Page 79 - Computational Fluid Dynamics for Engineers

64 2. Conservation Equations

St = ,„ ^ „ , (2.4.39b)
-
QCp(T w T e)u e
Whereas the integration of boundary-layer equations across the shear layer leads
to ordinary differential equations that are much easier to solve than partial-
differential equations, their solutions require auxiliary equations. The momen-
tum integral equation has three unknowns, #, H and cj, and the energy integral
equation has two unknowns, 6T and St; their solutions require at least two other
equations to solve the momentum equation and one for the energy equation.
While there are several useful integral methods developed for two-dimensional
flows that require much less computer time than the differential methods, they
are limited to simple flows [3, 7]. These limitations are not present in differential
methods and, although the partial differential form of the boundary-layer equa-
tions is more difficult to solve, its use is preferred in this book and in current
engineering practice and the solution procedures are described in some detail
in Chapter 7.

2.5 Stability Equations

As will be explained in detail in Chapter 8, a turbulent boundary layer can de-
velop from a laminar boundary layer by the gradual amplification of infinitesimal
disturbances within the boundary layer or introduced from the external flow or
by imperfections in the body surface. At first these disturbances are so weak
that they have practically no influence on the mean flow but they gradually
increase until the flow is distorted and the fluctuating velocity and pressure
characteristics of turbulent flow appear. Transition can result from the growth
of these disturbances in the boundary layer and takes place over a streamwise
distance that depends on the boundary conditions.
The behavior of these disturbances within a laminar boundary layer can
be studied by small-disturbance theory, as will be discussed briefly in this sec-
tion and in more detail in Chapter 8. To derive the stability equations for
two-dimensional mean flows with two-dimensional disturbances, assume that u,
v and p in the two-dimensional form of Eqs. (2.2.1) to (2.2.3) represent the
instantaneous components of the flow properties and, as in the discussion of
turbulence in Section 2.3, these components are divided into a mean-flow term
and a fluctuating term so that the instantaneous velocity components are u + v!
f
and v + v r and the instantaneous pressure is p + p . The mean-flow velocity and
pressure terms satisfy the boundary-layer equations for a steady laminar flow
given by Eqs. (2.4.33) and (2.4.34) with the Reynolds stress term neglected and
with overbars on u, v and p.
Next consider Eqs. (2.2.1) to (2.2.3) with instantaneous velocity components
and pressure expressed in terms of their mean and fluctuating components. Since
1
u ', v' and p' are small, their squares and products can be neglected. Noting that