R=19670024067 2020-03-12T11:12:03+00:00Z
f 
HEAT TRANSFER IN A PLANE INCOMPRESSIBLE LAMINAR JET c/ 
> 
(L. Knox Millsaps and Norman L. S o n g  7 1 
% D E P A R T M E N T  OF A E R O S P A C E  E N G I N E E R I N G  ‘9J 
I U N I V E R S I T Y  O F  F L O R I D A  
G A I N E S V I L L E ,  F L O R I D A  A 
231 
ABSTRACT 
Using  the ve loc i ty  components  derived b y  Bickley  from a phys ica l  formulation due  
to  Schlichting, the thermal distribution i n  a plane incompress ib le  laminar je t  i s  determined 
by reducing the energy equation to  a n  ordinary d i f fe ren t ia l  equat ion  b y  a s imi la r i ty  t rans-  
formation. The reduced equation i s  so lved  in terms of a s s o c i a t e d  Legendre  func t ions  for 
a l l  P r a n d t l  numbers, and so lu t ions  a re  derived for par t icu lar  Prandt l  numbers by  e lemen-  
ta ry  methods. Some numerical  r e s u l t s  a re  presented. 
NOMENCLATURE 
jT a n d  7 
P 
P 
k 
P 
M 
a 3 = p M / p 2  
x = a 3 E a n d y = a  3 
8 = c p 2  T / ( a 6  p 2 )  
u = p c / k  
p = (48)-1/3 = 0.2751606040 
5 =  y /x2 /3  
rectangular Ca r t e s i an  coord ina te s  [I] 
cor responding ve loc  it y c om pone n t s  [ l / t I  
temperature [“I 
e x c e s s  temperature [“I 
pressure [m A t  7 
cons tan t  dens i ty  [ m /  I I 
thermal conduct iv i ty  [ m l / ” t ]  
spec i f i c  hea t  [ 1 2 / / ” t 2 1  
coe f f i c i en t  of abso lu t e  v iscos i ty  [m/lt l  
momentum f l u x  per unit length [m/t21 
the  reciprocal of a cha rac t e r i s t i c  length [I-’] 
d imens ion le s s  coord ina te s  [OI 
c orres pon d ing d ime n s  ion le ss ve  loc it ie s [ O  I 
d imens ion le s s  e x c e s s  temperature LO1 
Prandt l  number [OI 
a numerical cons t an t  [OI 
similari ty parameter LO1 
d imens ionless  s t r eam function [ O ]  
transformed s imi la r i ty  parameter [OI 
dimension l e s s  auxi l ia ry  func t ions  [ O l  
assoc ia t ed  Lengendre  function of order p a n d  degree  v 
of the f i r s t  and second  kind respec t ive ly  [ O ]  
233 
IN TROD UC T I 0  N 
The theory o f  a plane incompress ib le  laminar jet,  i.e., the p ressu r i zed  s l o w  f l o w  
o f  a l i q u i d  through a l i nea r  s l i t  i n  a w a l l  i n t o  the same l i qu id ,  can  be t rea ted  by boundary 
layer  theory p rov ided that one i d e a l i z e s  the pob lem i n  the obv ious  way. T h e  hydrodyna- 
mic aspec ts  o f  the problem were f i r s t  formulated by Sch l i ch t i ng  [l] who showed tha t  the 
a n a l y t i r a l  treatment cou ld  be reduced b y  a s i m i l a r i t y  t rans format ion  to the so lu t i on  o f  an 
ordinary d i f f r r e n t i a l  equat ion  w i t h  p rescr ibed t w o  point  boundary cond i t i ons .  Subsequent- 
ly, EX- 
c e l l e n t  rev iews  [31 o f  the a n a l y s i s  a r r  read i l y  ava i l ab le .  
a c losed form so lu t i on  to the d i f f e r e n t i a l  sys tem was ob ta ined by B i c k l e y  [ 2 I .  
Y i h  [41 has inco r rec t l y  treated the assoc ia ted  heat t rans fer  problem b y  requ i r i ng  
the v i scous  d i ss ipa t i on  t o  van ish  i den t i ca l l y .  T h i s  requirement was  used  t o  der ive  a s i m i -  
l a r i t y  t ransformat ion i n  a p rec ise  analogy w i t h  Sch l i ch t ing ’s  t reatment o f  the hydrodynamic  
problem where the momentum f l u x  across  any plane normal  to the j e t  i s  constant.  Subse- 
qu rn t l y ,  Y i h  was ab le  t o  ob ta in  a c l o s e d  form s o l u t i o n  t o  the t runcated  energy equat ion  
u s i n g  h i s  improper ly obtained transformat ion.  Need less  to say, the who le  procedure used  
by  Y i h  i s  h igh l y  susper t ,  fo r  there i s  a dec ided d i f fe rence be tween n e g l e r t i n g  the d is-  
s ipa t i on  t r rm  i n  the energy equa t ion  a f te r  one has  de r i ved  the c o r r e r t  s i m i l a r i t y  t rans- 
format ion and der iv ing  an i nco r rec t  s i m i l a r i t y  t rans format ion  from the a r t i f i c i a l  requirement 
tha t  t h r  rnergy  f l u x  per un i t  depth  o f  the s l i t  b e  cons tan t  for each normal  plane (i.c., the 
d i ss ipa t i on  van i s  h r  s iden t i r a  I l y ) .  
I n  v i r w  of t h r  con t i nu ing  i n t r r e s t  i n  the plane jet ,  i t  appears wor thwh i l e  t o  p resent  
an r x a c t  t r ra tmrn t  for the thermodynamic s i t u a t i o n  w i t h i n  the f ramework  o f  boundary 
l a y r r  theory. I t  i s  i n l e w s t i n g  t o  note that  the exac t  t rea tmrn t  i s  comp le te l y  d i f fe ren t  
from t h r  i n ro r rec t  theory due t o  Yih. 
1. FOHMULATIUN O F  THE PROBLEM 
A m a t h r m a t i r a l  desc r ip t i on  o f  the per t inent  phenomena w i l l  b e  a ided  by reference 
t o  Fig. 1 ,  a s rhemat i c  diagram o f  a plane lam ina r  jet .  
Thr d i f f e ren t i a l  equat ions  tha t  descr ibe  the f low are the c o n t i n u i t y  equat ion,  the 
monivntum rqua t ion  w i t h  the boundary layer  approx imat ion  and the energy equat ion  i n  i t s  
boundtlry l a y r r  form; these three equat ions  are 
a ,  a ,  - f - = o  
a z  ay 
23 4 
a n d  
The  boundary condi t ions  a r e  
= 0, T (m,$ = 0 and  (;) = O  (1.4) 
- 
y = o  
- u (E, ? W )  = 0, T(3T;O) = 0, 
y = o  
In addition there a re  two o ther  important cons idera t ions :  the p re s su re  field i s  t aken  to  
be cons t an t  and the flux of momentum per unit depth of the j e t  a c r o s s  any plane normal 
to the x’ - a x i s  i s  a cons tan t .  Analytical  formulations of t hese  cons idera t ions  yield 
W 
a n d  2 J p U2 d y  = cons t .  = M dP - d j T =  0 
0 
3 2 
One immediately no te s  that a 
t i ons  may then be put in to  d imens ion le s s  forms: 
= pM/p i s  a reciprocal length ,  and the fundamental equa-  
a n d  
du a, d Z u  
ax a,. a y 2  
a e  ae 1 a 2 e  
ax dy u a y 2  
u -  + v - = -  
u - +  v -= - -  - 
The  transformed boundary condi t ions  a r e  
u (x ,+w ) = 0, v(x,O) = o,($) = 0, e(  m ,y) = 0 and 
y = o  
(1.6) 
(1 .7)  
(1.8) 
(1.9) 
11. SOLUTION OF THE H Y D R O D Y N A M I C  PROBLEM 
A s  previously remarked Schliching has shown tha t  the introduction of a s t ream 
function, \I’ = x F ( 0 ,  where 6 = Y / x ” ~ ,  r educes  the  so lu t ion  of Eqs .  (1.6) and (1.7) 
t o  t h e  integration of an ordinary differential  equat ion ,  and Bickley  h a s  integrated t h e  
r e su l t i ng  equat ion  in to  the  following c losed  forms: 
u = 6p2 [sech’  (Po] / x  1’3 (2 .1)  
23 5 
a n d  
The dimens ionless  s t reaml ines ,  $(x,y) = const., a re  shown in F ig .  2 (a)  for the c a s e  in 
which a-3  i s  equa l  to the  unit of length in terms of which the o ther  phys ica l  quan t i t i e s  
v = 26 [2p[ sech’ (fit) - tanh (/3t)I/x 2’3 (2.2) 
a r e  measured, i.e., it  i s  the d imens ion le s s  momentum, p M / p ’ ,  per unit depth of the l ine 
a long  which the unidimensional force is applied.  F ig .  2(b) s h o w s  the d imens ion le s s  
velocity components for the same  c a s e ;  the u-profiles a re  a l s o  shown. 
H I .  SOLUTION OF THE THERMODYNAMIC PROBLEM 
If Eqs. (2.1) and (2.2) are inser ted  into Eq. (1.8), the  energy equat ion  becomes 
6 p  2sech2((BQ a 0  2/3[2/35 sech’ ( (P0  - tanh  d o  - 
J Y  3 F+ 2 / 3  
(3.1) 
1 d 2  0 144p ‘ sech*(Pt )  tanh‘((P6) 
f - _ - -  
a dy2 X -  
Some algebraic manipulation s h o w s  that the s imi la r i ty  transformation, 8 = ,k34x-2/3 G (6). 
r educes  the partial  d i f fe ren t ia l  equation (3.1) to the ordinary d i f fe ren t ia l  equation: 
- 2 [ t a n h ( P n I  G’-4[sech2(/3~)1C; = G “/a  + 144sech4  (fit) tanh’ (fit) 
where the primes denote differentiation with r e spec t  to pt .  
to the symmetry of the thermal field about the a x i s  of the jet becomes  
(3.2) 
T h e  boundary condition due 
G ’(0) = 0 (3.3) 
The  form of Eq. (3.2) may be simplified further by putting 7 = t a n h  ( p t ) ;  the  s impl i f ied  
equation is 
( 1 - q 2 ) G ” -  2(1 - a )  q G ’ +  40 G = - 1 4 4  a q 2 ( 1 - q 2 )  (3 .4)  
where the primes now denote  d i f f e ren t i a t ion  with r e s p e c t  t o  q. Eq. (3.4) can  be e a s i l y  
transformed i n t o  the s tandard  form for a s s o c i a t e d  Legendre  func t ions ,  a n d  the genera l  
so lu t ion  for the homogeneous part i s  
h h e r e  v = [-I 2 (402+ 20a + l)’] /2  and where P:and 
funct ions  of  order ~1 and degree  u of the f i r s t  and s e c o n d  k inds  r e spec t ive ly  [SI. T h e  
complete solution may be obtained in terms of quadra tures  by the  method of variation of 
parameters ,  and one no te s  tha t  the  Wronskian h a s  a par t icu lar ly  s imple  form e / ( n 2  -1) [6]. 
a re  the a s s o c i a t e d  1,egendre 
236 
where 
Whenever the  Prandt l  number i s  a n  integer or whenever u i s  an in teger ,  the  a s s o c i a t e d  
Legendre functions become de r iva t ives  of Legendre func t ions  or s imple  polynomials in 
q [7]. However, the spec i f i c  proper t ies  a n d  the numerical v a l u e s  of the  a s s o c i a t e d  Le- 
gendre func t ions  a re  generally unknown; consequent ly ,  i t  s e e m s  appropriate to give a s  
i l lus t ra t ive  examples  some so lu t ions  of Eq. (3.4) for particalar 0 which can be obtained 
b y  elementary methods. 
The  complete so lu t ion  of Eq. (3.4) may be written in the  following form: 
G ( q )  = A G , ( q )  + B G ,(q) + Gp(q)  (3.6) 
where Gl(q) and G , ( q )  are  genera l  so lu t ions  of the homogeneous equat ion  a n d  where G p ( q )  
i s  a particular integral. A particular integral  in polynomial form may be found by the  
method of undetermined c o n s t a n t s ;  t h i s  integral is 
and  
36 (30 -2)  
=(30 - 5 )  ( 4 0 -  3)  
- 7 2 ~  ( 3 0 -2) 
r =  
(30-5)  ( 4 0 - 3 )  
(3.7) 
Obviously the Prandtl  numbers, 0 = 3/4 and 0 = 5/3,  are  not inc luded ,  a n d  these  v a l u e s  
require the more e labora te  treatment which w a s  previously outlined. If the parametric 
dependence  of the solution on the Prandtl  number i s  indicated by G ( q , o )  and if i t  i s  
a s s u m e d  that 
(3.8) 
where the even powers of q are  required by the symmetry of the  thermal field,  one can  
der ive  a recurrence relation between the coef f ic ien ts  by putting Eq. (3.8) into the homo- 
geneous  part of Eq. (3.4). T h i s  procedure y i e lds  
= [21(21 + 1 - 20) - 4 u  C Z i  (3.9) 1 [(21 + 2) (21 + 1q C z , +  , 
If t he  s e r i e s  in Eq. (3.8) i s  t o  terminate,  then 21(21 + 1-20)  - 40 = 0 or 
23 7 
i (2 i  + 1) 
2(i + 1 )  
u =  (3.10) 
Unfortunately, the c a s e s  i = 1 and i = 2 are  the except iona l ly  res t r ic ted  ones ,  u = 3/4 a n d  
(T = 5/3, s o  that the f i r s t  simply complete so lu t ion  occur s  for i = 3 or u = 2 1/8. T h i s  c a s e  
w i l l  be treated a s  a particularly s imple  and i l lus t ra t ive  example.  
for u = 21 /8 is 
T h e  cbmplete so lu t ion  
161  
64 
G ( q ,  21 /8) = A 1 - (  2 1  /4)  + (105/16)q 4-(-) q6]+ [3780q4-592212 + ,1281 /115 
(3.11) 
c 
The cons t an t  A is determined b y  the s t rength  of the  hea t  sou rce  a t  the origin or, equiva-  
lently, by ass igning  the value of the temperature a t  a point. F igs .  3,4 and 5 s h o w  the  
thermal distributions for A = 0, 10, and 100 respec t ive ly .  T h e  c a s e  A = 0 y ie lds  the  
thermal f i e l d  due to v i scous  d iss ipa t ion  alone. The  c a s e  A = 10 i s  one in which the  
e f f ec t s  of v i scous  d iss ipa t ion  and convec t ive  transfer a r e  comparable.  The  c a s e  A = 100 
i s  t h r  usua l  c a s e  i n  which v i scous  d i s s ipa t ion  may be neglec ted .  
Of course,  t h r r r  are many other c a s e s  which have s imple  s o l u t i o n s  in different 
When D = 1 /2 ,  the differential  forms; an in te r rs t ing  example of t hese  i s  the c a s e  u = 1/2. 
e q uat ion hr corn r s 
w h i c h  can be transformed to 
C’14) i- 2 G (4) = 0 (3.13) 
by the s u b s t i t u t i o n  7 = s i n  4 [8]. The  pertinently comple te  so lu t ion  i s  
G(q,1/2) = A c o s  (21’2 s i n -  ‘q) - 36  ( q 4  - q 2  + 1 ) / 7  (3.14) 
CON(: l , l lSIUN 
I t  is shown that the partial  d i f f e ren t i a l  equat ion  for the hea t  t ransfer  i n  a plane 
inc-omprrssiblc laminar j e t  can be reduced b y  a s imi l a r i t y  transformation to an ordinary 
d i f fe ren t ia l  equation that can be so lved  in terms of a s s o c i a t e d  Legendre  functions.  T h e  
particular ins tance  when the P rand t l  number of the f lu id  i s  i n  the  form, 0 =[i(2i + I ) ] /  
[2/i + 1)1, i s  trrated by e lementary  methods; the c a s e ,  0 = 1/2,  i s  a l s o  treated by another  
e I e me n t u r  y m r t h oil. 
A C h N 0 W I, EDGM E N  ‘I’ 
This work was  supported by the Nat iona l  Aeronau t i c s  and  Space  Administration 
under Grant NsC -542. .- . 4. *r,t...rl 
1 
/ 238 i ‘4, 
R E F E R E N C E S  
[11 H. Schlichting, “Laminare Strahlausbreitung,” Ztschr .  Plrr angew.  Math. und Mech., 
VOI. 13, pp. 260-263, 1933. 
[2] W. G. Bickley, “The Plane Je t ,”  Phil. Mag., Ser. 7, Vol. 23, pp. 727-731, 1937. 
[31 S. 1. Pai, F lu id  Dynamics of Jets ,  Chap. IV, See. 3, pp. 76-78, D. Van Nostrand 
Company, 1954, or L. Rosenhead ,  Laminar Boundary Laye r s ,  Chap. V, (Authors: C. 
W. Jones and E. J. Watson), Sec. 23, pp. 254-256, Oxford, 1963. 
141 C. s. Y i h ,  “Temperature Distribution in  a S teady ,  Laminar,  Prehea ted  Air Jet,” 
J o u r n .  Appl. Mech., Vol. 17 ,  pp. 381-382, 1950. 
[51 A. Erdelyi e t  a l ,  Higher Transcendenta l  Func t ions ,  Vol. I, Chap. 111, Sec. 3.2, pp. 
121-122, McGraw-Hill Book Co., 1953. 
[61 A. Erdelyi e t  a l ,  op. tit.* Chap. 111, See. 3.2, p. 123. 
171 A. Erde ly i  e t  a l ,  op. cit., Chap. 111, Sec. 3.6.1, p. 148. 
[8] H. Bateman, Differential  Equat ions ,  Chap. 111, See. 31, pp. 73-74, Longmans Green 
& Co., 1918. 
239 
Fig.  1 ,  A Schematic Diagram of a Plane Laminar Jet .  
240 
Io 
d 
n 
d 
rr) 
d 
Fig. 2(a),  Dimensionless Streamlines for a Plane Incompressible Laminar J e t  for a-3  
Equal to the U n i t  Length. 
24 1 
Y 
0 
7 
€ 
! 
1 
X 
Fig. 2(b), Se1ec.tc.d 1)imensionless Velocity Components and Corresponding Horizontal 
Dimensionless Velocity Profiles for a Plane Incompressible Laminar Jet for 
a-’Equal 10 the U n i t  Length. 
242 
0 . O I Y  I , , 
-200t 
F i g .  3, 
-r 
1.0 
T(X,Y 111.34 
T(X,Y)=I.O 
f 
T(X,Y )= 0.70 1 f 
2.0 
1 
0.01 x 
3 .O 
7-
The D i m e n s i o n l e s s  T h e r m a l  F i e l d  d u e  to V i s c o u s  D i s s i p a t i o n  i n  a P l a n e  
I n c o m p r e s s i b l e  L a m i n a r  Jet  for E q u a l  to t h e  U n i t  L e n g t h .  The P r a n d t l  
n u m b e r  i s  2 1 / 8 .  
24 3 
244 
I 
0 
t v) 
I 
0 
v) 
I 
F i g .  5, The D i m e n s i o n l e s s  T h e r m a l  F i e l d  N e g l e c t i n g  V i s c o u s  D i s s i p a t i o n  in a P l a n e  
I n c o m p r e s s i b l e  L a m i n a r  Jet for a - 3  E q u a l  to the U n i t  L e n g t h .  The P r a n d t l  
n u m b e r  is 2 1/8. 
245 


Heat transfer in a plane incompressible laminar jet

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