Page 174 - Bird R.B. Transport phenomena

158 Chapter 5 Velocity Distributions in Turbulent Flow

Having defined the time-smoothed quantities and discussed some of the properties
of the fluctuating quantities, we can now move on to the time-smoothing of the equations
of change. To keep the development as simple as possible, we consider here only the
equations for a fluid of constant density and viscosity. We start by writing the equations
of continuity and motion with v replaced by its equivalent v + v' and p by its equivalent
p + p'. The equation of continuity is then (V • v) = 0, and we write the x-component of the
equation of motion, Eq. 3.5-6, in the д/dt form by using Eq. 3.5-5:
(5 + v
+
+
+
+
h ъ ^ % v '*> iz ъ ^ = ° (5 2 4)
-"
| p ( v + v' ) = - £ ( ? + P') - [ £ Р & + v' ){v + v' ) + -^ {v + v' )(v + v' )
x x Х x x x 9 v y x x
+ ^ (v + vl)(v + v' )J + pN {v + v' ) + pg (5.2-5)
2
P 2 x x x x x
The y- and z-components of the equation of motion can be similarly written. We next
time-smooth these equations, making use of the relations given in Eq. 5.2-3. This gives

(5-2-7)

with similar relations for the y- and z-components of the equation of motion. These are
then the time-smoothed equations of continuity and motion for a fluid with constant density
and viscosity. By comparing them with the corresponding equations in Eq. 3.1-5 and
Eq. 3.5-6 (the latter rewritten in terms of д/dt), we conclude that
a. The equation of continuity is the same as we had previously, except that v is now
replaced by v.
b. The equation of motion now has v and p where we previously had v and p. In ad-
dition there appear the dashed-underlined terms, which describe the momentum
transport associated with the turbulent fluctuations.
We may rewrite Eq. 5.2-7 by introducing the turbulent momentum flux tensor т ш with
components
-<f> ^rn - ш ^r^i - ( 0 ^ r i ^ ^
= = = g o o n ( 5 2 g )
These quantities are usually referred to as the Reynolds stresses. We may also introduce a
symbol T {V) for the time-smoothed viscous momentum flux. The components of this ten-
sor have the same appearance as the expressions given in Appendices B.I to B.3, except
that the time-smoothed velocity components appear in them:

This enables us then to write the equations of change in vector-tensor form as
(V • v) = 0 and (V • v ) = 0 (5.2-10,11)
1
( 0
j t pv = -Vp - [V • pv v] - [V • (т (у) + T )] + pg (5.2-12)