Page 56 - Computational Fluid Dynamics for Engineers

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^fe# Conservation Equations

2.1 Introduction

This chapter considers the conservation equations upon which the structure
of fluid mechanics has been built. We assume that all significant aspects of an
incompressible or compressible flow can be adequately described by the solutions
of the conservation equations known as the Navier-Stokes equations. No serious
objection to this principle has ever been advanced, and the validity of these
equations has been established in so many instances that we may regard as an
act of faith and have full confidence in it.
The Navier-Stokes equations are based on the principles of conservation of
mass, momentum and energy and are presented in the following section for
three-dimensional flows in both differential and integral forms. Detailed deriva-
tions can be found in various textbooks such as [1-5], so Section 2.2 does not
attempt to provide detailed derivations. This brief presentation emphasizes the
equations formulated in both differential and integral forms. Their representa-
tion in vector form is also given.
Since most flows are turbulent with fluctuations of pressure, temperature
and velocity over a wide range of frequencies, the solution of the Navier-Stokes
equations of Section 2.2 presents a formidable challenge which has so far not
been met in a wholly satisfactory way and is unlikely to be achieved for the
boundary conditions of real engineering flows in the foreseeable future. As a
consequence, it is common practice to average the equations so that the equa-
tions lose their time dependence. The resulting equations include correlations of
fluctuation terms, as discussed in Section 2.3, and these require the assumptions
described in Chapter 3. The time-averaged equations, usually called Reynolds
Averaged Navier-Stokes (RANS) equations, are approximate representations of
flows, and the proper application of the numerical solutions requires that one
be familiar with the assumptions and approximations that have been made and
their effect on the accuracy of the numerical solutions.