Boundary Lyer Theory

292 XIT. Tl~er~nnl Imn~dary layers in laminar flow g. Thermal boundary layers in forced flow 293
evcn in the case of moderately-valued Prandtl numbers. At the stagnation point,
the corresponding cquation is
N, = 0.661 P1I3 1/% (stagnation point, P -+ oo) . (12.6213)
Analogous, simple asymptotic formulae can also be established for the case of frcc
convect,ion on n verlical flat plate, [73], see also eqns. (12.118a) and (12.1181)).
g. Thermnl bounilnry layers in forced Row
In the present section we shall consider several examples of thermal boundary
layors in forced flow. In solving thcso problems, uso will bo made of tho simplified
thermal boundary-layer equations. Just as in the case of a velocity boundary layer,
the general problem of evaluating tho thermal boundary layer for a body of arbitrary
shape proves to be extremely difficult, so that we shall begin with the simpler
example of the flat plate at zero incidence.
1. Parallel Row paet a Bat platc at zero incidence. We shall assume that the
x-axis is placed in the plane of the plate in the direction of flow, the y-axis being
at right ar~glcs to it and to the flow, with the origin at the leading cdgc. The boundarylayer
equations for incompressible flow and constant properties (i. e. independent
of temperature) have been given in eqns. (12.61 a, b, c): assuming that the buoyancy
forces are equal to zero as well as that dpldz = 0 [18, 941, we obtain
'I'hc: 1)ountlary ronclitions arc:
11 = 0 : u = v = 0 ; T == T,,, or aT/8g =O
'I'he vrlocit8y ficld is it~tlcpcndrnt, of t,hc tcmprraturc firltl so that tlrc two Ilow
equations (12.03a, b) can be solved first and the result can be employed to evaluatc
the tscmpcmtnrc field. An important rclatior~ship between the velocity distribution
and thc temperature distribution can bc obtained immediately from eqns. (12.63 b)
:md (I 2.fR c). Jf lhc hcat of friction p may be neglected in eqn. (12.63~).
the two rquat.ions, (12.03b) and (12.63c), become identical if T is rrplaced by 76 in
the sccond cqr~at~ion a.id if, in addition, the properties of the fluid satisfy the equation
If the frictional heat is neglected then a temperature field exists only if there is a
difference in temperature between thc wall and the extcrnal flow, e.g., if Tw - T, > 0
(cooling). Hence it follows that for a flat plate at zero incidence in psrallcl Row
and at small velocities the temperature arid velocity distributions arc idcr~tical
provided that the Prandtl number is equal to unity:
Thi~ result corresponds to eqn. (12.52) which Icd us lo thc f~rrnulat~ion of Llw
important Iteynolds analogy between heat transfer and skin friction.
11. Blasius introduced new variables for thc solution of the flow rquat.ions,
sce oqns. (7.24) and (7.26). (y) is 1110 slrcnm fnnclion):
'rhe diffcrcntial cquation for /(q), cqn. (7.28) bccornos
f f" + 21"' =0,
with thc boundary conditions: rl = 0 : f = f' = 0 ; 11 =- cm : 1' L- I . 'I'l~e solution
of these equations was given in Chap. VII, Table 7.1.
Including the eflect of frictional hcat, as seon from eqn. (12.63c), the temperature
distribution T(7) is given by the equation
It is convenient to represent thc general solution of eqn. (12.65) by tho super-
position of two solutions of the form:
ITcrc O1(7) dcnotes the grncml solution of thc hornogcncous cqu:~tion and 02(t7)
denotes a particular solution of the non-homogeneous equation. It is, further,
convenient to choose the boundary conditions for 01(7) and O,(q) so as to rnakc
01(7) the solution of the cooling problem with a prescribed temperature diKcrcnce
betwecn the wall and thc external stream, T, - T, with 02(7) giving the solution
for the adiabatic wall. Thus 01(7) and Oz(v) satisfy the following equations:
with 0, -1 1 nt 7 -1 0 arid O1 -- 0 nt 11 == oc> , r ~ l