Page 436 - Bird R.B. Transport phenomena

418 Chapter 13 Temperature Distributions in Turbulent Flow

Use of KY in place of y, and introduction of the dimensionless temperature function в(У,
в, z, t) = (T- TJ/CTQ - T ), enable us to rewrite Eq. 13.6-4 as
x

with boundary conditions as follows:
Inlet condition: at z = О, 0(Y, 0, 0, t) = 0 for У > 0 (13.6-8)
Wall condition: at У = 0, 0 ( 0 , 0 , 2 , 0 = 1 for 0 < z < L (13.6-9)
Equation 13.6-7 contains an unbounded derivative дВ/dt with a coefficient 1 indepen-
dent of a. Thus a change of variables is needed to analyze the influence of the parameter
a in this problem. For this purpose we turn to the Fourier transform, a standard tool for
analyzing noisy processes.
We choose the following definition 1 for the Fourier transform of a function g(t) into
the domain of frequency v at a particular position У, 0, z:

(13.6-10)
The corresponding transforms for the ^-derivative and for products of functions of t are

Г e~ l7TWt ^-git)dt = iTrivgiv) (13.6-11)

f e - 2 7 T l v l git)hit)dt = \ g i v M v - v )dv, = * h (13.6-12)
g
x
J — oo J — oo
and the latter integral is known as the convolution of the transforms g and h.
Before taking the Fourier transforms of Eqs. 13.6-7 to 9, we express each included
function git) as a time average g plus a fluctuating function g'it) and expand each prod-
uct of such functions. The resulting expressions have the following Fourier transforms:

8
®\g + g'\ = Mg + g'M (13.6-13)
= 8(v)gh + gh' + g'h + ~g'*h' (13.6-14)
Here biv) is the Dirac delta function, obtained as the Fourier transform of the function
git) = 1 in the long-duration limit. The leading term in the last line is a real-valued im-
pulse at v = 0, coming from the time-independent product gh. The next two terms are
complex-valued functions of the frequency v. The convolution term g' * h' may contain
complex-valued functions of v, along with a real-valued impulse 8iv)g'h r coming from
time-independent products of simple harmonic oscillations present in g' and h'.
_ Taking the Fourier transform of Eq. 13.6-7 by the method just given and noting that
d&/dt is identically zero, we obtain the differential equation

h

р ® \дред&' гдрёдв 1 We *дв'\ KY 2
0д
R дв dY Rdd dY R S6 dY R дв dY) 2

\ w dz dY dz dY dz dY dz dY