Boundary Lyer Theory

268 XII. 'I'hcrrnal borrnrlary lnycrs in laminar flow b. Temperature incren.w through ndinbntic cornprcssion; stagnation tempcratme 269
Here c,[d/kg (leg] represents the specific: heat at const,nnt, pressure per unit mass.
In general, c, clepcnds on t,ctnpcmtnre. In the casc of a constant thrmal contluo-
tivit,y, we obtain the simpler form
In thc ca.se of an incomprcssil)lc fluid, wc havc tliv rct = 0, antl cqll. (12.7) togrthcr
wit.11 dn .- c tb7' yields
r 7
I he tan~pnmtnre changes brought, about by thc dynamic: pressure variation in
a comprc:ssil)le flow arc important for its heat bala~ice. In particular, it appears useful
to compare t.11~ tc~nperaturc diffcrcnccs which result from the hcat due to friction
wit,ll those cattsctl by comprcssion. For this reason wo shall first cvnluatc the tcrnpcra.t.trrc
increaso due to compression in a frictionless fluid stream : 1 f the velocity
varies along a st.rm.tnlitlc t,l~o tcmpcr:~t.urc must vary also. In order to simplify - thi: ..
argumcnt it, is perrnissi1)lo to assumc that the process is adiabatic and rcvcrsiblc
bccaltsc the small value of conductivit.~ and the high rate of change in the thcrmodynamic
propcrtics of state will, in gcncral, prevctit, ally appreciablc cxchange of
hmt with the surroundings. In particrtlar wc propose to calculate the temperature
increase (AT),, - T, - 7', which occurs at the stagnation point of a body in a
stream anti wltich is due t,o compression from p,, t>o p,, Fig. 12.2.
l'ig. 12.2. Calc~~lation of bl~c tctnpcraturc incrmno
at stagnation point due to adiabatic comprc3sion
(A7'),,, = To - 7.m
For the casc of zero hcat conduc%ion in frictionless flow the energy equation
(12.11) givcs the following relation between temperature and pressure along a strcam-
liric (coordinate s)
I
where w(s) dcnotcs the vclocity along a streamline. Dividing by euy and integr,zting
along a strcamlinc we obtain
"
Mercury
1,uhr.
oil
Air
(ntrnooph.)
Temperature
6 T
Table 12.1. Physical constants
(1 d = 1 Nm; 1 IrJ/kg dcg - lo3 m2/sec2 dog)
Specific
hent
Cv
[lzJ/kg K]
Thernrnl Ther~nnl
condnrtivity difl~mivit.y
k n x 10'
[Jim noc K] [ni2/mc]
Vincoaity
/1 X lom
[kg/~n ncc
= r~ R]
Kinernntir
vinronity
v x 108
[rn2/uec]
Prnndtl
nurnber
In an analogous manner, thc complete Navicr-Stokcs r(~na1ions (3 26) lead to the
I%ernoalli equation when viscosity is neglected in them and whcn an intagrnl along a
streamlinc is talren:
so that the tcmpcrature increase
1
T - T, = -- (wW2 - w2), (12.14a)
%
antl, in particular, the temperature increase at the stngnat~ion point (w - 0) t111c to
adiabatic contprcssion becomes
Here w, dcnotes the free-stream vclocity (Fig. 12.2). The temperature T, assumrd 1)y
the fluid when the velocity is reduccd to zcro is known as the slagnntio~t te~npernlurr,
sometimes also referred to as the total lernperature. The difference (AT),, = To - l',
brtween the stagnation and the free-stmam temperature will hcre be called the
ndiabtrlic trmprralitre incrrosc
P
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