Environmental Engineering Reference
In-Depth Information
Equations (
11.1
) are completed by the continuity equation
@
@t
r r
ðÞ¼rQ
r
v
(11.2)
which represents the principle of mass conservation for the fluid. As outlined
in Chap. 3, the derivation of (
11.2
) is analogous to the derivation of the mass
conservation for species, utilizing the equation for fluid flux: j
v.
Q
represents
volume sinks and sources within the flow region. Not taken into account in both
(
11.1
) and (
11.2
) is the case in which the fluid phase does not cover the entire space.
That can be included by an additional factor, which represents the share of
the concerned phase on the entire volume. In the system of (
11.1
) and (
11.2
) the
number of equations equals the number of unknown variables
p
and v. From the
mathematical aspect, the most problematic term in the equations is the second term
of (
11.1
), which is nonlinear.
¼ r
The plot is produced by a
'viscosity_dyn.m'
, included in the accompanying
software.
There are few classical solutions for the complete set of Navier-Stokes (
11.1
)
and (
11.2
). Analytical solutions are mostly restricted to special circumstances. One
example is incompressible laminar flow through a pipe, see Sidebar 11.1. Incom-
pressible flow concerns fluids with constant density, for which the continuity (
11.2
)
simplifies to:
r
v
¼ Q
(11.3)
where the right hand side represents sources and sinks, measured as volumetric rate.
In absence of sources and sinks the equation becomes identical to the condition for
divergence-free vector fields:
r
v
¼
0
(11.4)
The
Reynolds
3
number Re
is defined by
Re ¼ v
char
, with a characteristic
velocity
v
char
, a characteristic length
H
char
and kinematic viscosity
H
char
=u
.In
situations in which the Reynolds number is above that value, the flow regime
becomes
turbulent
. In turbulent flow small disturbances are amplified, and the
assumption of zero velocity components perpendicular to the main flow direction
is not valid anymore. For turbulent flow there are no analytical solutions of the
Navier-Stokes equations.
u ¼ =r
3
Osborne Reynolds (1842-1912), English physicist.
