Page 175 - Bird R.B. Transport phenomena

§5.3 The Time-Smoothed Velocity Profile near a Wall 159

Equation 5.2-11 is an extra equation obtained by subtracting Eq. 5.2-10 from the original
equation of continuity.
The principal result of this section is that the equation of motion in terms of the
stress tensor, summarized in Appendix Table B.5, can be adapted for time-smoothed tur-
bulent flow by changing all v to v and p to p as well as r to ~r = rf + rf in any of the
{ { /y i}
coordinate systems given.
We have now arrived at the main stumbling block in the theory of turbulence.
The Reynolds stresses rf above are not related to the velocity gradients in a simple
way as are the time-smoothed viscous stresses r ^ in Eq. 5.2-9. They are, instead, com-
plicated functions of the position and the turbulence intensity. To solve flow prob-
lems we must have experimental information about the Reynolds stresses or else
resort to some empirical expression. In §5.4 we discuss some of the empiricisms that
are available.
Actually one can also obtain equations of change for the Reynolds stresses (see Prob-
lem 5D.1). However, these equations contain quantities like v'jVJvl. Similarly, the equa-
tions of change for the v\v v contain the next higher-order correlation v-vjvlvi, and so
} k
on. That is, there is a never-ending hierarchy of equations that must be solved. To solve
flow problems one has to "truncate" this hierarchy by introducing empiricisms. If we
use empiricisms for the Reynolds stresses, we then have a "first-order" theory. If we in-
troduce empiricisms for the v'jVJvl, we then have a "second-order theory," and so on.
The problem of introducing empiricisms to get a closed set of equations that can be
solved for the velocity and pressure distributions is referred to as the "closure problem."
The discussion in §5.4 deals with closure at the first order. At the second order the "k-E
empiricism" has been extensively studied and widely used in computational fluid
mechanics. 2

§5.3 THE TIME-SMOOTHED VELOCITY PROFILE NEAR A WALL
Before we discuss the various empirical expressions used for the Reynolds stresses, we
present here several developments that do not depend on any empiricisms. We are con-
cerned here with the fully developed, time-smoothed velocity distribution in the neigh-
borhood of a wall. We discuss several results: a Taylor expansion of the velocity near the
wall, and the universal logarithmic and power law velocity distributions a little further
out from the wall.
The flow near a flat surface is depicted in Fig. 5.3-1. It is convenient to distinguish
four regions of flow:

• the viscous sublayer very near the wall, in which viscosity plays a key role
• the buffer layer in which the transition occurs between the viscous and inertial
sublayers
• the inertial sublayer at the beginning of the main turbulent stream, in which viscos-
ity plays at most a minor role
• the main turbulent stream, in which the time-smoothed velocity distribution is
nearly flat and viscosity is unimportant
It must be emphasized that this classification into regions is somewhat arbitrary.

J. L. Lumley, Adv. Appl. Mech., 18,123-176 (1978); С G. Speziale, Ann. Revs. Fluid Mech., 23,
2
107-157 (1991); H. Schlichting and K. Gersten, Boundary-Layer Theory, Springer, Berlin, 8th edition (2000),
pp. 560-563.