Page 142 - Bird R.B. Transport phenomena

126 Chapter 4 Velocity Distributions with More Than One Independent Variable

§4.3 FLOW OF INVISCID FLUIDS BY USE
OF THE VELOCITY POTENTIAL 1
Of course, we know that inviscid fluids (i.e., fluids devoid of viscosity) do not actually
exist. However, the Euler equation of motion of Eq. 3.5-9 has been found to be useful
for describing the flows of low-viscosity fluids at Re > > 1 around streamlined objects
and gives a reasonably good description of the velocity profile, except very near the
object and beyond the line of separation.
Then the vorticity equation in Eq. 3D.2-1 may be simplified by omitting the term
containing the kinematic viscosity. If, in addition, the flow is steady and two-dimen-
sional, then the terms д/dt and [w • Vv] vanish. This means that the vorticity w = [Vx v]
is constant along a streamline. If the fluid approaching a submerged object has no vortic-
ity far away from the object, then the flow will be such that w = [V X v] will be zero
throughout the entire flow field. That is, the flow will be irrotational.
To summarize, if we assume that p = constant and [V X v] = 0, then we can expect
to get a reasonably good description of the flow of low-viscosity fluids around sub-
merged objects in two-dimensional flows. This type of flow is referred to as potential flow.
Of course we know that this flow description will be inadequate in the neighbor-
hood of solid surfaces. Near these surfaces we make use of a different set of assump-
tions, and these lead to boundary-layer theory, which is discussed in §4.4. By solving the
potential flow equations for the "far field" and the boundary-layer equations for the
"near field" and then matching the solutions asymptotically for large Re, it is possible to
develop an understanding of the entire flow field around a streamlined object. 2
To describe potential flow we start with the equation of continuity for an incom-
pressible fluid and the Euler equation for an inviscid fluid (Eq. 3.5-9):
(continuity) (V • v) = 0 (4.3-1)

(motion) p[^- + V y - [v X [V X v]]) = -V9> (4.3-2)
\dt I
In the equation of motion we have made use of the vector identity [v • Vv] = V\v 2 —
[v X [V X v]] (see Eq. A.4-23).
For the two-dimensional, irrotational flow the statement that [V X v] = 0 is

(irrotational) -r 1 - -r 1 = 0 (4.3-3)
dy dX

and the equation of continuity is
(continuity) -г 1 + -г 1 = О (4.3-4)

The equation of motion for steady, irrotational flow can be integrated to give
2
(motion) \p{v\ + V ) + & = constant (4.3-5)
That is, the sum of the pressure and the kinetic and potential energy per unit volume is
constant throughout the entire flow field. This is the Bernoulli equation for incompress-
ible, potential flow, and the constant is the same for all streamlines. (This has to be con-
trasted with Eq. 3.5-12, the Bernoulli equation for a compressible fluid in any kind of
flow; there the sum of the three contributions is a different constant on each streamline.)

1
R. H. Kirchhoff, Chapter 7 in Handbook of Fluid Dynamics (R. W. Johnson, ed.), CRC Press, Boca
Raton, Fla. (1998).
2
M. Van Dyke, Perturbation Methods in Fluid Dynamics, The Parabolic Press, Stanford, Cal. (1975).