Page 87 - Computational Fluid Dynamics for Engineers

72 2. Conservation Equations

familiar and less perplexing. In a real external-flow problem, u e(x) is obtained
from the solution of the inviscid flow equations. It the external flow is affected
by the displacement effect of the shear layer, iteration between the external-flow
and shear-layer calculations is necessary, as discussed briefly in Chapter 7 and
in more detail in [12]. The alternative is to solve the Navier-Stokes equations
throughout the flowfield.
Internal flows (Fig. 2.5d) consist of shear layer or layers filling part or all at
the space between two solid boundaries. In this case the pressure distribution is
set predominantly by the displacement effect of the shear layer. It is convenient
to distinguish flows in which the shear layers fill the cross section and flows such
as that near the entrance to a duct (the left-hand part of Fig. 2.5d) where a
region of effectively inviscid flow obeying Bernoulli's equation remains.
It is also convenient to distinguish the "entrance region", in which the ve-
locities change with x, and the "fully developed" region far downstream in a
constant area duct, in which the velocities do not, but of course the pressure
continues to decrease with x. As will be shown in Chapter 7, the relevant bound-
ary conditions in each case are similar to those for external flows, except that it
is also necessary to satisfy the requirement of constant mass flow between the
solid surfaces.
The boundary conditions for hyperbolic equations are rather involved in
comparison to those for elliptic or parabolic equations. For this reason, their
discussion is postponed to Chapters 4 and 10. For a detailed discussion, the
reader is referred to Hirsch [11].

References

[1] Anderson, DA., Tannehill, J.C. and Pletcher, R.H.: Computational Fluid Mechanics
and Heat Transfer, Hemisphere Publishing Co., 1984.
[2] Arpaci, V.S. and Larsen, P.S.: Convection Heat Transfer, Prentice Hall, New Jersey,
1984.
[3] Cebeci, T.: Convective Heat Transfer, Horizons Pub., Long Beach, Calif, and Springer,
Heidelberg, 2002.
[4] Anderson, J.: Aerodynamics, McGraw Hill, N.Y., 1988.
[5] Anderson, J.: Introduction to Computational Fluid Dynamics. McGraw Hill, N.Y.,
1995.
[6] Pulliam, T.H.: Efficient Solution Methods for the Navier-Stokes Equations, Lecture
notes for the Von Karman Institute for Fluid Dynamics Lecture Series: Numerical
Techniques for Viscous Flow Computation in Turbomachinery Bladings, Brussels, Bel-
gium, Jan 20-24, 1986.
[7] Cebeci, T.: Analysis of Turbulent Flows. Elsevier, London, 2004.
[8] Blottner, F.G.: Significance of the Thin-Layer Navier-Stokes Approximation. In Nu-
merical and Physical Aspects of Aerodynamic Flows III, p. 184 (ed. T. Cebeci),
Springer, N.Y., 1996.
[9] Schlichting, H.: Boundary-Layer Theory. McGraw-Hill, N.Y., 1968.