Page 33 - Introduction to Computational Fluid Dynamics

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INTRODUCTION
Before leaving this section, it is important to note that since p must equal p
in a continuum (see Equation 1.11), the former must essentially be the hydrostatic
pressure, irrespective of the flow considered. Mathematically, therefore, we may
define p as
1 1
p =− (σ x + σ y + σ z ) = (p + p + p ), (1.17)
x
z
y
3 3
2
2
2
2
where p is a solution to ∂ p/∂x = 0, p is a solution to ∂ p/∂y = 0, and p is
z
y
x
2
2
a solution to ∂ p/∂z = 0.
In effect, therefore, the equations of motion (also called the Navier–Stokes
equations) valid for both continuum and discrete space must read as
Du ∂(p − q) ∂τ xx ∂τ yx ∂τ zx
ρ =− + + + , (1.18)
Dt ∂x ∂x ∂y ∂z
D v ∂(p − q) ∂τ xy ∂τ yy ∂τ zy
ρ =− + + + , (1.19)
Dt ∂y ∂x ∂y ∂z
D w ∂(p − q) ∂τ xz ∂τ yz ∂τ zz
ρ =− + + + , (1.20)
Dt ∂z ∂x ∂y ∂z
where q is given by Equation 1.13 for incompressible (viscous or inviscid) flow and
by Equation 1.15 for compressible flow. In spite of this recognition, the equations
are further discussed (in conformity with standard textbooks) for a continuum only
with λ 1 = 0, but the existence of finite λ 1 will be discovered in Chapter 5 where
solutions in discrete space are developed.
1.6 Outline of the Book
The book is divided into nine chapters. Chapter 2 deals with one-dimensional (1D)
conduction in steady and unsteady forms. In this chapter, the main ingredients of a
numerical procedure are elaborately introduced so that familiarity is gained through
very simple algebra. Chapter 3 deals with the 1D conduction–convection equation.
This somewhat artificial equation is considered to inform the reader about the nature
of difficulty introduced by convection terms. The cures for the difficulty developed
in this chapter are used in all subsequent chapters dealing with solution of transport
equations.
Chapter 4 deals with convective transport through boundary layers. This is an
important class of flows encountered in fluid dynamics and heat and mass transfer.
The early CFD activity relied heavily on solution of two-dimensional (2D) parabolic
equations (a subset of the complete transport equations) appropriate to boundary
layer flows. In this chapter, issues of grid adaptivity and turbulence modelling are
introduced for external wall boundary layers and free-shear layers and for internal
(ducted) boundary layer development.