fluid_mechanics

306 Chapter 6 ■ Differential Analysis of Fluid Flow
Potential flow solutions
are always approximate
because
the fluid is assumed
to be frictionless.
V6.10 Potential
flow
Determining the velocity potential or stream function for a given body shape is a much more
complicated problem.
It is possible to extend the idea of superposition by considering a distribution of sources and
sinks, or doublets, which when combined with a uniform flow can describe the flow around bodies
of arbitrary shape. Techniques are available to determine the required distribution to give a
prescribed body shape. Also, for plane potential flow problems it can be shown that complex
variable theory 1the use of real and imaginary numbers2 can be effectively used to obtain solutions
to a great variety of important flow problems. There are, of course, numerical techniques that can
be used to solve not only plane two-dimensional problems, but the more general three-dimensional
problems. Since potential flow is governed by Laplace’s equation, any procedure that is available
for solving this equation can be applied to the analysis of irrotational flow of frictionless fluids.
Potential flow theory is an old and well-established discipline within the general field of fluid
mechanics. The interested reader can find many detailed references on this subject, including Refs.
2, 3, 4, 5, and 6 given at the end of this chapter.
An important point to remember is that regardless of the particular technique used to obtain
a solution to a potential flow problem, the solution remains approximate because of the
fundamental assumption of a frictionless fluid. Thus, “exact” solutions based on potential flow
theory represent, at best, only approximate solutions to real fluid problems. The applicability of
potential flow theory to real fluid problems has been alluded to in a number of examples considered
in the previous section. As a rule of thumb, potential flow theory will usually provide a reasonable
approximation in those circumstances when we are dealing with a low viscosity fluid moving at
a relatively high velocity, in regions of the flow field in which the flow is accelerating. Under
these circumstances we generally find that the effect of viscosity is confined to the thin boundary
layer that develops at a solid boundary. Outside the boundary layer the velocity distribution and
the pressure distribution are closely approximated by the potential flow solution. However, in
those regions of the flow field in which the flow is decelerating 1for example, in the rearward
portion of a bluff body or in the expanding region of a conduit2, the pressure near a solid boundary
will increase in the direction of flow. This so-called adverse pressure gradient can lead to flow
separation, a phenomenon that causes dramatic changes in the flow field which are generally not
accounted for by potential theory. However, as discussed in Chapter 9, in which boundary layer
theory is developed, it is found that potential flow theory is used to obtain the appropriate pressure
distribution that can then be combined with the viscous flow equations to obtain solutions near
the boundary 1and also to predict separation2. The general differential equations that describe
viscous fluid behavior and some simple solutions to these equations are considered in the remaining
sections of this chapter.
6.8 Viscous Flow
To incorporate viscous effects into the differential analysis of fluid motion we must return to the
previously derived general equations of motion, Eqs. 6.50. Since these equations include both
stresses and velocities, there are more unknowns than equations, and therefore before proceeding
it is necessary to establish a relationship between the stresses and velocities.
6.8.1 Stress–Deformation Relationships
For incompressible Newtonian fluids it is known that the stresses are linearly related to the rates
of deformation and can be expressed in Cartesian coordinates as 1for normal stresses2
s xx p 2m 0u
0x
s yy p 2m 0v
0y
s zz p 2m 0w
0z
(6.125a)
(6.125b)
(6.125c)