C HAPTER 3
Some Solutions for External Laminar Forced Convection
3.1
INTRODUCTION
T he purpose o f this chapter is to illustrate some o f the ways in which the equations derived in the previous chapter can b e u sed to obtain heat transfer rates for situations involving external laminar flows. External flows involve a flow, which i s essentially infinite in extent, over the outer surface o f a body as shown in Fig. 3.1. The problems chosen to illustrate the methods o f solution are not directly, in most cases, o f great practical significance but they serve to illustrate t4~ basic ideas involved in the solution procedures. Some o f the flows considered, although apparently highly i dealized,are, however, good models o f situations that are o f great practical importance. For example, flow pver a flat plate aligned with the flow will b e extensively considered. This flow is good model o f m any situations involving flow over fins that are relatively widely spaced. In all the solutions given in the present chapter, the fluid properties will b e assumed to b e constant a nd the flow will b e assumed to b e twodimensional. In addition, dissipation effecfs in the energy equation will b e neglected in most o f this chapter, these effects being briefly considered in a last section o f this chapter. Also, solution~ to the full NavierStokes and energy equations wi~l b e d ealt with only relatively tmefly, t he majority o f the solutions considered being based on the use o f the boundary layer equations.
a
. 3.2
S IMILARITY SOLUTION F OR F LOW OVER AN I SOTHERMAL PLATE
Consider the flow o f a fluid at a velocity o ( U l over a flat plate whose entire surface is held at a uniform temperature o f Tw whidh is different from that o f t he fluid a head. o f the body which is T 1 . T he flow situation 'lis thus as shown in: Fig. 3.2. 83
8 4 Introduction to Convective Heat Transfer Analysis
F low whose size is large compared to the dimensions o f the body
~
F low over surface o f b ody
F IGURE 3 .1 External flow.
I t will be assumed that the Reynolds number is large enough for the boundary layer assumptions to be applicable. I t will further be assumed that the flow is twodimensional which means that the plate is assur,ned to b e wide compared to its longitudinal dimension. As a result o fthese assumptions, the equations governing the problems are:
au + av = 0 ax ay au au u +vax ay aT aT u a x + v ay
= =
(3.1)
a2u vay2
(3.2)
(3.3)
(vr )a2T P ay2
In writing these equations it has been noted that the solution for inviscid flow over a flat. pl~te o f zero thickness is that the velocity is everywhere the same and equal to the undisturbed free stream velocity, U l, i.e., that, i f the effects o f viscosity are ignored, a f lat plate aligned with the flow will have no effect on the flow. As a result, the pressure gradient, d p /dx, is everywhere zero. But the conditions outside the boundary layer are assumed to b e those that exist in inviscid flow over the body considered and the pressure gradient in the boundary layer is assumed to be equal to that existing in this/outer inviscid flow. Therefore, in obtaining the boundary layer solution for flow over a flat plate, it will be assumed that the velocity outside the boundary is equal to U l and that the pressure gradient is everywhere equal to zero. In the case o f a real plate of finite thickness, a pressure gradient will exist but this should only be significant in the immediate vicinity o f the leading edge. Since the boundary layer equations themselves are not applicable very near the leading edge, the longitudinal gradients o f velocity and temperature being comparable to the lateral ones in this region, this effect will not be considered here.
1v
Uniform Flow
u x
Flat Plate at Uniform Temperature Tw
"'"
F IGURE 3.2 Flow over a flat plate.
C HAPTER
3: Some Solutions for External Laminar Forced Convection 85
I n writing Eq. (3.2), the kinematic viscosity, v , defined by:
v
=
JL p
(3.4)
has been introduced as before. I t h as also been noted in writing Eq. (3.3) that:
k
p Cp
v
Pr
(3.5)
Equations (3.1) to (3.3) must b e solved subject to the following boundary conditions: ' y = 0: U = 0, v = 0, T = T w (3.6) y large: U ~ U t, T ~ T l ,. A discussion o f how these conditions are derived was given i n C hapter 2. I t is noted that Eqs. (3.1) and (3.2), which together allow the velocity distribution to be determined, are independent o f the temperature because the fluid properties are being assumed constant. Therefore, as previously discussed, these two equations c an b e solved independently o f Eq. (3.3) to give the velocity distribution. O nce this velocity distribution has b een obtained, Eq. (3.3) c an b e solved for the temperature distribution. T he h eat transfer rate c an t hen b e d etermined from this distribution. The similarity solution uses as a starting point the assumption that the boundary layer profiles are similar a t all values o f x, i.e., that the basic form o f the velocity profiles at different values o f x, as shown i n Fig. 3.3, are all the same [1],[2],[3],[4],[5]. The velocity profiles a t a ll points i n t he boundary are therefore assumed to b e s uch that:
~ Ul
= function
(B ~)
(3.7)
~'
T he velocity profiles are thus a ssumed to b e s imilar in the sense that although B varies with x a nd althQugh the velocity a t a fixed distance y f rom the wall varies with x, the velocity a t fixed values o f ylB, w here B i s the local boundary layer thickness, remains constant. Th~ p roof that similar profiles do exist actually follows from the analysis given below. , I t w as shown i n tJ1e derivation o f t he boundary layer assumptions that for the boundary layer assumptions to apply i t i s necessary that:
(3.8)
"Edge" o f
Boundary Layer Flow ..
I
Velocity Profiles at Various Positions i n Boundary Layer
F IGURE 3.3 Velocity profiles at various positions iIi the boundary layer on a fiat plate.
86 Introduction'to Convective Heat Transfer Analysis
Ithe characteristic length having here be¢n taken as x. Rex is, o f course, the Reynolds
number based on x, i.e. (U1X/V). ! Therefore, the similar velocity profile assumption given i n Eq. (3.7) can b e written as:
I t is convenient to define:
Y rn:71 =  "Rex = y x
/fx1 vx
(3.10)
this variable, 7], being termed the "similarity variable". Eq. (3.9) can, therefore, b e written as:
 ' = F(7J)
u U1
_ (3.11)
Consider the continuity equation (3.1). Because the boundary conditions given in Eq. (3.6) give v as zero at the wall, this continuity equation can b e integrated w ith respect to y leading to the following expression for v at any point in the boundary layer.
v = i
B ut
yau d F a7J dy= f £V il1 U1d7J o ax U1 0 d 7Jax
(3.12)
~; = ~~j¥f =
so Eq. (3.12) can be written as:

2:
(3.13)
v
=
!
2
J
V (11 d F 71 d7J XU1 Jo d7J .
(3.14)
Because o f the form o f this integral, i t is convenient in the following work to define a new f unction,f o f 71, which is related to F by:
df d7J
=
f'
=
F
(3.15)
Rewriting Eq. (3.14) in terms o f f and carrying out the integration then gives:

v
U1
=
1[£ , "
2
 (7 ]f  f)
XU1
(3.16)
Next consider the momentum equation (3.2) which can b e written as:
( !!...)a(U/U1) a7J + (~)a(U/U1) a7J = U1 a7J a x U1 a7J a y
(~)a2(u/U1) (a7J)2
U1 a7J2 ay
(3.17)
C HAPTER
3: Some Solutions for External Laminar Forced Convection 87
but by virtue o fEq. (3.15): (3.18) so it follows using these and Eq. (3.16) that the momentum equation reduces to:
f'f"(!~ ~ X; 2x
i.e., to:
ful)+[!
! v(TJlIf)]f" 2 ~ X;;; ~
~ri= .!!..flll(~) X xv
Ul
2 flll
+ f f"
=0
(3.19)
The primes, o f course, denote differentiation with respect to TJ, i.e., Eq. (3.19) could have been written as:
d 3f 2 d TJ3
+
d2f f dTJ2 = 0
The problem o f determining the velocity profile i n the boundary layer has, therefore, been reduced to that o f solving the ordinary differential equation given in Eq. . (3.19). In terms o f the similarity function, the boundary conditions on the solution that are given in Eq. (3.6) become:
= 0, U = 0 becomes, using Eq. (3.18), when TJ = 0, f ' = 0 When y = 0, v = 0 becomes, using Eq. (3.16), when TJ = 0, f = 0 When y is large, U ~ U l becomes, using Eq. (3.18), when TJ islarge, f ' ~
When y
1 (3.20)
There are, therefore, two b<;mndary conditions at TJ = 0 and one boundary condition that applies at large values 'of TJ [1],[6],[7],[8]. The solution to Eq. (3.19) subject to these boundary conditions is relatively easily found. One way o f finding the solution involves~ basically, guessing the value o f f " at the wall and then, using the first two o f the boundary/conditions given in Eq. (3.20), numerically integrating Eq. (3.19), thus giving the/variation o f f with TJ. This solution will o f course not, in general, satisfy the outer boundary condition, i.e., the boundary condition for large . TJ. The $olution is, therefore, repeated for another guessed value o f f " at the wall. The solution that satisfies the outer boundary condition can then be obtained by combining these two solutions or by using an iterative procedure. Because the iterative ("shooting") method will' b:e used elsewhere in this book, it will be adopted here. The iterative method is based on Newton's method, i.e., i f f~' is the value o f f " a t .,11 = 0 and i f f~ i's the value o f f ' for large TJ obtained by using this value, then a better estimate o f f~r is:
0
+"
Jw
foo  1 + d(f~)ld(f.:v')
;
I n obtaining the solution, ~. (3.19) is c~nveniently written as three simultane,.. ous firstorder differential equations; i.e., by defining g = f ' a nd h = g' = f "
88
Introduction to Convective Heat Transfer Analysis
E'q. (3.19) can b e used to give the following set o f simultaneous equations:
2h' = t o.5lh
g'
=
h
I'
'1]
=g
T he boundary conditions on the functions involved in these equations are:
= 0, g = 0;
'1]
= 0, I = 0;
'1]
large, g
~
1
The first step o f the solution thus involves guessing the value o f h a t '1] 7 0 and then integrating the above set o f equations. The Newton equation is then \used to· progressively work towards the value o f h a t '1] = 0 that gives g ~ 1 for '1] large. In method, the value o f d(/~)/d(/~') is obtained by obtaining I~ using a applying particular value o f I~ and then incrementing I~ b y a small amount, then getting a new value o f I~ and then using a finite difference approximation for the derivative. I n this way the variation o fI a nd h ence I I with '1] c an b e found. The variation resembles that shown in Fig. 3.4. A simple program to obtain the solution is available as discussed in the Preface. The predicted velocity profile shown in this figure is in very good agreement with experimentally measured profiles. The fact that the original two partial differential equations governing the velocity components, u and v, were reduced to a n ordinary differential equation justifies the assumption o f similar velocity profiles. . W hile there i s no distinct " edge" to the boundary layer, i t is convenient to have .some measure o f the distance from the wall over which significant effects o f viscosity exist. For this reason, it is convenient to arbitrarily define the boundary layer thickness, 8"~as the distance from the wall at which u reaches to within 1% o f the freestream velocity, i.e., to define 8 as the value o fy at which u = 0 .99ul' Using the result given in Fig. 3.4 then shows that u = 0 .99ub i.e., I ' = 0.99, when '1] = 5 which indicates, in view o f the definition o f '1], that 8 u is approximately given by:
this
8u
v{iil ;X
=
5
6 r.,
I I"l
0 ...
4

:i'1~
...,
II !:'"
2
F IGURE 3.4
o.0 ~~~~.0 0 0.5 1
ulu1 =1'
Variation o f the velocity profile function l ' with the similarity variable 'YJ for the boundary layer on a flat plate.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
89
1.e.,
(3.21) Having established the form o f the velocity profile, attention can now be turned to t he determination o f the temperature profile, i.e., to the solution o f Eq. (3.3). Because the wall temperature is uniform in t he situation being considered, it is logical to assume that the temperature profiles are similar in the same sense as the velocity profiles. For this reason, the following dimensionless temperature function is introduced
( )=
T wT Tw  T}
(3.22)
The assumption that the temperature profiles are simil~ is equivalent to assuming that () depends only on the similarity vari;lble, TJ, b ecause the thermal boundary . layer thickness is also o f order x l Rex. Since both Tw a nd T} are constant, the energy equation (3.3) can b e written in terms o f () as
J
u  +v
a(} ax
a(} v a2 (} = ay P ra y2
(3.23)
while in terms o f this variable, the boundary conditions become:
y
= 0: () = 0
(3.24)
y large: () ~ 1
Because the temperature profiles are being assumed to b e similar, i.e., i t is being assumed that () is a function o f TJI alone, it follows using the relations for the velocity components previously derived, that Eq. (3.23) gives:
On rearrangement this equation becomes:
0" + Pr ()' f 2
TJ TJ large: ()
=
0
(3.26)
whfle the boundary conditions on the sol\ltion given In Eq. ( 3.24) can b e written as:
= 0: () = 0
~
(3.27)
1
Thus, as was the case with the velocity distribution, the partial differential equation governing the temperature distribution has, b een reduced to an ordinary differential equation. This confirms the .asSumption t hat the temperature profiles are similar.
90
Introduction to Convective Heat Transfer Analysis
Now Eq. (3.26) could b e solved, subje~t to the boundary conditions given in Eq." , (3.27), using a similar procedure to that us~d to solve Eq. (3.19) which was briefly outlined above. The solution is, however, more easily obtained i n t he following way. I t is noted that Eq. (3.26) can b e written as:
1d _P   ()'  _  rf
()' dTJ 2
(3.28)
This can b e integrated to give:
()' = C l exp
[7
fol1
f
dTJ
1
,(3.29)
where C l is a constant o f integration which has still to b e determined. Integrating Eq. (3.29) then gives: (3.30) where C2 is another constant o f integration. It will, o f course, b e equal to the value o f () when TJ is zero and so b y virtue o f the first o f the boundary conditions given in Eq. (3.27) C2 is zero. C l is determined from the second o f the boundary conditions given in Eq. ( 3.27rand is, therefore, given by '. (3.31) The boundary condition for large TJ has been applied by taking the integration to 0 0. The final equation obtained is, however, such that once the upper limit o f the integration islaken above a certain value, the result becomes independent of the actual value used for this u pper limit. This point is discussed further below. Substituting Eq. (3.31) into Eq. (3.30) then gives since C2 is zero
(3.32)
Since f is a known function o f TJ being given b y the solution for the velocity field, Eq. (3.32) c~ b e integrated to give the variation o f () with TJ. This solution is more easily obtained by noting that Eq. (3.19) gives:
=
\
f '"
f
f"
2
(3.33)
which can be integrated to give loge f " =
"2 Jo
1
r
11
fdTJ
+ constant
C HAPTER
3: Some Solutions for External Laminar Forced Convection
91
I.e.,
1 f" exp [   f71 fdTJ ] = 20 C3
C3
(3.34)
being, o f course, a constant o f integration. Integrating this equation again then gIves
Jo
r exp [ _ Pr Jo r
71
71
f dTJ] dTJ =
Jo
r (fll )pr dTJ = ~ Jr (f")Pr dTJ o
71 71
C3 C3 r
(3.35)
Substituting this result into Eq. (3.32) then gives
[1071 f',Pr dTJ ]
( )=
[1000 ! "'"dTJ _1,00 !"p,dTJ]

~; !"p,dTJ1
(3.36)
=:;::;= = 1  .;:."'"
[10
00
f " Pr dTJ ]
[I; f"P'd TJ ]
[1000 f "'" dTJ]
This allows the variation o f () with TJ to b e derived for any chosen value o f Pro Since the variation o f f " with TJ is given by the solution for the velocity profile, this equation is easily integrated to give the variation o f () with TJ. I t will b e noted that f ", which is equal to d(ulul)ldTJ, tends to zero outside the boundary layer, i.e., at large values o f TJ. T he actual value o f the upper limit o f t he integrals in Eq. (3.36) will not, therefore, affect the result provided it is sufficiently large. A simple computer program, SIMPLATE, written i n FORTRAN that implements the above procedures for obtaining the similarity profi1~s for velocity and temperature is available as discussed in the Preface. I n this program, the solution to the set o f equations defining the velocity profile function is obtained using the basic RungeKutta procedure. \ Some typical variations o f () with TJ for various values o f P r obtained using this . program are shown i n Fig. 3.5. The curve for a Prandtl number o f 1 is identical to that giving the variation o f f ' with TJ which was g;jven i n Fig. 3.4. This is discussed below.
6~~~
0_
"l
II
4
:fl~
.....
~
2
F IGURE 3.5
Variation o f t he temperature profile function, (J, with the similarity variable, 11, for various values o f P r for the boundary layer on a tlatplat~.
1
0.5 (J =( Tw  D/(Tw  Tt )
1.0
/
92
Introduction to Convective Heat Transfer Analysis
I t will be seen from the results given i n Fig. 3.5 that, i f the thermal boundary layer "thickness," 8T , is defined in a si~lar way to the velocity boundary layer "thickness" as the distance from the wall a t which 0 becomes equal to 0.99, i.e., reaches to within 1% o f its freestream value, then:
d (Pr) 8T x J Re x
but:

8u x
5
J Re x
I t follows from these that:

8T
8u
=
d (Pr)
i.e., the ratio of the two boundary layer thicknesses depends only on the value of the Prandtl number. I t will be noted from the results given in Fig. 3.5 t hatthe thermal boundary layer is thicker than the velocity bO\lndary layer when P r is less than one and thinner than the velocity boundary layer when P r is greater than one. The heat transfer rate at the wall is given by
qw =  k aaTI
Hence, using Eq. (3.22)
(3.37)
y y=o
_ _q_w_ _ k(Tw  Td
i.e.,
= aOI
ay y=o
= dOl
a17 d17 1)=0 ay
(3.38)
(3.39) i.e., (3.40)
N ux and Rex being, o f course, the local Nusselt and Reynolds numbers based on x. Because 0 depends only on 17 for a given P r, 0'11)=0 depends only on P r and its value can be obtained from the solution for the variation 0 with 17 for any value of P r. It is convenient to define:
A (Pr)
= 0'11)=0
(3.41)
In terms of this function A, Eq. (3.40) can be written as
N ux
=
A JRe x
(3.42)
Values o f A for various values of P r are shown in Table 3.1.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
93
T ABLE 3 .1
Values o f A f or various values o f P r
Pr
0.6 0.7 0.8 0.9 1.0 1.1 7.0 10.0 15.0
A = 6'1.,=0
0.276 0.293 0.307 0.320 0.332 0.344 0.645 0.730 0.835
O.332Prl/3
0.280 0.295 0.308 0.321 0.332 0.343 0.635 0.715 0.819
Over the range o f Prandtl numbers covered in the table, it has been found that A varies very nearly as P r 1l3 and, as will be seen from the results given in the table, is quite closely represented by the approximate relation:
A = O.332Pr 1l3
(3.43)
For values o f P r very different from those in the table, i.e., very different from one, the errors involved in the use o f this approximate equation may b e unacceptably large. Now, in practical situations, the concern is more likely to be with the total heat transfer rate from the entire surface than with the local heat transfer rate. Consideration is, therefore, now given to the total heat transfer rate from a plate o f length, L. Unit width o f the plate is considered because the flow is, by assumption, twodimensional. The total heat transfer rate per unit width Qw will, o f course, be related to the local heattrjUlsfer rate, qw by: Qw
:d
IOL qw d x
(3.44)
B ut Eq. (3.39) gives the local heat transfer rate as: (3.45)
Substi~uting
this result into Eq. (3.44) then gives on carrying the integration: (3.46)
I f a 'mean heat transfer coefficient for the whole plate, Ii, i s defined such that:
Ii
=
'.
Qw
L (Tw ~ T t)
\
(3.47)
then, since unit width o ( t he p lateis beiI1g considered, Eq. (3.46) gives:
.
 U,kJU1L
h=n ,;
(3.48)
9 4 Introduction to Convective Heat Transfer Analysis
T railing
qw
w q
i Plate
I
i
i
Edge of
Mdr~~~
_.L
LI
11risis . Twice . T his
F IGURE 3.6
Relation between local and mean heat transfer rates.
x
3 .r.
• Experimental
1T rMsition 1T o
~ B egins
l Turbulence
F IGURE 3.7
Comparison between predicted and experimental mean Nusselt numbers.
1
~~~~~
4.0
T he mean Nusselt number for the whole plate N ud = h Uk) is, therefore, given by: (3.49) where ReL is the Reynolds number based on the plate length,·L. I t will be. seen from the above that the average heat transfer rate for the entire plate surface is twice the local h eat transfer rate existing at the e nd o f the plate, i.e., a t x = L. T he relation between the local and average heat transfer coefficients is shown diagrammatically in Fig. 3.6. I t should be noted that i f the approximate expression for A given i n Eq. (3.43) is utilized, the local/c;md mean Nusselt numbers are given by:
Nux = 0.332Pr1l3 Re~2
N UL =
(3.50) (3.51)
0.664Pr 1l3 R et2
These expressions give results that are in reasonably good agreement with experimental results, a comparison o f some measurements o f m ean Nusselt number with the values predicted by Eq. ( 351) being shown in Fig. 3.7.
A ir flows at a velocity o f 5 m ls over a wide flat plate that has a length o f 20 cm in the flow direction. The air ahead o f the plate has a temperature o f 20°C while the surface o f the plate is kept at 80°C. Using the similarIty solution to the laminar boundary layer equations, plot the variation o f local heat transfer rate in W1m2 along the plate. Show the mean heat transfer rate from the plate on this plot. Also plot the velocity and temperature profiles in the boundary layer at the end o f the plate. Use the equations and graphical results given above to answer this q uestionit is not necessary to solve the governing equations.
E XAMPLE 3 .1.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
95
Solution. The mean temperature o f the air is:
80 A t this temperature for air:
k = 0.0278W/moC, v = 0.0000179 m 2 /s
+ 20 = 500C
2.
Hence, since here:
Ul
=
5 m ls
it follows that the Reynolds number based on the length o f the plate, i.e., 0 .2 m, is:
U IL
ReL
=  ; = 0.0000179 = 55,866
5 X 0.2
The boundary layer on the plate will therefore remain laminar (see discussion o f transition Reynolds number given later). A ir has a Prandtl number o f approximately 0.7 and for this value o f Pr, Table J .l shows that:
A
Hence, since:
= 0.293
it follows that:
qw
= 0.293k(Tw 
T.)
Hf
l
xv
Using the value o f k given above, it then follows that:
qw
= 0.293 X 0.0278 X (80  20) X
J
5 x 0.0000179
= 258.2 W /m2 jX
I n this equation, x is iq/m. T he variation o f qw with x given by this equation is shown in Fig. E3.1a. Because the length o f the plate is 0.2 m and because the mean heat transfer rate is twi~ the local value at the end o f the plate, it follows that:
qw
= 2 X 2 5;; =
1154.7 W/m2
, ,0.2 This value is also shown in Fig. E 3.la. Figures 3.4 and 3.5 show the variations o f I ' with 71 and (J with 71 for P r But: 71 so:
= 0.7.
= Y," v x
Jfx
l
/
9 ' Introduction to Convective Heat Transfer Analysis
2000
~r~~~
1500
"'s
I
~ 1000
J
500
0 ' __ '____ '__ 1_ _ _ _ 1 0.00 0.05 0.10 0.15 0.20 x m
F IGURE E3.1a
I n the present case, the velocity and temperature profiles at the end o f the plate, i.e., at x = 0.2 m, are required, so, using the value o f ]J given previously, i t follows t!Iat at the trailing edge o f the plate:
y = TJJO.OOOOl;9
x 0.2
=
0.OOO8462TJ m
Also, recalling that:
i t follows that:
u = 5 f'mls
Also:
8
from which it follows that:
=
Tw

Tw  Tl
T = 80  T 80  20
T = 80  608
'
. Using the variations of f ' with TJ and 8 with TJ for P r = 0.7 given in the above figures or by using the program discussed above and then using the equations for y, u, and T derived above, a table o f the following form can. be constructed:
'rI
f'
0 0.166 0.330 0.630 0.846 0.956 0.992 0.999 1.000
8
0 0.146 0.291 0.564 0.780 0.914 0.975 0.995 1.000
y (cm)
u (mls)
T eC)
0 0.5 1.0 2.0 3.0 4 .0 5.0 6.0 8.0
0 0.0423 0.0846 0.169 0.254 0.339 0.423 0.508 0.677
0 0.829 1.649 3.149 4.230 4.778 4.958 4.995 5.000
80.0 73.0 62.5 46.2 33.2 25.2 21.5 20.7 20.0
C HAPTER
3: Some Solutions for External Laminar Forced Convection
0.8
97
0.8
0.6
0.6
S u
,
;:.,
0.4
S u
,
;:.,
0.4
0.2
0.2
0.0
0
2
u m/s
4
6
0.0
0
20
40
60
80
TOC
F IGURE E 3.1b
F IGURE E3.1c
The variations o f velocity and temperature given in this table are plotted in Figs. E 3.lb and E 3.lc. More values than are given in the above table were actually used in constructing these figures.
I t should b e noted that for the particular case o f a fluid having a Prandtl number equal to 1, Eq. (3.23) reduces to
ForPr
=
ao ao a20 1: u  +v = v ax ay ay2
(3.52)
The boundary conditions on the solution to this equation are still those given in Eq. (3.24). Now the momegtum equation (3.2) can b e written as
U
a(U/Ul) ax
+v
a(U/Ul) a2(U/Ul) =v ay ay2
(3.53)
While the boundary conditions listed in Eq. (3.6) are:
y = 0: U/Ul = 0 . y large: U/Ul
~
(3.54)
1
A comparison o f Eqs. (3.52) and (3.53) and also o f t heir boundary conditions as given i n Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case o f Pr equal to one, the'·distribup.on o f 0 through the bpundary layer is identical to the distribution o f (U/Ul). In this par~ tjcular case, therefore, Fig. 3.4 also gives the temperature distribution and the ~wo boundary layer thicknesses are identical i n this case. Now many gases have Prand~ numbers which are not very diff~rent from 1 and this relation between the veloeity a nd temperature fields and the results dedqced from i t will be approximately correct for them. " , A s discussed above, the p eat transfer t he wall is given by:
~
at
I
.
.
'.k(Tw
·qw

Tl)=
0
'IT}=O ..;Rex ~
./
(3.55)
98 Introduction to Convective HeatTransfer Analysis
B ut since the distributions o f (J and U/Ul have been shown to b e identical for P r = 1 i t follows that for this case:
(J
'I ~o = _
d(U/Ul) d T]
I
~=o
= f "l ~o 
(3.56)
From the results given in Fig. 3.4 i t follows that
f"I~=o
= 0.332
(3.57)
Using this result then gives: For P r
= 1: Nux = k(T:w~ T d = 0 .332JRex
Tw ,
(3.58)
Nux being as before the Nusselt number based on x. I t is also worth noting that the shearing stress o n the wall, Tw = J.L ay y=o
This can be rearranged to give:
is given by  (3.59)
aul
TW ( J.L ) d (u/ud pur = pUt dT]
I
~=o
aT] 1 f" a y = J Re x
~=o
(3.60)
Combining this result with Eqs. (3.56) and (3.55) then shows that: For P r = 1: Nux = (Tw/pur)Rex (3.61)
This is, basically, the Reynolds analogy for laminar flow. I t relates the dimensionless heat transfer rate ,to the dimensionle~s shear stress. I t will not be pursued " further at this stage. Equation (3.58) which, it will be recalled, was deduced without solving the energy equation, gives results which agree reasonably well with the exact results for gases with Prandtl numbers near one. I f the Prandtl n umber o f air is assumed to be 0.7, then the value o f Nux given by Eq. (3.58) is about 12% greater than the true value.
3.3 SIMILARITY S OLUTIONS F OR F LOW O VER F LAT PLATES W ITH O THER T HERMAL BOUNDARY C ONDITIONS
The above similarity solution was for the case where the plate has a uniform surface temperature, i.e., for which:
Tw  TI = constant
Similarity solutions for a few cases o f flow over a flat plate where the plate temperature varies with x in a prescribed manner can also be obtained. In all such cases the solution for the velocity profile is, o f course, not affected by the boundary condition
C HAPTER
3: Some Solutions for External Laminar Forced Convection
99
on temperature and is the same as for the uniform temperature case. For example, consider the case where:
I.e.: (3.62) C a nd n being constants. T he following dimensionless temperature is again introduced:
(J
= Tw  T =
T wT 1
1 _ T  Tl T wT1
(3.63)
a nd it is again assumed that (J depends only on the similarity variable, T]. In the present case, Tw  T 1 is a function o f x because:
T  Tl = (1 ..". (J)(Tw  T 1)
t he energy equation (3.3) c an b e written in terms o f (J a s
a a(J  u [(1  (J)(Tw  T 1)]'+ v a (Tw  T 1) h y
=
[ v ] 2(J (Tw : T ) (3.64) P a2 1
r~
I n terms o f (J, t he boundary conditions again are:
y=
0: (J
=
0
(3~65)
y large: (J,  + 1
Using the relations for the velocity components previously derived, Eq. (3.64) gives:
f ' d(J aT] _ n f'(l  (J) dT] a x x
+
,[! j
2
v ( f ' _ f )] d(J aT] ...:.. ~ ,!:.. d 2(J [aT]]2 U IX T] dTJ ,ay . Pr U l dT]2 a y, ,
(3.66)
O n rearrangement, this equation becomes:
(J"
+ n Prf'(l
 (J)
+ 2 (J'f
Pr
=
0
(3.67)
while the boundary conditions on the solution given i n Eq. (3.65) can b e written as:
T] == 0: (J = 0 T] large: (J+ 1,
(3.68)
Thus, as was the case ~ith qniform plate" temperature~ t he partialdlff~rential equation governing the temperatpre disttibution h as b een r educed to an ordinary differential equation. This confirms the ass~mption t hat the t emperature profiles' are similar. For any prescribed values o f P r at1p n, ,the variation o f 8" with, T] c an b e obtained b y solving Eq. (3.67). A computer pr~gram, SIMVART, which is an extension o f t hat for the uniformtem~rature surface case, obtains this solutiOIi. T he p rogram
/
100 .·Introduction to Convective Heat Transfer Analysis
2r~~~~~
0 ''''
0.0 0.5 n 1.0 1.5
F IGURE 3.8 Variation o f 0'11/=0 with n for various values o f Pro
first obtains the velocity profile solution and then uses the same procedure to soJve Eq. (3.67). The program is available i n the way discussed in the Preface. The heat traQ.sfer r ate at the wall is as before given by
q
w
=
k
aTI y=o ay
(3.69)
Hence, using Eq. (3.63), i t a gain follows that: (3.70) For any prescribed values df P r and n, (J'l l1 =o will have a specific value. I t therefore follows that qw will b e proportional to ( Tw  T1)l xO.5. Hence, the case where the heat flux at the surface o f t he plate is uniform corresponds to the case where n = 0.5, i.e., a similarity solution e)(ists for flow over a flat plate with a uniform surface heat flux. Some typical variations o f (J'l l1 =o with n for various values o f P r obtained using the above program are shown in Fig. 3.8.
E XAMPLE 3 .2.
A ir flows at a velocity o f 3 m ls over a wide flat plate that has a length o f 30 em in the flow direction. The air ahead o f the plate has a temperature o f 20°C while the temperature o f the surface o f the plate is given by [20 + 40(xI30)]OC, x being the distance measured along the plate in cm. Using the similarity solution results, plot the variation o f local heat transfer rate in W1m2 along the plate.
Solution. T he plate temperature varies linearly from 20 to 60°C. Its mean temperature is therefore 40°C. T he mean temperature o f the air in the boundary layer is therefore, Tmean = Twmean + TI = 40 + 20 = 300C 2 2
/
A t this temperature for air:
k = 0.0264 W/moC, v = 0.0000160 m 2 /s
C HAPTER
3: Some Solutions for External Laminar Forced Convection
101
Hence, since here:
u,
=
3 mls
it follows that the Reynolds number based on the length o f the plate, i.e., 0.3 m, is:
ReL =
u v,L
=
3 X 0.3 0.0000160
=
56,250
T he boundary layer on the plate will therefore remain laminar (see discussion o f the transition Reynolds number given later) . . Because the plate temperature variation can be written:
40 oC Tw  20 = 0 .3x
x being in m, the plate temperature variation is o f the form: Tw  T,
= ex
Hence, in this case n = 1. Now, air has a Prandtl number o f approximately 0.7 and for . this value o f Pr, for n = 1, Fig. 3.8 gives:
8'111=0 = 0.480
Hence since:
it follows that:
Using the value Q fk given above, it follows that:
qw
= 0.480 XO.0264
X \(40/0.3)xJ
xO.00~0160
=
7 31.6jX W/m2
In this equation, x is in m. The variation o f qw.with x given by this equauon is shown in Fig. E3.2. In this case, because the plate temperature is equal to the air temperature at x the heat transfer rate is zero at the leading edge o f the plate and jncreases with x.
600 , .,.,..,
=0,
400
0.1
0.2
x m
0.3
FIGUREE3.2
102
Introd4~tion
to Convective H eat Transfer Analysis
Air flows a t a velocity of7\mls over a wide flat plate that has a length of 10 c m i n the flow direction. The air ahead q f the plate has a t emperature o f 20°C. There is a uniform heat flux o f 2 k W1m2 a t t he surface o f the plate. Using the similarity solution results, plot the variation o f local temperature along the plate.
E XAMPLE 3 .3.
Solution. Here, the plate temperature is not known so the mean film temperature at
which the air properties are determined is not initially known. The a ir properties will therefore first b e evaluated a t t he freestream temperature 20°C a nd the plate surface temperature variation will b e evaluated. This will allow an estimate o f the mean temperature to b e made and the a ir properties a t this mean temperature c an b e found and the calculation can then b e repeated using these air properties. I f necessary, a new mean air temperature can b e determined using this new temperature and the calculation again repeated. This third step is, however, seldom required. ' At 20°C, air has t he following properties:
k
Hence, since here:
= 0.0256 W/moC,
Ul =
v = 0.0000151 m 2/s
'1 mls
it follows that the Reynolds number based on the length o f the plate, Le., 0.1 m, is:
ReL
=  ; =
u lL
7 x 0.1 0.0000151
= 46,358
The boundary layer on the plate will therefore remain laminar (see discussion of the transition Reynolds number given later). Now, i t was shown that i f there is a uniform heat flux a t the surface:
Tw  Tl = CxO. 5
Le., when there is a uniform surface heat flux, n = 0.5. A ir has a Praridtl number o f approximately0.7 a nd for this value o f Pr, for n = 0.5, Fig. 3.8 gives:
fJ'I'Il=o
Hence, since:
= 0.406
k (T~w~ T 1) = 0'1'Il=0 j Re x
it follows that in the present case where qw
= 2000 W /m2 :
2000x 0.0256(Tw  Tl)
i.e., since Tl
= 0 .406
J
7x
0.0000151
= 20°C:
T w = 20 + 282.6xo. 5
The variation o f T w with x g iven by this equation is shown in Fig. E3.3. Now the mean plate temperature is given by:
Tw
=
±Io Tw dx
L
which using the expression for T w derived above gives:
TWmean = 2 0 + 5 9.6 = 79.6°C
CHAPTER
3: Some Solutions for External Laminar Forced Convection
103
150 . .          r      ,
100
50
o~~~
0.00
0.05
T ffi
0.10
F IGUREE3.3
T he mean air temperature is therefore
T
mean 
 Twmean + Tl = 79.6 + 20 = ' 49.80C 2 .' 2
A t this temperature for air:
k
= 0.0277 W/moC,
v
= 0.0000178 m2/s
Using these values for the air properties, the equation for the surface temperature becomes:
Tw = 2 0 + 283.7 x°.5
This gives values that differ from those derived before by less than 0.5%. T he results given in Fig. E3.3, therefore, adequately describe the surface temperature variation and there is no need to repeat the calculation with an improved mean air temperature.
Because o f the linearity o f the energy equation, solutions for different wall temperature variations can b e combined to get solutions for other temperature variations, i.e., i f TA a nd T B a re similarity solutions to the energy equations for two different wall temperature variations then T A + TB is also a solution to the energyenergy equation and applies to the case where the wall temperature is the sum o f those giving the solutions T A a nd TB, i e., since:
( T  Tl)A = (1  8 A )(Tw  Tl)A
and:
( T  Tt)B
= (1 
8B)(Tw  Tt)B
w here (J A and (J B a re the similarity solutions for the two wall temperature variations theq. the solution for the wall temperature variation: '
(Tw  Tl)A
+ (Tw 
T4JB Tt)B
/
is:
( T  Tt) = (1  8 A)(Tw  : Tt)A
+ ( 1 8B)(Tw 
F or example, i f 8 Ai s the solutionfor plate wit11 a uniform surface temperature, Two, and i f 8 B is the solution for T w =, T~ + C x n., then: (3.71)
a
104 Introduction to Convective Heat Transfer Analysis
S OLUTIdNA
~
Distance Along Wall
c:= :
Distance Along Wall Wall Temperature
~_ _   
S OLUTIONB
SOLUTION A + S OLUTIONB
Distance Along Wall
F IGURE 3 .9 Combination o f temperature field solutions for different boundary conditions.
w ill b e t he s olutionJor t he c ase w here:
(Tw  T 1 ) = (Two  TI)
T his i s i llustrated i n F ig. 3.9. T he h eat t ransfer r ate a t t he w all i s g iven b y:
+ Cxn
(3.72)
qw x k(Two  T 1) =
E XAMPLE 3 .4.
[(JA 71=0 + (JB 'I71=0 Two x n Tl Jv~ 'I C_ Rex
(3.73)
A ir flows at a velocity o f 3 mls over a wide flat plate that has a length o f 30 c m in the flow directiori. T he air ahead o f the plate has a temperature o f 20°C while the surface o f the plate is equal to [40 + 40(x/30WC, x being the distance measured along the plate in cm. Using the similarity solution results, plot the variation o f local heat transfer rate in W /m 2 along the plate.
Solution. T he plate temperature varies linearly from 40 to 80°C. Its mean temperature
is therefore 60°C. The mean temperature o f the air in the boundary layer is:
I
Tmean = Twmean+TI = 60 + 20 = 400C 2 2
At this temperature for air:
k
= 0.0271 W/moC, v = 0;0000170 m2/s
UI
Hence, since here:
=
3 mls
i t follows that the Reynolds number based on the length o f the plate, i.e., 0.3 m, is:
ReL =
V
u iL
=
3 x 0.3 0.0000170 = 52,941
CHAPTER
3: Some Solutions for External Laminar Forced Convection
105
2000
rr,r,
1500
'"§ .
~
I
1000
J
500
0.1
X ffi
0.2
0.3
F IGUREE3.4
The boundary layer on the plate will therefore remain laminar (see discussion o f the transition Reynolds number given later). The surface temperature variation is,made up o f a step j ump i n temperature at the leading edge o f the plate plus a linearly increasing temperature along the plate. The surface temperature can be written as: ( Tw

T 1)
= (60 
20)
+ 133.33x °C
where, in this equation, x is in m. In the present case therefore (Two  T 1) = 40°C and n = 1. As discussed above, the solution is obtained b y combining the similarity solutions for a plate with a uniform temperature and for a plate with a linearly increasing temperature. These two solutions give for air ( Pr = 0.7) (see Fig. 3.8):
6'1"1=0 = 0.293
and
6'1"1=0 = 0.406
respectively. Now it was shown that:
Il~x [, , c xn] ~ k(Two  T 1) = 6A 1"1=0 + 6B 1"1=0 Two _ Tl " ,Rex
He:qce, in the present case where C = 133.33 and n = 1:
qw x 0.0271 X 4 0
i,e.,
= [0 293
.
+
0.406 X 133.33X] 3x 40 ' 0.0000170
J
33.4 qw = 1xO.5 + 616.1X 0,5 The varia~on o f qw with x given by this equation is shown i n Fig. E3.4. I t will b e seen from Fig. E3.4 that ne~ the leading edge o f the plate the effect o f the temperature j ump a t the leading ,edge predf?minates and qw i s approximately proportional t o 11 j X whereas near the trailing edge of\the plate the effect o f the linearly increasing plate temperature predominates and qw i s 6nly very weakly d~pendent on x.
/
106
Introduction to Convective Heat Transfer Analysis
3.4
OTHER SIMILARITY S OLUTIONS
I n the preceding sections, the solution for boundary layer flow over a flat plate was obtained by reducing the governing set o f partial differential equations to a pair o f ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values o f x, ( UIUl) a nd (Tw  T)/(Tw  Tl) were functions o f a single variable, 1], alone. Now, for flow over a flat plate, the freestream velocity, U l, is independent o f x. The present section is concerned with a discussion o f whether there are any flow situations in which the freestream velocity, U J, varies with x a nd for which similarity solutions can still be found [1],[10]~ Consideration is again first given to the velocity profile solution. The problem is to determine for what distributions o f freestream velocity, U l, these profiles will be similar, i.e., for what distributions o f Ul is i t possible to assume that:
~ = function (1]*) = f '(r()
Ul
(3.74)
where 1]* is some similarity variable whose form has also still to be determined. Notice that the function relating ( UIUl) to 1]* has again been denoted by / " the prime now denoting differentiation with respect to 1]*. Now the form o f the similarity variable, 1], for flow over a flat plate was arrived a t by noting that i f the velocity profiles were similar then:
~ = function ( yIB)
Ul
(3.75)
where 8 was some measure o f the local boundary layer thickness. I t was then noted that, from the order o f magnitude analysis used in the derivation o f the boundary layer equations, the order o f magnitude o f B has to be x l j R ex and this was substituted for B in Eq. (3.75). This then gave 1] as y jRexlx. In the case being studied in the present section where Ul varies, the above result will be generalized by assuming that B will depend on Le(x)1 j ReL where x is equal to x lL, e is some function o f x , L is some convenient reference length, and ReL is the Reynolds number based on L and some convenient reference velocity U, i.e.,
ReL
=
U L/v
(3.76)
Substituting this as the measure o f the boundary layer thickness into Eq. (3.75) then gives the following as the form o f the generalized similarity variable: *
TJ
y~ = L e(x)
(3.77)
The function e (x) remains to b e determined. The basic procedure for arriving at the similarity solution is, o f course, the same as that for flat plate flow. The continuity equation is first used to express v in terms o f the similarity function f '. F or this purpose it is written as:
Jv Jy
au ax
CHAPTER
3: Some Solutions for External Laminar Forced Convection
107
i.e.,
av a11* = _ ~(ul/') = a11* a y ax
 I' d~
dx
_ u l/"
a~*
ax
(3.78)
where, as before
X =L
x
(3.79)
Using Eq. (3.77) in Eq. (3.78) then gives:
v~R av = /.\eLa11*
I ' eUl + U l11 * 1" dde
dx dx
(3.80)
This is integrated with respect to 11* noting that v = 0 when 11* = 0, to give:
v~ ReLv
I.e.,
=
I e dUl + dx
dx Ul deC 11* f' 
I)
(3.81) Now, since flows i n which Ul, the freestream velocity, i n general, varies with x . are being considered, the momentum equation to b e solved has the f orm
au Uax
+v
au dUl = Ul ay dx
+V
a2u ay2
(3.82)
Using the expressions for the variables given above, this equation becomes:
' a ( I') Lid ( ) * I ' de a I ' Ul I a x Ul  JReL d x eUl  11 Ul d x a y ( Ul)
(3.83)
With 11* as defined i n Eq. (3.77), this equation can b e rewritten as:
1,2
d~
dx
_
I f"!
d_(eul) _ d~ _ u 1'" e dx d x . e2
=0
(3.84)
I f t he following are defined, for convenience:
a
=
f3
ed U d x ( eur)
=
(3.8S) (3.86)
e2 dUl
t /dx
Eq. (3.84) can b e w ritten as:
I'" + a ll" + {3(1 
1,2)
=0
'
(3.87)
108
Introduction to Convective Heat Transfer Analysis
Now, i f similar velocity profiles do e~st, Eq. (3.87) must allow f to b e determined as a function o f the similarity variab1e 7]* alone. This will only b e possible i f ex and f3 a re independent o f x. S ince they d on't depend on 7]*, this means that i t is only for flows i n which:
a = constant
13
(3.88)
= c onstant
that similarity solutions c an b e found. This requirement on a a nd f3. t hen allows the freestream velocity distributions U l ( x) occurring i n the expression for t he similarity variable 7]* to b e found. For this purpose i t is noted that Eqs. (3.85) and (3.86) give:
!
I
1d 2 2 a  f3 = U d x(e U l)
(3.89)
Thus, the velocity distribution and the function, e, for similar solutions must be such that d (e2ul)ldx is a constant equal to ( 2a  f3)U. Now one possibility is t hat(2a f3) b e equal to zero. However, t he variation o f U l with x that provides this situation seems to have little practical significance and i t will not b e considered here. T\ierefore, (2a  f3) will be assumed to b e nonzero. Integrating Eq. (3.89) gives:
e2~
= ( 2a 
f3)x
(3.90)
Now i t will also be noted from Eqs. (3.85) and (3.86) that:
Ul d e a f3=e Udx
(3.91)
w hich can be rearranged using Eq. (3.86) to give:
( a  f3)'! dUl = !. dUl d e = 13 d e Ul d x U d x dx e dx
This, in turn, c an b e integrated to give:
U l a f3 =
(3.92)
K e f3
(3.93)
w here K is a constant o f integration. The function e is now eliminated between Eqs.(3.90) and (3.93) to give:
J (2a  f3)xU lUI =
U l ( af3)/f3 K 1/ f3
i.e.,
(3.94) Since ex a nd f3 are both constants and both contain the function e, i t is possible, without any loss of generality, to set ex equal to 1, any common factor between a and f3 then being incorporated into e. T he c ase a = 0 is, o f course, being excluded from the discussion. With a s et equal to 1 for this reason, Eq. (3.94) becomes: (3.95)
/
C HAPTER
3: Some Solutions for External Laminar Forced Convection
109
I t is convenient to express {3 in terms o f a n ew constant, m, w hich is such that:
m
=
(3/(2  (3)
(3.96)
This can, of course, be rearranged to give
{3 = 2m/(m
+ 1)
(3.97)
I n terms o f m, t he freestream velocity distributions for which similarity solutions can be found, these being given in Eq. (3.95), can be written as:
",
~ Kl+ [(m ~ 1) ~r x'"
m
(3.98)
Since K, U, a nd L are all constants this equation has the form:
UI
= c xm
(3.99)
where c is a constant. Therefore, similarity solutions to the boundary layer equations can b e found for flows over bodies whose freestream velocity varies as xm. Having established the form o f the free stream velocity distribution that gives similarity solutions, consideration can be given to the similarity variable function e(x). This is found by using Eq. (3.90). Noting that a has already been set equal to one, this equation gives:
e
=
J (2  (3) u I x L~U;
~
J(
m +l L~U;
2
)U
[X
(3.100)
Substituting this into the expression for the similarity variable 'Y/* then gives:
(3.101)
where Rex is the Reynolds number based on x a nd the local freestream velocity is With a s et equal to one and {3 defined i n t erms o f m b y Eq. (3.97), the reduced momentum Eq. (3.87) becomes
f "' + 1 + (
m+ 1
2m ) (1 
1,2)
=
0
(3.102)
T he boundary conditions on the solution to this equation being, o f course
= 0: 1 = 0, f ' = 0 1'/* large: I ' ~ 1
1'/*
(3.103)
Before discussing any solutions to this equation"~ i t is useful to consider what body shapes will give a velocity distribution o f t he typy for which similarity solutions c an b e found, i.e., o f the type given in Eq. (3.99), A velocity distribution o f this type will exist with flow over wedgeshaped ·bodies having an included angle, c/J, which i s equal to 1T{3 as shown in Fig. 3.10:' Because m i s related to (3 b y Eq. (3.~6) i t ~s related to the included angle cfJ lJy:
. I
/
m = c/J/(21T 
c/J)
(3.104)
· no
Introduction to Convective H eat T ransfer Analysis .
F IGURE 3.10
Wedgeshaped body for which similarity solutions exist.
m =l
F IGURE 3.11
Stagnation point flow.
It may b e noted that the case 4> equal to zero corresponds to flow over a flat plate while the case o f 4> equal to T r, i.e., m equal to one, corresponds to flow over a plate s et normal to the direction o f the undisturbed free stream as shown i n Ffg. 3.11. T he latter flow closely represents flow near the stagllation point o n a b luff body. For example, i n the case o f flow over a circular cylinder the freestream velocity is given by
u 1lU = 2 sin(xIR)
(3.105)
the variables being defined i n Fig. 3.12. For small values o f x l R, i.e., near the stagnation point, this equation reduces approximately to:
Utl(J = 2 xlR
(3.106)
Therefore, near the stagIl;ation point, U l is proportional to x so that the similarity solution with m equal to 1 WIll apply i n this region. O f course freestream velocity distributions o f the form U l = c xm c an also be obtained in other situations, e.g., i n a d uct with varying crosssectional area. Solutions to Eq. (~.l02) subject to the boundary conditions given i n Eq. (3.103) have been obtained for a number o f different values o f the velocity distribution index, m. Some typical results are shown in Fig. 3.13.
u
F IGURE 3.12
Flow over a circular cylinder.
CHAPTER
3: Some Solutions for External Laminar Forced Convection
111
4
m
Vl
f3
o~
I~
" o J ' '
2~IN .....
II
 0.091 0.199  0.065  0.14 0 0 113 0.5 1 2 4
* ~
0 0.0 0.5
u /u) =1'
1.0
F IGURE 3.13 Results for various values o f the index, m.
The profile for m equal to zero corresponds to flow over a flat plate and is identical to the solution given i n t he previous section for this type o f flow. In making a comparison with this solution it should b e noted that for m = 0, T/* is equal to Ull2xl v, i.e., T/* = T /I./i, T/ being the similarity variable used in deriving the flat plate solution. I t should also b e noted that when m is less than zero, d Ut/dx is negative i.e., there is an adverse pressure gradient (the pressure is rising). These solutions for m < : 0 do not, o f course, correspond to flow over real wedges, but such velocity distributions can Ife generated i n other ways. The velocity profiles for values o f m less than zero have an inflecti\On point as will b e noted from the results shown in Fig. 3.13. When m i s equal to  0.091, the slope o f the velocity profile at the wall is zero, i.e., d(ulul)/dT/* = 0 and so d uldy = 0, a nd the boundary layer is, therefore, on the point o f separating. Solutions for more negative values o f m have no meaning because they exhibit rey~rsed flow i n the region adjacent to t he wall and the boundary layer equations are not applicable to such flows. Having established that similarity solutions for the velocity profile can be found for cert~n flows involving a varying freestream velocity, attention must now be turned to the solutions o f t he energy equation corresponding to these velocity solutions. The temperature is expressed in.tenns o f t he same nondimensional variable that was used in obtaining the flat plate solution, i.e., in t enns o f 0 = ( Tw  T)/(Tw Tt) and i t is assumed that 0 is also a fuilction o f T/* alone. Attention is restricted to J low over isothermal surfaces, i.e., w~th T wa constant, and T 1 . o f course, i s also constant. I n terms of the variables introduced above, the eJ;1etgy equation is:
J
u
1
f'O'
aT/* ax
+ vO aT/*'., = (.!!....)O/l (aT/* )2.
'ay P r'
>{
ay
(3.107),
I
112 .Introduction to Convective Heat Transfer Analysis
I
B ut substitutingEq. (3.100) a nd (3.1010 into the expression derived for v gives: (3.108) Substituting this into Eq. (3.107) and using Eq. (3.99) then gives o n rearrangement
(J"
+ P r f (J'
= 0:
(J
=0
(3.109)
The boundary conditions on (J are the same as those for flow over a flat plate, i.e.,
'YJ*
=0
(3.110)
'YJ* large: (J ~ 1
For any selected values o f m and P r, the variation o ff w ith 'YJ* derived using the procedure previously outlined, can b e u sed to obtain the solution to Eq. (3.109) subject to the boundary conditions given in Eq. (3.110). To obtain this solution, the same procedure that was used to find the flat plate solution and which was discussed above is utilized. This solution will give the variation o f (J w ith 'YJ* corresponding to the selected values o f m a nd Pro With the temperature distribution determined in this way, the heat transfer rate can be determined b y noting that (3.111) This can be rearranged using Eq. (3.101) as (3.112) where N ux i s again the local Nusselt number and R ex is the local Reynolds number. I t is convenient to write (3.113) and Eq. (3.112) can then b e written as
N ux = a * J Re x
(3.114)
Values o f a * for various values o f m and P r have been derived from the calculated variations o f () with 'YJ*. Typical values are shown in Table 3.2. The values for m = 0 are, o f course, the same as the values o f a given i n the previous section for flow over a flat plate. For m = 1, w hich corresponds, as previously indicated, to stagnation point flow, the values o f a * given in the table can b e closely represented by (3.115)
C HAPTER
3: Some Solutions for External Laminar Forced Convection
113
T ABLE 3 .2
Values o f a * f or v arious values o f m a nd P r
Pr
m
 0.0753 0.0 0.111 0.333 1.0 4.0
0.7
0.242 0.293 0.331 0.384 0.496 0.813
0.8
0.253 0.307 0.348 0.403 0.523 0.858
1.0
0.272 0.332 0.378 0.440 0.570 0.938
5.0
0.457 0.585 0.669 0.792 1.043 1.736
10.0
0.570 0.730 0.851 1.013 1.344 2.236
Substituting this into Eq. (3.114) then gives the local heat transfer rate in the region of the stagnation point as
Nux = 0.57Pr°.4Re~.5
(3.116)
I n calculating stagnation point heat transfer rates, it is more convenient to use a Reynolds number based on the undisturbed freestrearn velocity ahead o f the body, U, rather than one that is based on the local freestrearn velocity, U l. T he actual relation between U l and U will depend on the body shape. For the case o f flow over a ~ylinder i t is, as previously given in Eq. (3.106):
Ul
= 4 UxlD
(3.117)
This relationship only applies for small values o f x. D is the diameter o f the cylinder. While this relation is strictly only applicable for flow over a cylinder, it closely describes the freestrearn velocity distribution in the region o f the stagnation point for flow over any rounded body, D, then being twice the radius o f curvature o f the leading edge. Substituting this value o f U l into Eq. (3.116) then gives:
Nux = 1.14Pr°.4Re~\xID)
where ReD is the Reynolds number based on U and D, i.e.,
ReD
(3.118)
= =
U Dlv
(3.119)
I f the Nusselt number is also based on D , i.e., i f the following is defined
N UD
h Dlk
(3.120)
h being the local heat transfer coefficient which, by virtue o f Eq. (3.118), is a constant i n the stagnation point region, then Eq. (3.118) gives
(3.121) The values o f N u Dpredicted by this equation are in good agreement with measured heat transfer rates in the region o f the stagnation point on rounded bodies.
E XAMPLE 3 .5.
Consider twodimensional, air flow over/a circular cylinder with a diameter o f 4 c m and a surface temperature of 50°C. I f t he temperature in the air stream ahead o f the cylinder is 10°C a nd i f the air velocity i n this stream is 3 mis, determine the heat transfer rate i n the 'vicinity o f the stagnation point.
114 Introduction to Convective Heat Transfer Analysis
Surface Temperature
T = lOoe
~ ~ ~
= 50oe
~ ~
U =3m/s
FIGUREE3.5
Solution. T he flow situation being considered is shown in Fig. E3.5.
T he mean temperature o f the air is:
T
m ean
 Tw
+ Tl
2
= 50
+ 10
2
= 300C
A t this temperature for air:
k
= 0.0264W/moC, v = 0.0000160 m2/s
N UD
I t w as shown above that near the ,stagnation point:
=
1.14Pr°.4Re~5
i.e.,
q~D
Here, D = 0.04 m a nd U
= 1. 14PrM [ U:f5
= 3 mls so this equation gives:
qw
which gives:
x 0.04 = 1 14 07°.4 [ 3 x 0.04 ]0.5 0 .0264 . x. 0.0000160
qw
= 56.5 W /m2
Hence, in the stagflation point region, the heat flux is 56.5 W /m2 • T he heat flux is constant in this region, i.e., i n this region qw does not vary with x.
3.5
I NTEGRAL EQUATION SOLUTIONS
Similarity solutions o f t he type discussed above cannot, o f course, b e obtained for arbitrary distributions o f freestream velocity and surface temperature. Therefore, while the similarity solutions do give results which are o f considerable practical significance and while the exact results they give are very useful for checking the accuracy o f approximate methods o f solving the boundary layer equations, other methods o f solving the boundary layer equations for arbitrary distributions o f freestream velocity and surface temperature must b e sought. Approximate methods, based on use o f the integral equations developed in the previous chapter, can b e applied to flows with such arbitrary boundary conditions and they will b e discussed in the present section [1]. As mentioned in Chapter 2, these integral equation methods have largely been superceded by purely numerical methods. Howev~r, they are still sometimes
C HAPTER
3: Some Solutions for External Laminar Forced Convection
115
used and are therefore briefly discussed here. Attention will b e restricted to twodimensional constant fluid property flow. I n order to illustrate the main ideas involved in the procedure, attention will first b e given to flow over an isothermal flat plate. T he variation of velocity boundary layer thickness, 5 , with the distance along the plate from the leading edge, x, is first determined by solving the momentum integral equation. I n order to obtain this solution, it is assumed that the velocity profile can b e represented b y a thirdorder polynomial, i.e., by
U
=
a+ by + c i + d l
(3.122)
There is, of course, no special reason for adopting a thirdorder polynomial. Experience has, however, shown that it is the lowestorder polynomial that leads to a solution o f acceptable accuracy. T he values o f t he coefficients a, b, c, a nd d i n t he assumed velocity profile are obtained, as mentioned in Chapter 2, by applying the boundary conditions on velocity a t the inner and outer edges o f t he boundaryJayer. Three such boundary conditi~ns are: at:
y y y
= 0: = =
U=o
u=
Ul
5: 5:
(3.123)
au
=
0
ay
T he first of these conditions follows from the "noslip" a t the wall requirement while the second simply follows from the fact that the velocity m ust b e continuous a t the outer edge o f t he boundary layer. T he t hird condition follows from the requirement that the boundary layer profile blend smoothly into the free stream velocity distribution i n which t he viscous stresses are zero. Equation (3.123) provides three boundary conditions which c an b e u sed to find three o f the coefficients i n Eq. (3.122). A fourth boundary condition is obtained by applying the momentum equation (2.136) for the boundary layer to conditions at the wall. Since U = v = 0 a t the wall, this equation gives:
1 dp y = 0: ay2 = p dx
,
\
a2 u
(3.124)
T his relation can also b e derived b y applying the consiervation o f m omentum principle to a control volume that is in contact with the wall and is o f s mall size !l.y measured nonna! to t he wall. Since the velocities will b e negligibly small. the momentum flux through this control volume will b e negligible and t he forces acting ol)/it· m ust balance. Applying this force balance requirement and then taking/the limiting form of t he r esultant equation as the size o f t he control volume goes to zero leads to Eq. (3.124). S ince, for the moment, only the flow over a fl~t p late for which d p /dXi = 0 i s being considered, Eq. (3.124) ~educes in this case:
\
A ty
= 0: \ =.0 ay'2
a2u
( 3.125) ,
116 Introduction to Convective Heat Transfer Analysis
Applying the boundary conditions given',in Eqs. (3.123) and (3.125) to Eq. (3.122) then gives the following three equations:·
O :a
Ul = a + bS + cS 2
+ dS 3
(3.126)
,
o=
b + 2cS + 3dS 2
. Q:;:: 2c
This is a set o f four algebraic equations i n the four unknowns a,' b, c, and d. Solving for these gives:
a
= 0,
b
= 3ull2S,
c = 0,
d
=
 u l l2S 3
(3.127)
Substituting these values into Eq. (3.122) and rearranging gives the velocity profile as: .(3.128) It should b e realized that there is no real purpose in comparing this velocity profile with that given by the exact similarity solution since the integral equation method does not seek to accurately predict the details o f the velocity and temperature profiles. The method seeks rather, by satisfying conservation o f mean momentum and energy, to predict with reasonable accuracy the overall features o f the flow. Now for the case o f flow over a flat plate for which Ul is constant, the momentum integral equation (2.173) can b e written as
dx
d [s J( (1  Ul o
1
U )( U )
Ul
d("Y)~~ _ pur 8
Tw
(3.129)
Using the velocity distribution given in Eq. (3.128) gives the following:
Jo
Also:
Tw
e(l )(u )d(Y) _2380 9 S
Ul
U
Ul
(3.130)
=
iJu J.L iJy
I
y =o
=T
(J.LUl) iJ(U/Ul) iJ(y/S)
I
y =O
(3.131)
so that again using Eq. (3.128) gives:
Tw
~ ~ (1';')
=
(3.132)
Substituting Eqs. (3.130) and (3.132) into Eq. (3.129) then gives:
~SdS
140
~
PUl
dx
(3.133)
This equation can b e directly integrated using the following initial condition: . W hen x = 0: S = 0 ( 3.134)
CHAPTER
3: Some Solutions for External Laminar Forced Convection
117
The integration o f Eq. (3.133) then results in: 1 IL 3 2 =  x 8
280
P Ul
(3.135)
This is conveniently rearranged as
  ==
x J Re x
8
4.64
(3.136)
where Rex is again the local Reynolds number ( ulxlv). It may be recalled that it was deduced from the similarity solution for flow over a flat plate that (8Ix) = 51 J Re x . The difference between the value o f the coefficient in this equation, i.e., 5, and the value in Eq. (3.136), i.e., 4.64, has no real significance since, in deriving the similarity solution result, it was arbitrarily assumed t4at the boundary layer thickness was the distance from the walJ a t which U became equal to 0.99 U l. Having obtained the solution to the momentum integral equation, attention"must now be turned to the energy integral equation. To solve this, the form o f the temperature profile must be assumed. As with the velocity, a thirdorder polynomial will b e used for this purpose, i.e., it will b e assumed that:
T=e
+ fy + gl + hl
(3.137)
The coefficients i n this equation, i.e., e ,f, g, and h, are determined by applying the boundary conditions on temperature at the inner and outer edges o f the thermal boundary layer. Three such boundary conditions, which are analogous to those given for the velocity in Eq. (3.123), are:
.y = 0: :y' = 8T :
y = 8T :
T = Tw
T = Tl
(3.138)
a Tlay
=
0
8T is, o f course, the thic~ess o f the thermal boundary layer. The first o f these conditions follows from t berequirement that the fluid in c ontact with the wall must attain the same temperature as the wall. The other two conditions follow from the requirement that the boundary layer temperature profile must blend smoothly into the freestr~am temperature distribution at the outer edge o f the boundary layer. A fourth boundary condition is obtained b y applying 'the boundary layer energy equation (2.146) to conditi{ms at the wall. Since dissipation is being neglected this gives: (3.139) At Y = 0:
Applying the four boundary conditions given i n Eqs. (3.138) and (3.139) to Eq. (3.137) then gives:
Tw = \e \ 2 3 T l = e + f 8 T +\g8T + h8T
/
o=
(3.140)
f 2g8T f\ 3h8'f. 0 = 2g
118 Introduction to Convective Heat Transfer Analysis
T his is, o f course, again a set o f four equations i n the four u nknown coefficients. Solving between them then gives
e = T w,
g
f
=  3(Tw  Tl)l2B T
= 0,
(3.141)
h = (Tw  Tl)/25j.
Substituting these values into Eq. (3.137) and rearranging gives the t emperature profile as: (3.142) T he h eat t ransfer rate a t t he wall is given b y F ourier's l aw as:
qw
=
 k aT a y y =o
1
=
 k(Tw  Tl) a BT a (yIBT )
[
T  Tl T w  Tl
II
y =O
(3.143)
Using Eq. (3.142) to obtain the derivative then gives:
qw = 3k(Tw  Tl) 2BT
(3.144)
Now, since Ul a nd T w a re constants i n t he problem being considered, the energy integral equation (2.183) c an b e w ritten as:
u l(Tw  Tl) d x
d
[1f0
8 ( U )(
T
Ul
Tw  Tl
T T
1) ]
2P!B
qw d y = p Cp
(3.145)
Using Eq. (3.144) and rearranging then gives:
:x
[r (~)(£~~[)dY]
y s;
=
T : ..[
(3.146)
I n e valuating the integral, care has to b e e xercised because the velocity and temperature profiles are really discontinuously described, one relation being used inside the boundary layer and another outside the boundary layer. T he velocity profile, for example, is actually described by:
5: .!!.Ul
U
=
~ ~ _ !(~)3
25 2B
(3.147)
y > 5: 
Ul
=
1
This offers no difficulty w hen BT < B b ecause ( T  Tl)/(Tw  T 1) i s zero for y > BT • H owever w hen 5 T > 5 , t he integral in Eq. (3.146) has to b e e valuated in t he following way:
r )(£
(~
~ ~[) dy = L' (~)(£ ~~[) dy + I.: (;)(£ ~ ~1) dy
(3.148)
T he solutions for 5T
< B and 5T > 5 will, therefore, b e separately obtained.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
119
Consid,er, first, the case or < O. Here
Jo
(8 T
(u)( TT Tl ) = Jf8 [328" "2 (y)3] [1 y 18" TIdy "
T
Ul
w
0
"2 Or +"2
3y
1Or )3]
(y
dy
~8
T
I: [~ Ir 8;  ~ (JJ (8; )]
[1  ~L + !2 (Or Or L)3]d(L) 2 8T
X
~ 88;)[;0 (
i 2~0 (8;)]
_
(3.149)
I f the following is defined for convenience
(3.150) tJ. = 8r18 substituting the value o f t he integral given in'Eq. (3.149) b ack into Eq. (3.146) leads to
dx
d [8 ( 3 tJ.2 3 tJ.4)~ 3 ( IL ) 20  280 ~  2Pr/l8 P UI
(3.151) .
B ut t he momentum integral equation result given i n Eq. (3.133) c an b e written as:
: :0
~! ~ (~
)(:.., )
(3.152)
Comparing Eq. (3.151) and (3.152) then shows that since 8 a nd 8T are both equal to zero at the leading edge, i.e., when x = 0, tJ. is a constant and is given by:
./ . tJ. 3 tJ.3 3 tJ.5 39 F or OT < 0, I.e.: < 1: 2 0'  280 = 280Pr
(3.153)
This equation allows the variation o f tJ. w ith P t to b e found for values o f tJ. that are less than 1. Next, consider the solution o f t he energy integral equation for the case where 8T > 8 . I n this case si~e(ulul) is equal to 1 for values o f y greater than 8 , the integral is, as previousfy discussed, evaluated in two parts as follows:
8r
fa
(UJ(Tw  TJ
+
U
T  Tl
dy =
3Y [28 Io
8
  
2(8)3] [1 1y

3y 1y   if  
28T .
28T )3] (
dy
+
=
f
8 r [ 1  ' 3Y 828T
1 +  (Y )3] dy
2 8T
8
~)3 / .]d(~) J(1[~~ ! (8 o 2 8 2 ~)3][1 ~ ~~.!. + ! (8 Jl3 8 2 8 tJ. 2
_
+ 8T
II [ ,
lit:..
1 1 +  3  y +  ( Y 2 8T 2 8T
)3] d8T (y) (3.154)
3! 3 3] 3 = 8 [ g /l"8 + 20tJ.  280tJ.3
1 20 Introduction to Convective Heat Transfer Analysis
Substituting this result back into Eq. (3.146) then gives:
:x [ (~Ll
8

~ + 2~Ll  28~Ll3 )] ~ (2P~M ) (:.,)
 gA + 20
3 3 3 2_ 39  280~  280Pr
(3.155)
Comparing this with the momentum integral equation result given in Eq. (3.152) shows· that ~ i s again a constant and is given by . .3 2 For BT > B, I.e., ~ > 1. g~
(3.156)
This equation then allows the variation o f A with P r to be found for valutrs o f ~ that are greater than 1. Since the variation of B with x has been derived by solving the momentum integral equation, Eqs. (3.153) and (3.156) together constitute the solution o f the energy equation. The variation of ~ with P r that they together give is shown in Fig. 3.14. When P r i s equal to 1, ~ is, o f course, equal to 1 because the form o f the assumed velocity and temperature profiles are identical and when P r is equal to one the momentum and energy integral equations have the same form for flow over a flat plate. I t will also be noted from Eq. (3.153) that because this equation only applies for A < 1, the second term on the lefthand side can, i f an approximate solution will suffice, be neglected when compared to the first term. Therefore: For ~ < 1:
20~ = 2 80Pr ~3=~
1 4Pr
3
3
39
(3.157)
I t is seen, therefore, that for values o f ~ less than and equal to 1, A is approximately given by: (3.158)
I t has been found that this equation also gives results o f sufficient accuracy for most purposes for values o f ~ greater than 1 provided that P r is not very small. This can be seen from the results in Fig. 3.13.
2.5 , . .       ,      ,      .       , 2.0 1.5 1.0 0.5 0.0
L _ _ "_ _ _ L._ _ _  ' _ _  l
Eqs. (3.153) and (3.156)
....... ~ = I IPr 1l3
 1.0
0.5
0.0
log 10 P r
0.5
1.0
F IGURE 3.14 Variation o f boundary layer thickness ratio L\ with Prandtl number.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
121
Now the heat transfer rate as given by Eq. (3.144) can be written in terms o f ~ as: (3.159) Substituting the variation o f 0 with x given b y Eq. (3.136) into this equation and rearranging then gives:
Nux
=
0.343 JRex/~
~
(3.160)
I f the approximate expression for becomes:
given in Eq. (3.158) is used, Eq. (3.160) (3.161)
This result can be compared with that deduced from the similarity solution and given in Eq. (3.50). I t will b e seen that the integral equation method gives the correct form o f the result, i.e., Nux ex: Re;/2 Pr 1l3 b ut that the coefficient is somewhat in error. This is typical o f what can be expected o f t he integral equation method. The problem to which the integral equation method was applied in the above discussion, i.e., flow over an isothermal plate, is, o f course, one for which a similarity solution can b e found. The usefulness o f t he integral equation method, however, arises mainly from the fact that it can be applied to problems for which similarity solutions cannot easily b e found. In order to illustrate this ability, consider flow over a flat plate which has a n unheated section adjacent to the leading edge as shown in Fig; 3.15. The plate is, therefore, unheated up to a distance o f Xo downstream o f the leading edge. Beyond this i t is heated to a uniform temperature o f T w ' T he solution o f the momentum integral equation is, o f course, unaffected by the surface conditions o n temperature and is still, therefore, given b y Eq. (3.136). Since the thermal boup.dary layer pnly starts growing downstream o f the leading edge, attention w illbe restricted to the solution o f the energy integral equation for OT < O. I t is, o f course, possible, i f P r is less than one, for OT to eventually exceed 0 at large values o f x. This will Bot, however, b e discussed here.

Fluid Flow
Edge o f Velocity Boundary,Lay~r B oundary Layer
ri!!!!ll• ••••••~• ••aiture~ r
Unheated
xo
Plate _.._ ,.!,' Temp. Tw Freestreanl Fluid Temperature TI
o~~~
xo
Distance Along Plate x
F IGURE 3.15 F lat plate with unheated leading edge section.
122 Introduction to Convective Heat Transfer Analysis
Now, for 8 r < 8 , i.e., A < 1, i t was shown [see the derivation o fEq. (3.151)] that the energy integral equation becomes the (ollowing w hen thirdorder polynomials are assumed to describe the velocity and teihperature profiles:
:x [8 (iO Ll :x
2

2~OLl4)] = (2P~M )(:..,)
B ecause only the case o f A < 1 i s being considered, the factor ( 3A41280) will· b e neglected compared to (3A2/20). T his equation then reduces to: (8A2) =
[P!18 ][;J
x
280 JL 13 P Ul
(3.162) I
B ut the momentum integral equation gives
8=
(3.163)
Substituting this into Eq. (3.162) then gives: 2A2280 ~x d A 13 P Ul d x
+ A3 280 ~
2 13
P Ul
=
10 ~
P r p Ut
1.e.:
(3.164) Now, since:
u
A 2dA _ 1 d A 3

dx
3 dx
Eq. (3.164) can be written as:
i,
~xdA3
3
= 13 ~_
A3
dx
14 P r
(3.165)
This equation can t hen.hedirectly integrated to give: 13 A3 _ K .75 14Pr  xO w here K is a constant o f integration. Now since: when x = Xo, A = 0 this constant, K, is given by: (3.167) (3.166)
K
Eq. (3.166) therefore gives:
=
13 x 14Pr
°
0.75
(3.168)
A3
=
~ [1 _(XO )0.75]
1 4Pr x
(3.169)
CHAPTER
3: Some Solutions for External Laminar Forced Convection
123
This shows that for large x, Ll tends to a constant, this constant being of course, the same as that previously derived for flow over an isothermal plate. Substituting Eq. (3.169) into Eq. (3.160) then gives: Nux
~ O.352Re,/zPr 1l3 [1  (~)
075]ID
(3.170)
The integral equation method can also easily be extended to situations that involve a varying freestream velocity.
3.6
NUMERICAL S OLUTION O F T HE L AMINAR BOUNDARY LAYER EQUATIONS
Similarity and integral equation methods for solving the boundary layer equ~tions have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range o f problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors o f uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense o f a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. Another way o f solving the boundary layer equations involves approximating the governing partial differential equations by algebraic finitedifference equations [11]. The main advantages of this type of solution procedure are: • T he method can be applied to problems involving arbitrary surface thermal conditions and arbitrary freestream velocity and is easily extended to cover the effects o f variable fluid properties and dissipation effects. • The errors involved in the procedure are purely numerical and their magnitude can be estimated and can, in general, always be reduced to an acceptable level by reducing the numerical step size. The disadvantages o f finitedifference and other numerical methods are that quite a considerable amount o f computational effort is usually required in order to obtain the solution and that they do not, in general, reveal certain unifying features of the solutions, such as the fact that the profiles are similar under certain conditions. The widespre~d availability o f modern computer facilities has, however, made these disadvantages relatively unimportant. There are a number o f schemes for numerically approximating the boundary layer equations and many different solution procedures based on these various schemes have been developed. I n the ipresent section, one of the simpler finitedifference schemes will be described. The solution procedure based on this scheme should give quite acceptable results for\most problems. The scheme is .easily extended to deal with turbulent flows as will b e discussed in Chapter 6. Before outlining
124 Introduction to Convective Heat Transfer Analysis
this solution procedure, some general points concerning the numerical solution o f t he boundary layer equations will b e discus~ed. I n order to solve the boundary layer equations b y means o f finite difference approximations, a series o f nodal points is introduced. The values o f t he varjables are then only determined a t these nodal points and not continuously across the whole flow field as is the case with a n analytical m~thod. I n order to describe the position o f ~ ~ / t he nodal points, a series o f g rid lines running parallel to the two coordinate directions as shown in Fig. 3.16 is introduced, the nodal points lying at the intersection of these grid lines. By introducing finite difference approximations to the derivative terms in the  boundary layer equations, a set o f algebraic equations can b e obtained. Beca~se o f t he socalled parabolic nature o f t he governing equations, this set o f equations i s o f such a form that i f the values o f t he variables are known at the nodal points on one ygrid line, then the variables a t the nodal points on the next ygrid line c an b e found. Having determined the conditions on this grid line, the same procedure can b e used to determine the conditions at the points on the next yline and so on, the solution advancing from grid line to grid line i n t he xdirection as shown in Fig. 3.17. ., T he finite difference approximations can either be applied to the derivatives on the line from which the solution is advancing or on the line to which i t is advanc: ing, the former giving an explicit finite difference scheme and the latter an implicit scheme. The type o f solution procedure obtained with the two schemes is illustrated i n Fig; 3.18. In the explicit scheme, the values o f the variables at the point under consideration are directly determined from conditions at the points on the preceding line. I n the implicit scheme, the values o f the variables at adjacent points on the line on
y  Grid Lines
F IGURE 3.16
Grid lines and nodal points.
S olution Advances I n This W ay Values Known O n This L ine
F IGURE 3.17
Forward "marching" solution procedure used.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
125
.... , ..
~.....
... '"
.... ,......
/
...... ..... ~
...... ",~"'"
Q) Solution :E ~ At This f' ~ Point 4'~I· g Determined d .. :.::l From ~  ... ' ~ Conditons ~ " On ~ 8 Preceding .g  iIIJ<.. §
_ 11. &
1l
8
i
Line]
U
EXPLICIT IMPLICIT
~
§
U
F IGURE 3.18 Implicit and explicit solution schemes.
which the solution is being sought are r elated to each other and to the values o f the variables o n the preceding line. B y considering each nodal point i n turn 'on the line to which the solution is advancing, a set o f equations is obtained which must b e simultaneously solved to give the values at all the nodal points. I t is possible to obtain such a solution because the values o f the variables a t t he point lying on the wall and at the outermost grid point, which is always selected to lie outside the boundary layer, are given b y the boundary conditions. Explicit finite difference schemes lead to comparatively simple solution procedures which are usually very easy to implement. However, with an explicit scheme, i t is possible for the solution to become unstable, i.e., for small errors such as those resulting from numerical roundoff to become magnified as the solution progresses leading to a useless solution. In order to avoid instability, there is a maximum numerical step size t hat c an b e used in the xdirection for any given minimum ystep size. Since, particularly w hen dealing with turbulent flows, i t is necessary to use a smallystep size near the wall in order to obtain an accurate solution, small step sizes will have to be used in the x direction in order to avoid instability which will mean that long computer t imes w ill be required. Solution procedures based on implicit finite difference schemes are_usually somewhat more complex than those based on explicit schemes. However, they suffer from no inherent stability problems and consequently much larger xwise steps can b e u sed and the computing time required is usually ~uch less than with an explicit scheme. In the present section, therefore, a  simple implicit finite difference scheme will be described. F or the purpose o f illustrating the ideas involved, attention will initially b e restricted to twodiniensional constant fluid property flow over a flat surface and the dissipation term in the energy equation will b e neglected. T he equations to b~ solved are, therefore, a s before:
au + av
=
0
ax ay , aT . aT , 1 ' dp a2u u +v =  ++vax a y p dx ay2 2T aT a T!' V a
(3.171) (3.172) (3.173)
u +vax
a y.
= 
P r ay2
126· Introduction to Convective Heat Transfer Analysis
The boundary conditions on these equ~tions are, o f course,
y '= 0: u y large: u
= v =\ O ,T
~ U I (x),
= Tw(x)
~
(3.174)
T
Tl
Blowing or sucking at the wall ~hich gives a nonzero value o f v at y equal to zero is easily incorporated into the solution scheme, but will not b e considered here. Also, i n some cases, the heat flux distribution at the wall rather than the wall temperature distribution is known and this is also easily incorporated into the solution scheme b ut will, for the moment, not b e considered. Although not necessary, i t is convenient, iJ;l most cases, to write these equ.ations i n dimensionless form before deriving the finite difference approximations to \them. F or the case o f l aminar flow here being considered, the following dimensionless variables are convenient to use:
u = u/uo
X = x /L
P = ( p  Pl)/pu5
v
Y.
= v JRerJuo
= y JRerJL
(3.175)
o=
( T  TI)/(TwR  TI)
where Uo is some convenient reference velocity, L is some convenient reference length, and T wR some convenient reference wall temperature. ReL i s the Reynolds number based on Uo a nd L, i.e., ( uoUv) and Tl is, as before, the freestream temperature. In terms o f these variables, Eqs. (3.171) to (3.173) become:
au + av = 0 ax ay au au dP a2u u ax + v ay =  dX + ay2 \ ao ao 1 a2 0 u ax + v ay = P r ay2
a nd the boundary conditions given in Eq. (3.174) become:
(3.176)
(3.177) (3.178)
y = 0: U y large: U
V = 0, 0 ~ UI (X), 0
=
= Ow(X)
~
0
(3.179)
I f t he waIl temperature is unifonn, it will usually be convenient to set T wR equal to this uniform waIl temperature and Ow will then be equal to 1. In order to express the above set o f equations in finite difference form, the grid lines are labeled as shown in Fig. 3.19! i lines running in the Ydirection nonnal to t he surface and j lines running in the X direction parallel to the surface. Although not necessary, a uniform grid spacing, d Y in the Ydirection, will b e used here for simplicity. The conditions at nodal point lying at the intersection o f the i a nd j grid lines are denoted b y the subscript i, j . I n order to derive the finite difference equations, attention is given to conditions at the four grid points shown in Fig. 3.20. Consider first the finite difference approximation for au/ay at the point i, j . Iri order to derive this finitedifference approximation it is noted that the values o f U at
CHAPTER
3: Some Solutions for External Laminar Forced Convection
127
i.+++, j + 2 j i2,j + 2 i il,j+ 2 j i,j + 2 ji + l ,j + 2 U + 2 ,j + 2
 ........... J + 1
. .......... J
!
! i j i2,j+l j il,j+l j i,j+l I : !
l il,j
i
:
.
i
j i+l,j+l i i+2,j+l
:
.
....~..... j  1
:
l i2,j
:
!
i i,j
l i+l,j :
:
l i+2,j
..~...,...... j  2
l i2,j 2 lil,j2Ji,j2 l i+l,j2 i i+2,j2 i +2
j i2,jl i il,jl i i,jl
j i+l,jlli+2,jl
I
i 2
!
i I
Iii
i i +l
F IGURE 3.19 i and jgrid lines.
··········i~~j"~i.,· .....
...... _._ _ ._.. ... ! i  1,j
J'
t1X
!
i ,j i
•.
.:
I
t1y
 Ay
....~/  1 • ._ _,
I
!
!
F IGURE 3.20 Grid points used in deriving finitedifference approximations.
points i, j + 1 a nd i, j  1 can be related to the value at point i, j by Taylor expansions as follows, terms o f order A.y3 and higher being ignored:
U i,j+l = U i,j
+
au a y . . A.y
I ,)
I'
I ,)
+
a2 u a y2
2
A.y2 i ,j
2!
(3.180)
.' a UI A V ·I,)' 1 = U,· ,).   y. . uY I a
\
au +  y2 a
i ,j
2!
(3.181)
Subtracting these twoequa~ons and dividing the result by 2 dY gives:
au a y I.,) .
I
U i,j+l  Ui,jl
2A.Y
(3.182)
This i~the required finite'difference approximation for a Ulay a t point i, j . To obtrun t he difference approximation for a2Ulay? a~ point i, j , Eq. (3.1,81) is added to Eq. (3.180) and th~. result i s divided b y d y2. This gives: .,
U i,j+l
+ U i,jl
A.y2
 2l!i,j
(3.183)
I n the solution procedure here being us~d, the Xderivatives are approximated to a lower order in A.X than t he Yderivativesare in dY,·i.e., the following backward difference approximation i ll the Xdirection ~eing used: , \
a UI
a x..
I ,)
= U··..:,.. u · : 1,). I ,)·, 1
li
(3.184) ,
128 Introduction to Convective H eat Transfer Analysis
Now the derivative terms on the le~hand s ide o f the momentum equation have coefficients U and V. To illustrate how these a te dealt with consider u au/ax. Because: (3.185) and since, i n deriving the approximation given i n 134. (3.184), terms involving AX and higher have been neglected, i t is seen· that i t is consistent to a dopt the following approximation:
UiI,j], [u aU,]1 . . = u._ I,}. [Ui,j AX ax
I
(3.186)
I ,}
Similarly, using Eq. (3.182), the following is obtained:
[V aU]1 . . ay
I ,}
=
v._ I,}.[ Ui,j+I2AYUi,jI] , I
(3.187)
I f the finite difference approximations given i n Eqs. (3.183), ~3.186), a ndO.187) a re substituted into the momentum equation, an equation that has the following form is obtained on rearrangement:
AjUi,j + B j Ui,j+l where the coefficients are given by: Aj
=
+ C jUi,jl
=
Dj
(3.188)
( Uil,j) + Ay2 AX
Vil,j) B j = ( AY
(2) (1 )  Ay2·
(3.189) (3.190) (3.191)
: Cj =
r:;~j)(~~)
Dn =
C!1j) ~~ ,
(3.192)
Now the boundary conditions give Ui,l, the value o f U at the wall, and Ui,N the value o f U a t the outermost grid point which, as previously mentioned, is always chosen to l ie outside the boundary layer. Therefore, since dP/dX!; is a known quantity, the application o f Eq. (3.188) to e ach o f the points j = 1, 2, 3, . .. , N  1, N gives a s etofN equations in the N unknown values o f U, i.e., Ul, U2, U3, U4, . .. , UNI. UN. This set of equations has the following form since U i,1 is zero
U;,1 =
0
A2 U;,2 + B2 U;,3 + C2Ui,1 A3 Ui,3 + B3 U;A + C3 Ui,2
D2 = D3
=
(3.193)
A N 1 Ui,Nl + B N 1 Ui,N + CN I Ui,N2 : D N 1 Ui,N = U I
C HAPTER
3: Some Solutions for External Laminar Forced Convection
129
T he set o f equations thus has the fonn:
1 C2 0 0
0 A2 C3 0
0
B2 A3
0 0
B3 A4
0 0 0
B4
C4
0 0
0 0 0 0
0 0 0 0
0 0 0 0
VI'I , V '2 I, V '3 I, V '4 I,
0
D2 D3 D4

0 0
0 0
0 0
0 0
CN I 0
A N I
0
B N I 1
V i,NI V 'N I,
D NI VI
i.e., has the fonn:
Q UUi,j
= Ru
(3.194)
where Qu is a tridiagonal matrix. This equation c an b e solved using the standard tridiagonal matrix solver algorithm which is often t enned the Thomas algorithm. Consider next the energy equation. T he t enns in this equation have the same f onn as those i n t he momentum equation and they are, therefore, approximated in the same way. The following finite difference approximations are therefore used:
80 ( V 8X
)1 .. = '~,_
,(Oi,j  OiI,j)
"II,} l ,j
I lX
(3.195)
 Oi,jd (V 8 Y " = V,_ t ,),(Oi,j+1 1lY l 2 l ,}
80)1
(3.196)
(3.197) Substituting these into the energy equation then gives, on rearrangement an equation that has t he fonn:
E jOi,j
+ F jOi,j+l + G jOi,jl
, (Vil
= Hj
(3.198)
w here the coefficients ~this equation are given by:
Ej =
2 Ai j) + ( rlly2 ) P
(3.199)
Fj
1 =....,. ( Vil,j)  ( Prlly2 ) I lY
(3.200) (3.201)
(3.202)
/
;Because t he boundary conditio~sgive\ 0 0 a nd Oi,N, the o utepnost g rid p oint' being chosen to lie outside both the velocity a nd temperatUre boundary layers,
130 Introduction to Convective Heat Transfer Analysis
/
the application o f Eq. (3.188) to each o f the internal points on the iline, i.e~, j ; ; 1, 2, 3 ,4, . .. , N  2, N  1, N, again gives a set o f N equations i n the N unknown values of~. Because (Ji,N i s zero, this'setiof equations has the followingform:
.
,
= 8w E 2 8 i,2 + F 2 8 i,3 + G 28i,1 E 3 8 i,3 + F 3 8 i,4 + G 3 8 i,2
8 i,1
E N1 8 i,Nl
= H2 = H3
(3.203)
= H N l
+ F N1 8 i,N + G N  18 i,N2
8 i,N =
0
. This set o f equations qas the Same form as that derived from the momentum equation, i.e., has the form:
,
1
0
E2
0
O2
0 0 0 0
F2
E3
0 0
F3 E4
G3
0 0 0
F4
0 0 0
G4
0 0 0 0
0
0 0 0
0 0 0 0
8 '1 J, 8 '2 r, 8 '3 r, 8 '4 I,
8 w· Hz
H3.
H;"
0 0
0 0
0 0
G Nl
E Nl
F Nl
0
0
1
8 i,Nl 8 'N I,
H Nl
0
i.e., which has the form:
QT 8 i,j = R T
I
(3.204)
where 'QT i s again a tridiagonal matrix. Thus, the same form o f equation is obtained as that obtained from the momentum equation. This equation can also b e solved using the s tandard tridiagonal m~trix solver algorithm. To deal with tl),e,continuity equation, the grid points shown in Fig. 3.21 are used. The continUIty equation is applied to the point (not a nodal point) denoted by ( i, j  112) which lies on the iline, halfway between the ( j  1) and thejpoints.
aX
. +:. . . ._.;. . . . .. . .~::. .
I, .'
!
i
, I I,}
! I ,J
i ,!
!
i
I
I
i
! i
1 i,j1I2
ay  .i  !
...•
L 1
i
! .........._._...__............
i i ,jl
i
F IGURE 3.21
Nodal points used in finitedifference solution of the contiIiuity equation,
" i l,jl .
CHAPTER
3: Some Solutions for External Laminar Forced Convection
131
Now, to the same order of accuracy as previously used:
V i,j = Vi,jl12
+
a VI dY ay . . T l,jl12
+ +
a2v ay2 a2y ay2
i,j1I2
(3.205)
a VI dY V i,jl = Vi,j1I2  ay . . T l,jI12 i,j1I2
Subtracting these two equations gives:
~~ 1"jlf2 
y..y.. I I ,j I ,j
dY
(3.206)
It is alSo. assumed that the Xderivative at the po.int (i, j  112) is equal to the average of the Xderivatives at the po.ints ( i,j) and (i, j  1), i.e., it is assumed that:
~n,jlf2 ~ Hn.j + ~~I"J ~
=
(3.,207)
The finitedifference approximatio.n fo.r a u/ax given in Eq.(3.184) is used to determine the righthand side o.f this equatio.n. Therefo.re, since the co.ntinuity equatio.n can be written as:
av ay au ax
(3.208)
the fo.llo.wing finite1.lifference approximatio.n to. this co.ntinuity equation is o.btained:
V',j ~~,jl
/
~ _~ [(U"j ~i1,j)+ (U"jl ~'1:jl)l(3'209)
This equatio.n can be rearranged to. give
V i,j = Y i,jl  ( 2ll.X ( Uu  Uil,j
dY)
, , + U i,jl  Uhl,jl)
(~.210)
Therefo.re, i f the distributio.n o.f U across the iline is first determined, this equatio.n can be used to. give the distributio.n o.f V across this line. This is possible because Vi,l is given by the bo.undary conditions and the equation can, therefore, be applied progressively o.utward acto.ss the iline starting at point j = 2 and then go.ing to point , / j = 3 and so. o.n across the line. Thus, to. summarize, ,the co.mputational proced,ure based on the above finite difference equatio.n~ inyol\Te~ the fo.llo.wing steps. ,"
",
.
1. The co.nditio.ns along So.meinitial i = 1 line must b e sp~cified. 2. Eq. (3.194) can then be used to. find U at all thevno.dal pOints;ontbe i ~ 2 line. 3. Eq. (3.204)ocan then b e used to. f indlJat all the nodal points::o.n,the' i ' = 2 line.
,
'
,i'
"
132
/
Introduction t o Convective Heat Transfer Analysis
4. Starting at the first point away from \the wall, i.e., j = 2 a nd working progressively outward, Eq. (3.210) can b e used to determine the distribution o f V on the i = 2 line. '. '5. Having in this way determined the values o f all the variables on this i = 2 line, the same procedure can then b e u sed to find the values on the i = 3 line and so on. I t should be noted that at any stage o f the procedure, i t is only necessary to deal with the values o f the variables on two adjacent grid lines. Therefoie~_1t i s only necessary to allocate storage spaces for the values on the line on which conditions are . known, i.e., the J i  I)line, and the line on which conditions are being ca\lculated, I i.e., the iline. A s soon as the values on the iline have been calculated, they can be transferred into the storage spaces previously holding the values on the ( i  1)line. The storage spaces that held the values on the iline are then used to store the values on the ( i + I)line as they are calculated and so on. I n the above procedure, i twas assumed that the 'outermost grid point, i.e., the Npoint, was always outside both the velocity and thermal boundary layers. One way o f ensuring this is the case is, o f course, to simply estimate t he m aximum boundary layer thickness expected and then select the number o f grid points and their positioning such that the outermost grid line is at a greater distance from the surface than this maximum boundary layer thickness. This is illustrated i n Fig. 3.22. There are two disadvantages to, this approach. Firstly, it may not be possible to estimate the boundary layer thickness with any degree o f certainty and i t may then b e found during the calculation that the boundary layer grows out to the Nline and the whole calculation has then to b e started again. Secondly, i f t he boundary layer grows at all appreciably during the calculation, a considerable number o f grid points lie outside the boundary layer during the initial part o f the calculation and computer time is wastedin carrying out the calculation at these points since they will lie i n the freestream and the values q f the variables are actually known at these points. O ne w ay o f overcoming these disadvantages is to start with a small number o f jgridlines and monitor the boundary layer growth. Then, when the boundary layer has reached to within a few nodal points o f the outermost point, the number o f j lines is increased. Sjnce the additional points so generated initially lie outside the boundary layer t he ; 'alues o f t he variables on these points are initially known. The procedure i s illustrated in Fig. 3.23.
"Edge" o f Boundary Layer
NLine
Flow
...
F IGURE 3.22 Choosing outermost grid line to be always outside qoundary layer.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
" Edge" o f Boundary Layer
133
F low
...
N umber of Grid Lines
FIGURE 3.23
Adding points to keep outermost grid point outside the boundary layer.
Additional savings in computer time can b e achieved if, instead o f increasing the number o f g rid lines, their spacing is incre~sed a nd i n this way the outermost grid point is moved further from the surface. I n t he computer program discussed later i n this section this has, however, not been done because o f the added complication involved, the number o f grid points simply being increased. O nce t he distributions o f U, V, and 0 have been determined using the procedure outlined above, any other property of the flow c an b e determined. I n the present discussion, t he h eat transfer rate a tthe wall, qw, is the most important such property. This is, o f course, given b y Fourier's law as:
qw
=
k
aTI
ay y=o
(3.211)
In terms of t he dimensionless variables being used i n t he present w ork this gives:
qli>,L  JReL (TwR  T})k a y y=o
aOI
(3.212)
This c an be rearranged as: (3.213) _ w here Nux i s the local Nusselt number and Ow i s the local dimensionless wall temperatU,fe. :Now since point j = l lies on the wall:
a y y=o =
aOI' . · (ao)
ay
i ,l
(3.214)
I n order to determine this from the values o f 0 calculated a t the nodal points, i t i s noted that to t he s ame approximation as previqusly used: .
( h2 = O i,l .
+ (;~). .
. 41
,~Y+(:~ l,l t.;:
\
(3.215) .
134 Introduction to Convective Heat Transfer Analysis
B ut t he application o f t he boundary layer energy equation to conditions a t the wall gives
:~ y=o = (:~)
Therefore Eq. (3.215) c an be rearranged to give
i,1
=0
(3.216)
ay
and Eq. (3.213) then gives
aOI
i,1
=
(0;,2  0;,1)
aY
(3.217)
~=
i, e1;/i,2 )(f:)
(3.218)
Ow being, o f course, the same as Oi,l. Some consideration must b e given to the conditions exi&ting along the initial i = 1 line which were assumed to b e known i n the above' discussion. T he actual conditions will depend o n t he nature o f the problem. For flow over a flat plate, because the boundary layer equations are parabolic in form, the use o f these equations requires that the plate have no effect on the flow upstream o f the plate. Hence, i n this case, the variables will have their freestream values at all the nodal points (except a t t he point which lies on the surface) on the initial line which 'is coincident w itli the leading edge. A t t he nodal point on the surface, the known conditions at the surface must apply. This is illustrated i n Fig. 3.24. The reference velocity is taken as the freestream velocity U I, which/is, o f €ourse, constant for flow over a flat plate. Hence, the conditions on this initial line are as follows: Y
= 0,
i.e., j
=
1: UI ,1
= 0,
01,1
1,
=
Ow(O)'
0
(3.219)
Y > 0, i.e., j > 1:
U I,j =
O l,j =
The values o f V at the points On this initial line must also b e specified. I t is usually adequate to P~L
/
F or all Y, i.e., all j :
u=
Ul
V I,j =
0
(3.220)
T =T1 At These _ _J,.
FIGURE 3.24
Conditions on first grid line.
C HAPTER
3: Some Solutions for External Laminar Forced Convection
135
This is actually considerably in error because V, in fact, has its maximum value at the leading edge. However, the effects o f this erroneous assumption quickly die out and i f the initial spacing of the ilines is chosen to be small, it has a negligible overall effect on the solution. These initial conditions are incorporated into the computer program discussed below. A computer program, LAMBOUN, written in FORTRAN that is based on the above procedure is available as discussed in the Preface. The program, as available, will calculate flow over a plate with a varying freestream velocity and varying surface temperature. These variations are both assumed to be described by a thirdorder polynomial, i.e., by:
U1 = A v
+ B vX + CvX2 + D v X3
and:
Ow
=
AT + B TX + CTX 2 + DTX 3
I t follows from the assumed form of the freestream velocity distribution that since:

dp
dx
=  P U1 
dUl
dx
that:
dP d Ul d x =  U1 d X = U1(Bu
+ 2 CuX + 3 DuX
2
)
E XAMPLE 3 .6. Numerically detennine the heat transfer rate variation with twodimensional laminar boundary layer air flow over an isothermal flat plate. Compare " the numerical reslllts with those given b y the similarity solution.
Solution. B ecause flow over a ,flat plate with an isothermal s wface is being considered,
the freestream velocity and surface temperature are constant. Therefore, the inputs to the program are:
Xmax = 1, P r = 0.7, AT, B T, CT, DT = 1, 0 ,0,0
A v , Bv, Cv, D v = 1 ,0,0,0
T he program, when r un with these inputs, gives the values o f NuxIRe~.5 a t various X values along t he plate. These are shown i n Fig. E3.6. F or a P randtl number o f 0.7, t he similarity solution for flow over an isothermal flat plate gives:
N ux = constant = 0.293 J Re x
This i s also shown i n Fig. E3.6. I t will b e s een that while the numerical results are i n excellent agreement with the similarity solution a t larger X values,there are considerable ,differences between t he t wo solutions a t ,small X. These differences could have been reduced by using spuiller AX a nd AY val~s~
\
E XAMPLE 3 .7.
A ir flows a t a velocity o f 4!,mls o ver a wide flat plate that has 'a length o f
2 0 c m in t he flow direction. The air ahead o f the plate has a temperature o f 20°C while
136 Introductionto Convective Heat Transfer Analysis
0.6 . r, o Numerical Similarity Solution
0.0 L _ _ _ _. L..._ _ _    ' 0 .0 0.5 1.0
X
F IGUREE3.6
the surface temperature o f the plate is given by:
x < l Ocm: Tw
=
40°C
l Ocm < x < 2 0cm: Tw = 80°C B y numerically solving the twodimensional laminar boundary layer equations, determine how the local heat transfer rate i n W1m2 v.JUies along the plate.
Solution. T he mean plate temperature is (40 + 80)/2
o f the a ir in the boundary layer is therefore:
=
60°C. The mean temperature
Tmean
=
Twmean + Tl = 6 0 + 20 = 400C 2 2
A t this temperature for air:
k = 0.0271W/moC, v = 0.0000170 m 2 /s
Hence, since here:
Ul
=
4 m!s
it follows that the Reynolds number based on the length of the plate, i.e., 0.2 m, is:
=
4 xO.2 0.0000170 = 47,059
The boundary layer on the plate will, therefore, remain laminar (see the discussion o f the transition Reynolds number given later). The plate surface temperature variation is such that: x < 10 cm: Tw  Tl = 20°C l Ocm < x < 2 0cm: Tw  Tl = 60°C Hence, using the plftte length, 0.2 m, as the reference length, L, and the value o f the temperature difference Tw  TI on the first 10 cm o f the plate as the reference temper"'ature difference T wR  Tl, and recalling that the dimensionless temperature used in the computer program is:
C HAPTER
3: Some Solutions for External Laminar Forced Convection
137
it follows that the surface temperature difference is such that: X < 0.5: Ow
=
I
0.5 < X < 1.0: Ow = 3 A simple although not very elegant way o f modifying the available computer program L AMBOUN to deal with the situation being considered in this example is to delete the lines:
WRITE(6,6002) READ(S,*) AT,BT,CT,DT
and to change the subroutine T EMP to the following:
SUBROUTINE TEMP(X,TW) COMMON AT,BT,CT,DT,AV,BV,CV,DV
*
*********** THIS DETERMINES THE WALL
T~MPERATURE
*******************
*
*
I F (X.LT.O.S) THEN TW = 1 .0 ELSE TW = 3 .0 ENDIF
RETURN
END
RuIliling the program with these modifications gives the variation o f N uxlRe~·5 along the plate. T he local heat transfer rate can then b e calculated from this variation by recalling that: . lAT Vc VU _ qwx v )0.5 ( J Re x  (Tw  Tl)k UIX where T w is the local wall temperature. This can be rearranged to give:
qw
=
Nux (Tw _ J Re x
Tl)k(~)0.5
vx
Le.:
Le.:
But:
(Tw  T 1) = Ow(TwR  TI)
Therefore, because TwR  Tl was chosen to b e 4 0  20 = 20°C, i t follows that: ,
138 Introduction to Conv~ctive Heat Transfer Analysis
4000 . ..,...,.,
•
N
~
I
2000
J
• •
\ ....... ..
~.... .....
0.2
0.1
x m
FIGUREE3.7
The program with these modifications when run with the following inputs: Xmax = 1, P r = 0.7, A v, Bv, Cv, D v = 1 ,0,0,0 gives the values o f Nux/Re~·5 at various X values along the plate. Since the dimensionless plate temperature, Ow, is known at these points, the above equation then .allows qw to be calculated at these X values. Results are shown in Fig. E3.7. Only some of the calculated points are shown in this figure.
In the above discussion, i t w as assumed that the surface temperature variation o f t he p late was specified. T he p rocedure i s e asily extended to deal w ith o ther thermal boundary conditions at t he surface. F or e xample, i f t he h eat flux distribution at the surface is specified, i t is convenient to define the following dimensionless temperature:
(3.221)
w here q wR i s s ome convenient reference wall heat flux. T he e nergy equation in terms o f this dimensionless temperature has the same form a s t hat obtained w ith t he d imensionless temperature used w hen t he s urface temperature i s specified, Le., the dimensionless energy equation i n this c ase is:
ao* ao* u ax + v ay
I = Pr
a2 0* ay2
(3.222)
Using F ourier's law, the boundary condition o n t emperature a t t he w all in the specified h eat flux case is:
(3.223) I.e.,
Y = 0:
ao* ay
(3.224)
CHAPTER
3: Some Solutions for External Laminar Forced Convection
139
I n order to express this i n finite difference f onn it is recalled that i f t he wall point and the point adjacent to the wall on a given iline are considered, then a Taylor expansion gives to order o f accuracy ~y2:
(J *
' I, 2=
(J*
I' 1 + ,
a(J*
ay
2!
(3.225)
But since U andV are zero at the wall, Eq. (3.222) gives when applied at the wall:
a 2 (J*
af'Z
=0
i,1
(3.226)
Substituting this into Eq. (3.225) then gives the following finitedifference approximation:
a(J*
ay
=
i,1
(J~2  (J~1 I, I,
dY
(3.227)
Substituting this result into Eq. (3.224) then gives:
(J*
 =,""'::':"::;'
i,2 
(J* i,1
~Y
= QR
(3.228)
l.e.: (3.229) T his b oundary condition is easily incorporated into the solution procedure that was outlined above, the set o f equations governing the dimensionless temperature in this case having t he f onn:
+ Q RdY E2(J\i,2 + F2(Ji,3 + G2(Ji,1 = E3(Ji,3 + B3(Ji,4 + C3(Ji,2 =
. (Ji,1 = (Ji,2 E N"':'l(JiNl ,
H2 D3
(3.230)
= H Nl .
+ F Nl(JiN + G Nl(JiN2 , ,
(Ji,N =
0
the coefficients h aving the same values as previou~ly defined. T he m atrix equation that gives the dimensionless temperature therefore has the form.
I G2 0 0
I
E2
0
F2 E3
0 0
F3 E4
G3
0
0 0 0
F4
G4
0 0 0 0
0 0 0 0
0 0 0
9~1 I,
(J~2 I, (J~, 3 I
(J~4
Qr~Y
H2 H3 H4
0
FN1..f'· 1
"

.
6tN_'l
~i.N
' HNl
0 0
0 0
0 Q.
0 0
0 0
G N'l\
E Nl
,0
0
0
Thus, as before, a tridiagonal matrix i s obt3.ined.
140
Introductio~
to Convective Heat Transfer Analysis
. . T he program discussed above is therefpre e asily modified to deal with the specified wall heat flux c ase a nd a p rogram, L AMBOUQ, for this situation c an b e obtained in the way indicated i n t he Preface.
3.7
VISCOUS DISSIPATION E FFECTS O N L AMINAR BOUNDARY LAYER F LOW O VER A FLAT PLATE
T he effects o f viscous dissipation on the temperature field have b een i gnored i n t he discussion up to this point i n t he chapter. However, viscous dissipation effects c an b e important particularly i f t he viscosity o f t he fluid is high o r i f t he velocities are high. A n e xample o f t he first case would b e the flow o f t he lubricant through a journal bearing. In this case the lubricant temperature rise caused b y viscous dissipation can be quite high. T he second case usually involves the flo~ o f gases a t r elatively high Mach numbers a nd i t is to this case that att~ntion will b e g iven i n t he present section. In t he present section, then, a n analysis o f t he effects o f dissipation on the laminar boundary layer flow over a flat plate which is aligned with the flow will b e p resented [9],[12],[13],[14],[15],[16]. T he flow situation considered is, therefore, as shown i n Fig. 3.25. T he effects o f fluid property variations will again b e ignored. I t should, however, be noted that i f viscous dissipation effects are important, fluid property changes are usually quite large. These changes c an u sually b e a dequately accounted for b y evaluating the fluid properties at some suitable mean temperature a nd t hen treating them as constant in the analysis o f t he flow as is being done here. This will b e d iscussed in the next section. The equations governing the flow were discussed i n C hapter 2. Because twodimensional flow"over a flat plate is being considered, these equations are:
ax
au + av
ay
=
0
(3.231) (3.232)
(3.233) The last term i n t he energy equation, i.e., i n Eq. (3.233), is the viscous dissipation term.
Laminar
Plate at Temperature T w o r Plate Adiabatic
F IGURE 3.25 Laminar boundary layer flow with viscous dissipation.
CHAPTER
3: Some Solutions for External Laminar Forced Convection
141
T he boundary conditions on the velocity distribution are: At Y = 0: For large y:
U =v=O
U
~
(3.234)
U t(x)
Two possible boundary conditions on temperature at the wall will b e considered i n the present section, i.e., it will be assumed that either: At Y = 0: o r that: At Y = 0:
aT
ay
T
=0
(3.235)
=
Tw
(3.236)
where Tw is the temperature o f the wall which is assumed constant. The first o f these boundary conditions, Le., Eq. (3.235), applies when the wall is adiabatic because Fourier's law gives the heat transfer rate at the wall, i n general, as:
qw
=
k
. aTI
ay
y =o
(3.237)
Therefore, since qw = 0 i f the wall is adiabatic, the gradient o f temperature at the wall must be zero i f the wall is adiabatic. Outside t he boundary layer, the boundary condition on temperature is: For large y: (3.238)
Dissipation has no effect on the continuity and momentum equations because the fluid properties are being assumed constant. As a result, Eqs. (3.231) and (3.232) are t he same as those considered earlier in this chapter i n Section 3.2. As discussed i n t hat section, a similarity solution to these equations can be obtained. To do this, the following " similarity" varia~le was introduced:
y ;:;:11 = ....;rR ex = y
x
ffxl

vx
(3.239)
I n tenns o f this variable; the distribution o f U i n the boundary layer is assumed to be:
> //
U =
Ul
f ' (11)
(3.240)
usilng the continuity equation, i t was then shown that:
v
Ul
,
=
2"
1~ I XU;(11f  f)
(3.241)
T he m omentum equation could then be written i n terms o f 'YJ as:
2 f ill
+f
f" = 0
(3.242)
T he primes, o f course, denote differentiation with respect to 'YJ. I n terms o f t he similarity function, ~e boundary conditions o n t he solution become: 'YJ = 0, I ' = 0 'YJ = 0, f = q
<
large,
f'
~
1
142 Introduction to Convective Heat Transfer Analysis
T he solution o f E q. (3.242) subject to t hese b oundary conditions w as d iscussed in Section 3 .2 a nd g iven i n g raphical f onn ip. Figs. 3.4 and 3.5. This solution, it must b e stressed, also applies w hen viscous dissIpation is included. It was also shown in Section 3.2 that a similarity solution could be obtained for the temperature distribution for the c ase w here the wall t emperature is constant. To do this, a dimensionless temperature defined as follows was introdpced: (3.243) a nd it was assumed that 4> d epended only on 1]. I n t enns o f this variable, the energy equation without viscous dissipation became:
4>" + P r cfJ'f
2
=0
(3.244)
W hile the boundary conditions on the solution c an b e w ritten as:
= 0: ' 4> = 0 1] large: 4> ~ 1
1]
Thus, as was the case with the velocity distribution, the partial differential equation governing the temperature distribution was shown to reduce to a n ordinary differential equation. I t s eems reasonable to assume that similar temperature profiles will also exist when viscous dissipation is important. Attention will first b e g iven to the adiabatic wall case. I f the wall i s adiabatic and viscous dissipation is neglected, t hen t he solution to t he energy equation will b e T = T 1 e verywhere i n t he flow. H owever, when viscous dissipation effects are important, the work done b y t he viscous forces leads to a rise in fluid temperature i n t he fluid. This temperature will b e r elated to the kinetic energy o f t he fluid i n th~ freestream flow, i.e., will b e r elated to I2cp . F or this reason, the similarity profiles in the adiabatic wall case when viscous dissipation is important are assumed to h ave t he fonn:
ur
T  Tl 2/2 = (Ja(1])
ul
cp
(3.246)
Substituting this and the velocity profile results into the energy equation, i.e., Eq. (3.233), then gives:
u lf'()' a1J a ax
I.e.:
+
Ul
2
J
v ( l' XUl 1]

f)(J' a1J = ~(J" a1J2 a ay pCp a a y
+ ~(f")2
pCp
(a"fJ)2 u il2c p a y
ur
_f'(J~y [ iil + !
2x
V X;
2
J
v ( 1]f' XUl
f)(J~ { ii;
V;v
=
~(J~ ~ ~ + ~(f")22cp [~]
pCp
xv
Ul
pCp
Ul
xv
Le.:
C HAPTER
3: Some Solutions for External Laminar Forced Convection
143
I.e.:
(J~ + ~ f(J~ + 2 P r(f")2 =
0
(3.247)
This is an ordinary differential equation which shows that a similarity solution does, indeed, exist. This equation contains the Prandtl number as a parameter, i.e., the variation o f (J a with 17 depends on Pro T he boundary conditions on (J a are: 17 = 0: 17 large:
(Jf =
a
0
T he solution o fEq. (3.247) subject to these boundary conditions can b e obtained b y numerical integration or an analytical solution can b e obtained by introducing an integrating factor. T he latter procedure leads to the following equation for the value o f (J a a t the wall: '
iTJ exp(~ iTJ f
(J (0) =
a
0
r
d17)2 p r(f")2 d17
(3.248)
exp(~ fa" f d1J )d1J
0
This equation can be integrated numerically to give the value o f (J a(O) for any chosen value o f the Pr~dtl number. Some typical results are shown in Fig. 3.26. It will b e seen from the results given in Fig. 3.26 that for Prandtl numbers between approximately 0.5 and 10, the variation o f (Ja(O) with P r is approximately described by:
(J a ( 0) = P r1l2
(3.249)
10r~~~
  CalculiKed
8
•  _ . P rO5
6
F IGURE 3.26 Variation of dimensionless adiabatic
\
JOO
PI'andtl Number  Pr
w all temperature with Prandtl
number.
./
144 Introduction'to Convective Heat Transfer.AJialysis
Considering the definition o f (}a a s given i n Eq. (3.246), i t will b e s een that:
8a (0) =
Tw~ 
Tl u /2cp
(3.250)
where T Wad is the adiabatic walltemperature. This equation gives:
T Wad
=
Tl
+ 8a (0)uIl2cp
(3.251)
Now for a perfect gas:
R
= cp

Cv
= cp (1
 1/1')
where I ' = CpICv is the specific heat ratio o f the gas involved and R i s the b s c on· stant. From this i t follows that:
yR cp =  1'1
Hence: (3.252) But:
a
=
J YRT
where a is the speed o f sound i n t he gas. Therefore Eq. (3.252) gives:
    11
u I _ y 1 TM2 2cp 2
(3.253)
where the M ach l!!Ul1ber, M , i s defined by:
M=!!.=
a
Therefore, Eq. (3.251) gives:
T ;;
u J yRT
(3.254)
= 1 + 8a (0) y ~ 1M r
(3.255)
It is conventional to write this equation for the adiabatic wall temperature as:
Tl T
where:
W ad
=
1+r
(y  1 )Mr 2
(3.2561 "
(3.257)
r being termed the recovery factor. The results given above therefore show that for Prandtl numbers between approximately 0.5 and 10, r is approximately given by:
r = P r 1l2
(
(3.258)
This result agrees very well with experimental results for air las shown in Fig. 3.27.
I
C HAPTER
3: Some Solutions for External Laminar Forced Convection
145
1.0 . .__~rr_,
:~:;
R ange o f Experimental
r
0.9
Data
r = {Pr = 1(0 = 0 .84
0.8
'_'_~_ _ '_.....l
1
2
3
M
4
5
F IGURE 3.27 TYpical effect o f Mach number on the recovery factor for a laminar boundary layer.
E X A M PL E 3 .8.
Consider the flow o f air which has a freestream temperature o f O°C over an adiabatic flat plate a s shown in Fig. E3.8. I f the flow in the boundary layer can be assumed to b e laminar, determine how the temperature <?f the plate surface varies with Mach number.
Solution. S ince t he boundary layer flow is laminar and since it can be assumed that for air Pr = 0.7, i t follows that here:
r = 0.7 112 = 0.837
I n this case then:
T;~
T Wad
=
1 + 0. 167Mr
B ut here, Tl = O°C = 273 K so:
= 273(1
+ 0.167Mr)
U sing this equation, the following are obtained:
M
0
Tw,,/Tt
Tw""K
TWad C
1
2 3 4
1.000 1.167 1.669 2.506 3.677
273 319 456 684 1004
0 46 183 441 731
I t w ill b e seen, therefore,that high surface temperatures can exist at Mach numbers above about 2 . These surface temperatures may, in fact, b e so high that special hightemperature materials &uch a s titanium alloys rather than conventional aluminum/alloys must b e used for structural components i n aircraft designed to fly at high Mach numbers.

M
+
T =O°C
=273 K
Adiabatic Surface
<
F lGUREE3.8
146 Introduction to Convective Heat Transfer Analysis
The above analysis dealt with the effects o f dissipation on the flow over an adiabatic flat plate. Attention is now turned t9 flow over a plate that is kept at a uniform temperature o f Tw. Now, as discussed earlier in this section, when viscous dissipation effeCts are neglected, i f a dimensionless temperature defined as follows is introduced:
cf>
=
Tw  T Tw  Tl
the energy equation without viscous dissipation becomes:
cf>" + Pr cf>' f 2
11 = 0: 11 large:
=0
While the boundary conditions on the solution can be written as:
4> = 0
The variation o f cf> obtained by solving this equation was discussed earlier.in this , chapter. The temperature distribution i n flow over a plate kept at a uniform temperature when there are no dissipation effects and the temperature distribution in flow over an adiabatic plate accounting for dissipation effects have thus now been obtained. These two solutions were denoted by cf> and Oa, respectively. Because the energy equation is linear, it is to b e expected that the solution for flow over a plate kept at a uniform temperature when dissipation is accounted for will be some form o f linear combination o f these two solutions [9]. To investigate this possibility, the following dimensionless temperature will be introduced:
0(11) = T  Tl u r/2c p
I
(3.259)
Now it will be noted thdt:
cf>=
I.e.:
Tw

Tl + T l  T T  Tl () =11Tw  Tl Tw  Tl 0(0)
o=
(1  cf> )0(0)
(3.260)
Hence, it seems reasonable to assume that in the case o f flow with viscous dissipation over a plate kept a t a uniform temperature the solution will have the form:
( }T =
Oa
+ Cl(1
 4>Wr(O) + C2
(3.261)
where Or is the actual solution for flow with viscous dissipation over a plate kept at a uniform temperature and where C 1 and C2 are constants. I f the derivation o f Eq. (3.247) is considered, it will be seen that the boundary conditions do not influence the form of the equation obtained. Hence, the temperature profile for flow with viscous dissipation over a plate kept at a uniform temperature must satisfy the following equation:
()~ +
Pr fO~ + 2 Pr(f")2 2
=
0
(3.262)
CHAPTER
3: Some Solutions for External Laminar Forced Convection
147
Therefore, to check that Eq. (3.261) does represent a solution, it is substituted into the lefthand side o f Eq. (3.262) to give:
O~ + C10r(0)cp" +
i.e.,
7f(8~ +
C 10r(0)4>') + 2 p r(f")2
[8~ + ~ f(O~ + 2 pr(f"f] C10r(0) [4>" + ~ f4>']
(3.263)
Now the first term i n this equation is zero b y virtue o f Eq. (3.247) while the second term is zero by virtue o f Eq. (3.244). Hence, Eq. (3.261) is a solution o f the energy equation. The constants Cl a nd C2 are found b y applying the boundary conditions. These give:
TI = 0: Or = 1, O~ = 0, cp = 0 TI large: Or ~ 0, Oa ~ 0, 4> = 1
Hence, applying these boundary conditions i n Eq. (3.261) gives:
(3.264)
8 r(0)
= 8 a(0) + C 18r(0) + C2
0 = 0 + C2
(3.265)
and:
(3.266) These equations give:
C2
= 0,
8 a(O) Cl = 1  8r(0)
(3.267)
Substituting t hese into Eq. (3.261) then gives:
Or
=
Oa + [Or(O)  8 a(0)](1
4»
(3.268)
Now consider t he heat transfer rate from t he surface. It is given by Fourier's law
as:
qw =  k
~TI w y
In term~ o f the variables introduced above this gives:
qw   2cp
i.e.,
 kut [d8 T oTl]1 d'1l o y
y =O
_ ut 1~UI8' q w k    r 2cp ' x v
'71=0
/
which c an b e rearranged to give:
qw~/k _ _ \ u1 8 ' I
~
\2

~
LCp
r.,, 0 
148 Introduction to Convective Heat Transfer Analysis
Using E q.(3.268),this gives: '
qwxl k rn::, v Rex
But:
=
~ c~
u rI[ Ba'I1]=0 + «(JT(O) 
"J.,.' (JiO».,.. I 1 ]=0] ,
(3.269)
and:
u2 C1 (OT(O) Ba(O» 2
p
= (Tw 
Tl)  (Twad  T 1 )
=
Tw  TWqd
,
Hence, Eq. (3.269) becomes:
qw n::rx1k v Rex
= (Tw 
T wad )4> 'I 'IJ=0
(3.270)
But it was shown i n Section 3.1 that 4>'1 17 =0, which is only dependent on the p nmdtl number, is approximately given by:
4>'I'IJ=o = 0 .332Pr1l3
so, Eq. (3.270) gives, using this approximation:
J;/!;
Rex
I.e.:
=
0 .332Pr Il3 (Tw  T Wad)
(3.271) I f, therefo~, a local Nusselt number based on Tw the following is defined: qw x Nux = ( Tw  TWad ) k then Eq. (3.271) ca~be written as: Nux = 0.332Re~2 P r 1l3 (3.272)

T Wad is introduced, i.e., i f
This is identical to the equation that applies when dissipation effects are negligible except that the heat transfer coefficient is now based on the difference between the wall and the adiabatic wall temperatures, i.e., i n gas flows i n which viscous dissipation effects are important, the same results as obtained by neglecting dissipation can be used to find the heat transfetcoefficient provided that the heat transfer coefficient is defined by:
qw
=
h(Tw  T wad )
where h i s the heat transfer coefficient. This equation is really the same as that used when dissipation effects are neglected, i.e.: qw
=
h(Tw  T I )
=
because i f dissipation effects are negligible, TWad
TI •
CHAPTER
3: Some Solutions for External Laminar Forced Convection
149
Equation (3.272) can be integrated as shown in Chapter 2 to give the following expression for t he average Nusselt number for a plate o f length L as:
NUL = O.664ReL 112P r 1l3
(3.273)
where NUL a nd ReL are the mean Nusselt number and the Reynolds number based on L, respectively.
Consider the flow o f air which has a freestream temperature ofO°C over a flat plate that is kept at a temperature o f 30°C. I f the Reynolds number based on the length o f the plate is low enough for the flow in the boundary layer to b e a ssumed to b e laminar and i f the freestream Mach number is 0.9, find whether heat is being transferred to or from the plate.
E XAMPLE 3 .9.
Solution. T he flow situation being considered is shown i n Fig. E3.9. It will b e assumed that the Prandtl number for the air i s 0.7. Hence, using Eq. (3.258):
M j =O.9
T j =O°C
.. .. .. ..

FIGUREE3.9
r = (0.7)112 = 0.837
Therefore, since Eq. (3.256) gives:
Tl T
T \
Wad
= 1+ r
('Y  1 )Mt
2
i t follows that, s ince for air y = 1.4:
;~= 1 + 0.837 X 0.2M; = 1 + 0 .167 X 0.92 = 1.135
H ence, since the freestream temperature is O°C, i t follows that:
T Wad
= 1.135 X 273 = 3 09.9 K
Therefore, t he adiabatic wall temperature i s 36.9°C. Hence, since the wall is kept a t ~ t emperature o f 30°C, i.e., since TW  TWad is negative,heat is being transferred from I . the \air to t he plate.
3.8
E FFECT O F F LUID P ROPERTY VARIATIONS / W ITH V ISCOUS DISSIPATION E FFECTS O N L AMINAR BOUNDARY L AYER F LOW O VER A F LAT PLATE
As mentioned above, because o f the larg~ temperature variations that often exist i n flows in w hich viscous dissipation is imWrtant, t here can b e large variations in the fluid properties across such flows. fu dealibg with external flows i n which the effects  of viscous dissipation are not important, flhld property variations can usually b e adequately accounted for b y evaluating the properties at t he·mean film temperature,
/
/
1 50 Introduction to Convective Heat Transfer Analysis
y
No Viscous Dissipation
J
Adiabatic Wall Viscous Dissipation
F IGURE"3.28 Temperature distributions in a laminar boundary layer with and without viscous dissipation.
i.e., at the average o f the surface and freestream temperatures, i.e.; at ( TW i.e., at:
+ Tl)l2,
(3.274)
When viscous dissipation effects are important, however, there are three temperatures that have a n influence on the temperature dis,tribution and therefore on the fluid properties, these three temperaturesbeing the wall temperature, the freestream temperature, a nd the adiabatic wall temperature, i.e., Tw, T l , a nd T W ad' T his i s ilb,lstrated in Fig. 3.28. W hen viscous dissipation effects are important i t is to b e e xpected therefore, that fluid property variations could be accounted for by evaluating these properties at a mean fluid temperature that is given b y an equation o f the form [9],[12],[15]: (3.275) where Cl and C 2 are constants. Because in flows in which viscous dissipation effects are negligible, T Wad = T l, a comparison of Eqs. (3.274) and (3.275) shows t hatCl = 0.5. Therefore, i n flows i n which viscous dissipation is important, the fluid properties should be evaluated a t a temperature that is given by: (3.276) I n order to find the " best" value to use for the constant C2 i n t his equation, numerical solutions in which the effects o f property variations are accounted for can be obtained for some simple situations such as for flow over a flat plate and these numerical results c aathen be used to deduce the value o f C2 t hat leads to the best agreement between the results obtained accounting for property variations and results obtained assuming constant fluid properties evaluated at T prop. T his procedure indicates that C2 should b e taken a s 0.22 [17],[18],[19],{20],[21],[22], i.e., that: (3.277)
3.9
SOLUTIONS TO T HE FULL GOVERNING EQUATIONS
Solutions based on the use o f the boundary layer approximations to the full governing equations have been discussed i n the above sections. These boundary layer approximations are, however, often not applicable. For example, the boundary layer equations do not apply i f there are significant areas o f reversed flow o r i f t he Reynolds number is low. Even with twodimensional flow over a circular cylinder, the bound
CHAPTER
3: Some Solutions for External Laminar Forced Convection
151
ary layer equations can only be used to predict the heat transfer rate near the stagnation point. They do not apply after flow separation occurs and even well ahead of this separation point there is significant interaction between the outer inviscid flow and the wake with the result that the pressure distribution on the cylinder cannot be obtained by ignoring the boundary layer and calculating the inviscid flow over the cylinder as discussed before. When the boundary layer and other related approximate solutions cannot be applied, it is necessary to solve the full governing equations. In almost all cases it is necessary to do this using numerical methods [23],[24],[25]. However, because the full equations are a set of nonlinear, simultaneous partial differential equations, this usually requires a significant computational effort. Commercial computer packages based on finite element methods or on various types of finitedifference methods are available for obtaining such solutions. These packages usually contain preprocessing and postprocessing modules that make it easy to set up the required nodal system for complex geometrical situations and easy to display the calculated results in a convenient form. As a simple example o f a flow situation in which the boundary layer equations do not apply, consider twodimensional steady flow over a square cylinder placed in a uniform flow as shown in Fig. 3.29. This flow situation is not o f great practical significance. I t i s used purely to illustrate the type o f solution that can be obtained. The flow in the wake of the cylinder will, in general, be unsteady due to the shedding o f vortices from the rear portion o f the cylinder as shown schematically in Fig. 3.30. This shedding causes the wake flow to be asymmetric about an axis that is parallel to the upstream flow and drawn through the center of the cylinder as shown in Fig. 3.30. However, for many purposes it is adequate to assume that the flow is steady and symmetricaL T he solution obtained by making these assumptions will in many circumstances provide a good description o f the timeaveraged flow about the cylinder. Some results obtained using the program EXTSQCYL that obtains a solution for this flow and that is available in the way discussed in the Preface are shown in Fig. 3.31.
Forced Flow \
~
Square Cylinder
   1....  .
~
F IGURE 3.29
Flow over a square cylinder.
Wake Flow is Unsteady and Unsymmetrical
I~"
I~"
 Vortex Shedding
F IGURE 3.30
Unsteady flow i n the wake o f a square cylinder.
152 Introduction to Convective Heat Transfer Analysis
1
10 ._ _ _ ,_ _ _ _,
Nu
1 ''''
1
10 Re
100
F IGURE 3.31 Predicted variation o f mean Nusselt number for a square cylinder with Reynolds number for Pr =
0.7.
3.10 CONCLUDING REMARKS
Some o f the commonly used methods for obtaining solutions to problems involving laminar external flows have been discussed i n this chapter. M any s uch problems can b e treated with adequate accuracy using the boundary layer equations and similar,. ity; integral and numerical methods o f solving these equations have been discussed. A brief discussion of the solution of the full governing equations has also been presented.
PROBLEMS
3.1. A ir flows a tavelocity o f 9 m ls over a wide flat plate that has a length o f 6 c m i n the flow direction. The air ahead o f the plate has a temperature o f 10°C while the surface o f the plate is kept at 70°C. Using the similarity solution results given in this chapter, plot the variation o f local heat transfer rate in W/m2 along the plate and the velocity and temperature profiles in the boundary layer on the plate at a distance o f 4 e m from the leading edge o~!lte plate. Also calculate the mean heat transfer rate from the plate. 3.2. A ir at a temperature of 20°C flows at a velocity o f I mls over a surface which can be modeled as a wide 30mm long flat plate. The entire surface o f this plate is kept at a temperature o f 60°C. Plot a graph showing how the local heat transfer rate in W 1m2 and the boundary layer thickness in mrn varies along the plate. Also plot the temperature profile at the trailing edge o f the plate. Assume twodimensional flow. 3.3. A ir at a temperature of 40°C flows at a velocity o f 5 mls over a surface which can b e modeled as a wide lOOmrn long flat plate. The entire surface o f this plate IS kept at a temperature o f O°c. Plot a graph showing how the local heat transfer rate varies along the plate. Also plot the temperature profile in the boundary layer on the plate at a distance o f 6 0 mrn from the leading edge o f the plate. 3.4. A ir at 3 00 K and 1 atm flows at a velocity o f 2 m ls along a flat plate which has a length o f 0.2 m . The plate is kept at a temperature o f 3 30 K. Plot the variations o f the velocity and thermal boundary layer thicknesses along the plate. '
CHAPTER
3: Some Solutions for External Laminar Forced Convection
153
3.5. Glycerin at a temperature of 30°C flows over a 30cm long flat plate at a velocity o f 1 mls. The surface of the plate is kept at a temperature o f 2 0°e. Find the mean heat transfer rate per unit area to the plate. 3.6. Air flows at a velocity o f 6 mls over a wide flat plate that has a length o f 2 c m in the flow direction. There is a uniform heat flux o f 3 k W/m2 at the surface o f the plate. Using the similarity solution results, plot the variation o f local temperature along the plate. The air ahead o f the plate has a temperature of 1 0°e. 3.7. Air flows at a velocity o f 8 mls over a wide flat plate that has a length o f 10 c m in t he flow direction. The air ahead of the plate has a temperature o f 10°C while the surface of the plate is equal to [10 + 50(x/lO)]OC, x being the distance measured along the plate in cm. Using the similarity solution results, plot the variation o f local heat transfer rate in W /m 2 along the plate. 3.8. Air flows at a velocity of 4 mls over a wide flat plate that has a length o f 20 c m in the flow direction. The temperature o f the surface o f the plate is given by [30 + 30(xI20)O.7]oC, x being the distance measured along the plate in cm. The air ahead o f the plate has a temperature of 20°C. Using the similarity solution results, plot the variation of local heat transfer rate in W /m 2 along the plate. 3.9. Air flows at a velocity o f 2 mls normal to the axis of a circular cylinder with a diameter of 2.5 cm. The surface o f the cylinder is kept at a uniform surface temperature o f 50°C and the temperature in the air stream ahead o f the cylinder is 10°e. Assuming that the flow is twodimensional, find the heat transfer rate in the vicinity o f the stagnation point. 3.10. Consider twodimensional air flow normal to a plane surface. I f the initial air temperature is 20°C, t he surface temperature 80°C, and the air velocity in the freestream ahead o f the plate is 1 mis, plot the variation o f h eat transfer rate in the vicinity o f the stagnation point. 3.11. Consider laminar forced convective flow over a flat plate at whose surface the heat transfer rate p er u nit area, qw is constant. Assuming a Prandtl number o f 1, use the integral equation method to derive an expression for the variation o f surface temperature. Assume twodim.ensional flow. 3.12. A ir with a temperature o f  10°C flows steadily at a velocity o f 8 mls parallel to a flat Iplate t hat is 6 ern long i n the flow direction. The first 2 e m o f the surface o f the plate is adiabatic a nd t he remainder of the plate surface is kept at a temperature o f 50°C. Assuming twodimensional flow and using the integral equation result for a plate with an unheated leading edge section, plot the variation o f the heat flux along the heated portion o f the plate.
I.
3.13. A ir at SOC a nd 70 kPa flows over a flat plate at 6 mls. A heater strip 2.5 c m long is placed on the plate· at a distance o f 5 c m from the leading edge. Calculate the heat lost from the strip per unit depth o f p lateJor a heater surface temperature o f 65°C. Use the ~ppropriate i ntegral equation result. . 3.14. A ir flows parallel to the surface o f a flat plate which is unheated (adiabatic) up/to a distance of Xo from the leading edge. Downstream o f this point, there is a uniform heat
154 Introduction to Convective Heat Transfer Analysis
flux at the surface o f the plate. Assuming steady, twodimensional, constant property flow, use the integral method to find the s urface temperature along the heated portion o f the plate. Show t hat i f Xo = 0, the local wall temperature is propOrtional to xD.5.
3.15. Consider the twodimensional laminar boundary flow o f air over a wide I 5cm long flat plate whose surface temperature varies linearly from 20°C at the leading edge to 40°C a t the trailing edge. This plate is placed i n an airstream with a velocity o f 2 mls and a temperature o f lOoC. Numerically determine how the surface heat flux varies along this plate.
3.16. A ir flows at a velocity o f 3 rnJs over a wide flat plate that has a length o f 3 0 cnyt in the flow direction. The air ahead o f the plate has a temperature o f lOoC while the surface temperature o f the plate is given by:
x < 5 cm: Tw 5 c m < x < 2 5 cm: Tw 25 c m < x < 3 0cm: Tw
= 40°C
= 60°C
= 40°C
By numerically solving the twodimensional laminar boundary layer equations, determine how the local heat transfer rate in W1m2 varies along the plate.
3.17. Numerically determine the heat transfer rate variation with twodimensional laminar boundary layer a ir flow over a flat plate with a uniform heat flux at the surface. Compare the numerical results with those given by the similarity solution.
3.18. Liquid films are used for cooling in a number o f industrial situations. Consider·.the following simple case: Assuming the flow remains laminar a nd h as a boundary layerlike characteristic, write down the governing equations together with the boundary and initial conditions. I f the y coord~ate is replaced by the stream function derived by:
y = (Y al/!
Jo
u
show that the xmomentum equation becomes:
u~: + vu;; ~ v(;:~ )u
2
and write down the boundary conditions in this case. Discuss how you could numerically solve these equations using the finite difference method.
FIGURE P3.t7
C HAPTER
3: Some Solutions for External Laminar Forced Convection
155
3.19. An implicit finitedifference procedure for solving the laminar boundary layer equations was discussed in this chapter. Discuss how these boundary layer equations could be solved using a n explicit procedure. In such a procedure, the terms a ulay and a2 ulay2 are evaluated on the " (i  1)" line. The continuity equation is treated in the same way in both procedures. Write a computer program based on this procedure and show by numerical experimentation with this program that instability develops if:
A x> K tll
where K is a constant. Estimate the value o f K. 3.20. Show how the numerical method for solving the laminar boundary layer equations discussed in this chapter can be modified to allow for viscous dissipation. U se a computer program based o n this modified procedure to estimate the importance o f this dissipation on t he heat transfer rate along an isothermal flat plate in low speed flow. 3.21. Air at a Mach number o f 3 and a temperature o f  30°C flows over a flat plate .that is aligned with the flow. The plate is k~pt at a temperature o f 25°C. The flow in the boundary layer is laminar. I s heat transferred to or from the plate surface? 3.22. Air a t a p ressure o f 5 kPa and a temperature o f  30°C flows at a Mach number o f 2 5 over a flat plate that is aligned with the flow. T he plate is kept at a uniform temperature of soc. F ind the heat transfer from the plate surface to the air. 3.23. A flat plate o f length L is heated to a uniform surface temperature and dragged through water which is a t a temperature of 10°C at a velocity, V. Plot the variation o f p ower required to pull the plate through the water and o f the total heat transfer rate from the plate with plate surface temperature for surface temperatures between 10 and 95°C. Comment o n the results obtained. T he boundary layer on the plate can b e assumed to remain laminar. 3.24. In o rder to measure the velocity o f a s tream o f air, a flat plate o f l ength 2 c m i n the flow direction is placed in the flow. This plate is electrically heated, the heat dissipation rate being uniform over the plate surface. The plate is wide so a twodimensional laminar boundary layer flow C&ll b e assumed to exist. The velocity is to b e deduced b y measuring the temperatuJ:e~of the plate at its trailing edge. I f this temperature is to b e at least 40°C w hen the a ir temperature is 20°C a nd the air velocity is 3 mis, find the required rate o f h eat dissipation in the plate p er u nit surface area.
REFERENCES
1. Schlichting, H., Boundary Layer Theory, 7 th ed., McGrawHill, New York, 1979. 2. Howarth, L., " On t he Solution to the Laminar Boundary Layer Equations", Proc. R. Soc. London Ser. A , Vol. 1 64,1938, p. 547. 3. Howarth, L., Editor, M odem Developments in Fluid Dynamics High S peed Flow, Vols. 1 a nd 2 , O xford University Press, London, 1953.
4. White, F.M., Viscous Fluid Flow, 2nd. ~ed., McGrawHill, New York, 1991. S. S herman, E S., Viscous Flow, McGrawHill, New York, 1990. 6. B ejan, A., Convection Heat Transfer, 2 nd ed., Wiley, New York, 1995.
/
156 Introduction to Convective Heat Transfer Analysis
7;. Fox, L., The Numerical Solution o f TwoPoint Boundary Problems, Oxford University Press, London, 1957. 8. Van Dyke, M., Perturbation Methods in F luid Mechanics, Parabolic Press, Stanford, CA, 1975. 9. Kays, W.M. and Crawford, M.E., Convective Heat a nd Mass Transfer, 3 rd ed., McGrawHill, New York, 1993. 10. Falkner, V.M. and Skan, S.W., "Some Approximate Solutions o f the Boundary Layer Equations", Phi/os. Mag., Vol. 12, 1931, p. 865. 11. Patankar, S.Y. and Spalding, D.B., Heat a nd M ass Transfer in Boundary Layers, 2d. ed., International Textbook Co., London, 1970. 12. Dejarnette, ER., Hamilton, H.H., Weilmuenster, K.L., and Cheatwook, E M., " A Review o f Some Approximate Methods Used i n Aerodynamic Heating Analyses", J. Tfzermophys. Heat Transj, Vol. 1, N o.1, pp. 512, 1987. 13. Dorrance, W.H., Vlscous Hypersonic Flow, McGrawHill, New York, 1962. 14. Kaye, J., "Survey o f Friction Coefficients, Recovery Factors, and Heat Transfer Coefficients for Supersonic Flow", J. Aeronaut. Sci., Vol. 21, N o.2, pp. 117229, 1954. 15. Truitt, R.W., Fundamentals o f Aerodynamic Heating, Ronald Press, New York, 1960. 16. van Driest, E.R., " The Problem o f Aerodynamic Heating", Aeronaut. Eng. Rev., Vol. 15, pp. 2641, 1956. 17. Beckwith, I.E. and Gallagher J.J., "Local Heat Transfer and Recovery Temperatures on a Yawed Cylinder at a Mach Number o f 4.15 and High Reynolds Numbers", NASA TRRI04, 1962. 18. Cohen, C.B. and Reshotko, E., "Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient", NASA Report 1293, Houston, TX, 1956. 19. Eckert, E.R.G., "Engineering Relations for Heat Transfer and Friction in HighVelocity Laminar and Turbulent Boundary Layer Flow over Surfaces with Constant Pressure and Temperature", Trans. ASME, Vol. 78, pp. 12731284, 1956. 20. Fischer, W.W. and Norris, R., "Supersonic Convective Heat Transfer Correlation from Skin Temperature Measurement on V2 Rocket in Flight", Trans. A SME, Vol. 71, p p.457469,1949. 21. Levy, S., "Effect o f Large Temperature Changes (Including Viscous Heating) upon Laminar Boundary Layers with Variable FreeStream Velocity", J. Aeronaut. Sci., Vol. 21, N o.7, pp. 4 59474,1954. 22. Rubesin, M.W. and Johnson, H.A., "Aerodynamic Heating and Convective Heat T ransferSummary o f Literature Survey", Trans. ASME, Vol. 71, pp. 383388, 1949. 23. Patankar, S. Y., Numerical Heat Transfer a nd Fluid Flow, Hemisphere Publ., Washington, D. C., 1980. 24. Gosman, A.D., Pun, W.M., Runchal, A.K., Spalding, D.B., and Wolfshtein, M., Heat and Mass Transfer in Recirculating Flows, Academic Press, New York, 1969. 25. Chow, L .c. and Tien, C.L. " An Examination o f Four Differencing Schemes for Some EllipticType Convection Equations", Numerical Heat Transfer, Vol. 1, 1978, pp. 8 7100.