SOME SPECIAL CASES, COUEITE AND POISEUILLE FLOWS 681
The case shown here is for the upper wall at a higher temperature than the lower wall. The heat transfer at the lower wall is obtained from Eq. (16.24) with a negligible u,,:
At Y =0:
(16.29)
The heat transfer at the upper wall is similarly obtained as
At y=D:
.
=~ Ihehwl
(16.30)
qw
Pr
D
Equations (16.29) and (16.30) are identical; this is no surprise, since we have already shown that the heat flux is constant across the flow, as shown by Eq. (16.26), and therefore the heat transfer at both walls should be the same. Equations (16.29) and (16.30) can also be written in terms of temperature as
qw = k ITe; Twl
(16.31)
Examining Eqs. (16.29) to (16.31), we can make some conclusions which can be generalized to most viscous flow problems, as follows:
1.Everything else being equal, the larger the temperature difference across the viscous layer, the greater the heat transfer at the wall. The temperature difference (TeTw) or the enthalpy difference (hehw) takes on the role of a "driving potential" for heat transfer. For the special case treated here, the heat transfer at the wall is directly proportional to this driving potential.
2.Everything else being equal, the thicker the viscous layer (the larger D is) the smaller the heat transfer is at the wall. For the special case treated here, qw
is inversely proportional to D.
3.Heat flows from a region of high temperature to low temperature. For negligible viscous dissipation, if the temperature at the top of the viscous layer is higher than that at the bottom, heat flows from the top to the bottom. In the case sketched in Fig. 16.4, heat is transferred from the upper plate into the fluid, and then is transferred from the fluid to the lower plate.
16.3.2Equal Wall Temperatures
Here we
assume that Te = Tw; i.e., he = hw. The enthalpy profile for this case,
from Eq.
(16.16), is
h=h +1Pru.2(Y) 1Pru~"(y)2
w 2
'D 2
e D
or
= h,+.!.
2
(16.32)
Pr u
[L_ (L)2]
w 2
e
D D
682 FUNDAMENTALS OF AERODYNAMICS
In terms of temperature, this becomes
T= Tw+ P;c:; [~(~rJ
(16.33 )
Note that the temperature varies parabolically with y, as sketched in Fig. 16.5. The maximum value of temperature occurs at the midpoint, y = D/2. This maximum value is obtained by evaluating Eq. (16.33) at y = D/2.
Pr u 2
(16.34)
Tmax = Tw + __e
8cp
The heat transfer at the walls is obtained from Eqs. (16.24) and (16.25) as
At y = 0:
(16.35)
At y=D:
(16.36)
Equations (16.35) and (16.36) are identical; the heat transfers at the upper and lower walls are equal. In this case, as can be seen by inspecting the temperature distribution shown in Fig. 16.5, the upper and lower walls are both cooler than the adjacent fluid. Hence, at both the upper and lower walls, heat is transferred from the fluid to the wall.
Question: Since the walls are at equal temperature, where is the heat transfer coming from? Answer: Viscous dissipation. The local temperature increase in the flow as sketched in Fig. 16.5 is due solely to viscous dissipation within the fluid. In turn, both walls experience an aerodynamic heating effect due to this viscous dissipation. This is clearly evident in Eqs. (16.35) and (16.36), where tiw depends on the velocity, ue • Indeed, tiw is directly proportional to the square of ue • In light of Eq. (16.9), Eqs. (16.35) and (16.36) can be written as
(16.37)
u, 1
y
G
\ __ T=
'(
!
D
)
[
T.V
T•. + Pru; [~_(~)21
2cp D D
;;;:
FIGURE 16.5
Couette flow temperature profile for equal wall temperature with viscous dissipation.
SO~lI·. SPECIAL CASES. COLJr.TIE AND POISELJILLE FLOWS 683
which further emphasizes that qw is due entirely to the action of shear stress in the flow. From Eqs. (16.35) to (16.37), we can make the following conclusions that reflect general properties of most viscous flows:
1.Everything else being equal, aerodynamic heating increases as the flow velocity increases. This is why aerodynamic heating becomes an important design
factor in highspeed aerodynamics. Indeed, for most hypersonic vehicles, you can begin to appreciate that viscous dissipation generates extreme temperatures within the boundary layer adjacent to the vehicle surface and frequently makes aerodynamic heating the dominant design factor. In the Couette flow case shown herea far cry from hypersonic flowwe see that qw varies directly
J
as u;.
2.Everything else being equal, aerodynamic heating decreases as the thickness of the viscous layer increases. For the case considered here, qw is inversely proportional to D. This conclusion is the same as that made for the above case of negligible viscous dissipation but with unequal wall temperature.
16.3.3Adiabatic Wall Conditions
(Adiabatic Wall Temperature)
Let us imagine the following situation. Assume that the flow illustrated in Fig. 16.5 is established. We have the parabolic temperature profile established as shown, and we have heat transfer into the walls as discussed above. However, both wall temperatures are considered fixed, and both are equal to the same constant value. Question: How can the wall temperature remain fixed at the same time that heat is transferred into the wall? Answer: There must be some independent mechanism that conducts heat away from the wall at the same rate that the aerodynamic heating is pumping heat into the wall. This is the only way for the wall temperature to remain fixed at some cooler temperature than the adjacent fluid. For example, the wall can be some vast heat sink that can absorb heat without any appreciable change in temperature, or possibly there are cooling coils within the plate that can carry away the heat, much like the water coils that keep the engine of your automobile cool. In any event, to have the picture shown in Fig. 16.5 with a constant wall temperature independent of time, some exterior mechanism must carry away the heat that is transferred from the fluid to the walls. Now imagine that, at the lower wall, this exterior mechanism is suddenly shut off. The lower wall will now begin to grow hotter in response to qw, and Tw will begin to increase with time. At any given instant during this transient process, the heat transfer to the lower wall is given by Eq. (16.24), repeated below:
(16.24)
At time t = 0, when the exterior cooling mechanism is just shut off, hw = he. and
684 FUNDAMENTALS OF AERODYNAMICS
qw is given by Eq. (16.35), namely,
At time t = 0:
However, as time now progresses, Tw (and therefore hw) increases. From Eq. (16.24), as hw increases, the numerator decreases in magnitude, and hence qw decreases. That is,
At t> 0:
Hence, as time progresses from when the exterior cooling mechanism was first cut off at the lower wall, the wall temperature increases, and the aerodynamic heating to the wall decreases. This in turn slows the rate of increase of Twas time progresses. The transient variations of both qw and Tw are sketched in Fig. 16.6. In Fig. 16.6a, we see that, as time increases to large values, the heat transfer to the wall approaches zerothis is defined as the equilibrium, or the adiabatic wall condition. For an adiabatic wall, the heat transfer is, by definition, equal to zero. Simultaneously, the wall temperature, Tw, approaches asymptotically a limiting value defined as the adiabatic wall temperature, Taw' and the corresponding enthalpy is defined as the adiabatic wall enthalpy, haw.
The purpose of the above discussion is to define an adiabatic wall condition; the example involving a timewise approach to this condition was just for convenience and edification. Let us now assume that the lower wall in our Couette flow is an adiabatic wall. For this case, we already know the value of heat transfer to the wallby definition, it is zero. The question now becomes, What is the value of the adiabatic wall enthalpy, haw' and in turn the adiabatic wall temperature, Taw? The answer is given by Eq. (16.23)' where qw = 0 for an
,
!! U,'
Adiabatic wall limit
20
T..... = 7;,
Time

Time
Adiabatic wall limit
(a)
(h)
FIGURE 16.6
Illustration for the definition of an adiabatic wall and the adiabatic wall temperature.
f
SOME SPECIAL CASES; COUEITE AND POISEUILLE FLOWS 685
adiabatic wall.
Adiabatic wall:
qw=o~(ah)
=(aT) =0
(16.38)
ay w
ay 'w
Therefore, from
Eq. (16.19), with ah/ ay = 0, y = 0, and
hw = haw' by definition,
or
(16.39)
In turn, the adiabatic wall temperature is given by
(16.40)
Clearly, the higher the value of Un the higher is the adiabatic wall temperature. The enthalpy profile across the flow for this case is given by a combination of Eqs. (16.16) and (16.40), as follows. Setting hw = haw in Eq. (16.16), we obtain
(16.41)
From Eq. (16.39),
(16.42)
Inserting Eq. (16.42) into (16.41), we have
U2e ( Y )2
h=hPr 
a" 2 D
Equation (16.43) gives the enthalpy profile across the flow. profile follows from Eq. (16.43) as
(16.43)
The temperature
T= TPr u~
(L)2
(16.44)
a" 2cp
D
This variation of T is sketched in Fig. 16.7. Note that Taw is the maximum temperature in the flow. Moreover, the temperature curve is perpendicular at the plate for y = 0; i.e., the temperature gradient at the lower plate is zero, as expected for an adiabatic wall. This result is also obtained by differentiating Eq. (16.44):
:;= Prc~;(~)
which gives aT/ay=O at y=O.
686
FUNDAMENTALS OF AERODYNAMICS
u,.
y
1  \ T= T _ Pru..'(1')2
D
UIl' 2cfI D
FIGURE 16.7
Couette flow temperature profile for an adiabatic lower walL
16.3.4Recovery Factor
As a corollary to the above case for the adiabatic wall, we take this opportunity to define the recovery factora useful engineering parameter in the analysis of aerodynamic heating. The total enthalpy of the flow at the upper plate (which represents the upper. boundary on a viscous shear layer) is, by definition,
h
u 2
(16.45)
=h +~
o
e
2
(The significance and definition of total enthalpy are discussed in Sec. 7.5.) Compare Eq. (16.45), which is a general definition, with Eq. (16.39), repeated below, which is for the special case of Couette flow:
=
U2e
(16.39)
haw
he+ Pr 2
Note that haw is different from ho, the difference provided by the value of Pr as it appears in Eq. (16.39). We now generalize Eq. (16.39) to a form which holds for any viscous flow, as follows:
(16.46a)
Similarly, Eq. (16.40) can be generalized to
(16.46b)
In Eqs. (16.46a and b), r is defined as the recovery factor. It is the factor that tells us how close the adiabatic wall enthalpy is to the total enthalpy at the upper boundary of the viscous flow. If r = 1, then haw = ho. An alternate expression for
,,
SOME SPECIAL CASES; COUETTE AND POISEUILLE FLOWS 687
the recovery factor can be obtained by combining Eqs. (16.46) and (16.45) as follows. From Eq. (16.46),
(16.47)
From Eq. (16.45),
u2
(16.48)
~=h h
e
2
0
Inserting Eq. (16.48) into (16.47), we have
(16.49)
where To is the total temperature. Equation (16.49) is frequently used as an alternate definition of the recovery factor.
In the special case of Couette flow, by comparing Eq. (16.39) or (16.40)
with Eq. (16.46a) or (16.46b), we find that
r= Pr
(16.50)
For Couette flow, the recovery factor is simply the Prandtl number. Note that, if Pr< 1, then haw < ho ; conversely, if Pr> 1, then haw> ho .
In more general viscous flow cases, the recovery factor is not simply the Prandtl number; however, in general, for incompressible viscous flows, we will find that the recovery factor is some function of Pr. Hence, the Prandtl number is playing its role as an important viscous flow parameter. As expected from Sec. 15.6, for a compressible viscous flow, the recovery factor is a function of Pr along with the Mach number and the ratio of specific heats.
16.3.5Reynolds Analogy
Another useful engineering relation for the analysis of aerodynamic heating is Reynolds analogy, which can easily be introduced within the context of our discussion of Couette flow. Reynolds analogy is a relation between the skin friction coefficient and the heat transfer coefficient. The skin friction coefficient, cl, was first introduced in Sec. 1.5. In our context here, we define the skin friction coefficient as
(16.51)
From Eq. (16.9), we have, for Couette flow,
(16.52)
_.,,
688 FUNDAMENTALS OF AERODYNAMICS
Combining Eqs. (16.51) and (16.52), we have
(16.53)
Let us define the Reynolds number for Couette flow as
PeueD
Re= 
/L
Then, Eq. (16.53) becomes
2
(16.54)
c 
r Re
Equation (16.54) is interesting in its own right. It demonstrates that the skin friction coefficient is a function of just the Reynolds numbera result which applies in general for other incompressible viscous flows [although the function is not necessarily the same as given in Eq. (16.54)].
Now let us define a heat transfer coefficient as
(16.55)
In Eq. (16.55), CH is called the Stanton number; it is one of several different types of heat transfer coefficient that is used in the analysis of aerodynamic heating. It is a dimensionless quantity, in the same vein as the skin friction coefficient. For Couette flow, from Eq. (16.24), and dropping the absolute value signs for convenience, we have
(16.56)
Inserting Eq. (16.39) into (16.56), we have for Couette flow
(16.57)
Inserting Eq. (16.57) into (16.55), we obtain
C = (/L/Pr)[(haw 
hw)/ D]
/L/Pr = _1_
H
hw)
(16.58)
Peue(haw 
PeueD Re Pr
Equation (16.58) is interesting in its own right. It demonstrates that the Stanton number is a function of the Reynolds number and Prandtl numbera result that applies generally for other incompressible viscous flows [although the function is not necessarily the same as given in Eq. (16.58)].
SOME SPECIAL CASES, COUEITE AND POISEUILLE FLOWS 689
We now combine the results for cf and CH obtained above. From Eqs.
(16.54) and (16.58), we have
CH
(
1
) Re


Cf
Re Pr
2
or
C H
1
_I
(16.59)
 =  Pr
Cf
2
Equation (16.59) is Reynolds analogy as applied to Couette flow. Reynolds analogy is, in general, a relation between the heat transfer coefficient and the skin friction coefficient. For Couette flow, this relation is given by Eq. (16.59). Note that the ratio CHI cf is simply a function of the Prandtl numbera result that applies usually for other incompressible viscous flows, although not necessarily the same function as given in Eq. (16.59).
16.3.6Interim Summary
In this section, we have studied incompressible Couette flow. Although it is a somewhat academic flow, it has all the trappings of many practical viscous flow problems, with the added advantage of lending itself to a simple, straightforward solution. We have taken this advantage, and have discussed incompressible Couette flow in great detail. Our major purpose in this discussion is to make the reader familiar with many concepts used in general in the analysis of viscous flows without clouding the picture with more fluid dynamic complexities. In the context of our study of Couette flow, we have one additional question to address, namely, What is the effect of compressibility? This question is addressed in the next section.
Example 16.1. Consider the geometry sketched in Fig. 16.2. The velocity of the upper plate is 200 ft/ s, and the two plates are separated by a distance of 0.01 in. The fluid between the plates is air. Assume incompressible flow. The temperature of both plates is the standard sea level value of 519°R.
(a)Calculate the velocity in the middle of the flow.
(b)Calculate the shear stress.
(c)Calculate the maximum temperature in the flow.
(d)Calculate the heat transfer to either wall.
(e)If the lower wall is suddenly made adiabatic, calculate its temperature.
Solution. Assume that }L is constant throughout the flow, and that it is equal to its value of 3.7373 x 10 7 slug/ft/s at the standard sea level temperature of 519°R.
(a) From Eq. (16.6),
u = (200) G) =1100 ftls 1
690 FUNDAMENTALS OF AERODYNAMICS
(b) From Eq. (16.9),
where D = 0.01 in = 8.33 x 104ft
_ (3.7373 x 107)(200) _I
2 1
T" 
8.33 x 104

0.09Ib/ft
Note that the shear stress is relatively smallless than a tenth of a pound acting over a square foot.
(c) From Eq. (16.34), for equal wall temperatures, the maximum temperature, which occurs at y/ D = 0.5, is
T = T" + ~: u2~[;  (; YJ = T" + P:c:;
For air at standard conditions, Pr=0.71 and cp =6006 (ft ·lb)/(slug· OR). Hence,
(0.71)(200)2
T=519+ 8(6006) 519+0.6=1. 519.6°R 1
Notice that the maximum temperature in the flow is only sixtenths of a degree above the wall temperatureviscous dissipation for this relatively lowspeed case is very small. This certainly justifies our assumption of constant p, 11,and k in this section, and gives us a feeling for the energy changes associated with an essentially incompressible flowthey are very small.
(d) From Eq. (16.35),
• 11
(u~)
(3.7373 x 1O7)(20W
r:
2
q" ="2
D =
(2)(8.33 x 104)
= L8.97
(ft· lb)/ (ft Is) 1
Since there are 778 ft . lb to a Btu (British thermal unit), then
q" = 8.97 (ft· Ib)/(ft2 /s) = 0.0115 Btu/(ft2 /s)
(e) From Eq. (16.40),
Pr (u;)
T . =
(0.7I)(20W
T + 
 = 519+'
'''
a"
e
cp
2
(2)(6006)
=519+2.36=/521.36°R I
Note in the above example that the adiabatic wall temperature is higher than the maximum flow temperature calculated in part (c) for the cold wall case. In general, for cold wall cases, the viscous dissipation in the flow is not sufficient to heat the gas anywhere in the flow to a temperature as high as the adiabatic wall temperature. Also, we again note the comparatively low temperature increase Taw is only 2.360 higher than the upper wall temperature. In contrast, for the compressible flow to be treated in the next section, the temperature increases can be substantialthis is one of the major aspects that distinguishes compressible viscous flow from incompressible viscous flow. Note that, in the present problem, the Mach number of the upper plate is