Environmental Engineering Reference
In-Depth Information
Once a particular discretization method has been
adopted a set of nonlinear algebraic equations is obtained.
This nonlinearity and the lack of an explicit equation
for pressure are the main difficulties that have to be
addressed in devising a robust solution algorithm for
these equations. The SIMPLE algorithm of Patankar and
Spalding (1972) used a derived, approximate pressure
equation that was solved separately from the velocity
equations. In essence, the equations for the momen-
tum equations and the pressure equation were solved
sequentially - they were segregated. It rapidly became the
most popular technique andwas themainstay of commer-
cial packages until recently. During the 1980s researchwas
conducted (Schneider and Raw, 1987; Wright, 1988) on
alternatives that solved for velocity and pressure simulta-
neously and removed the necessity for deriving a pressure
equation. Unsegregated, or coupled, solvers have now
been adopted for some commercial codes and have deliv-
ered significant increases in computational efficiency and
robustness, although often at the expense of increased
memory requirements.
resolve and track the smallest eddies in the energy cascade.
The amount of computation time needed scales with the
number of cells in the mesh, so for practical flows this
approach requires computing power that is not available
at present and may not be available for many decades.
A first level of approximation can be made through
the use of large eddy simulations (LES) (Versteeg and
Malalasekera, 2007), which make use of a length scale to
differentiate between larger and smaller eddies. The larger
eddies are resolved directly through the use of an appro-
priately refined grid. The smaller eddies are not directly
predicted, but are accounted for through what is known
as a sub-grid-scale model (Smagorinsky, 1963). This
methodology can be justified physically through the argu-
ment that large eddies account formost of the effect on the
mean flow and are highly anisotropic whereas the smaller
eddies are less important and mostly isotropic. Care is
needed in applying these methods as an inappropriate
filter or grid size and low accuracy spatio-temporal dis-
cretization can produce spurious results. Otherwise, LES
is not much more than an inaccurate laminar flow simu-
lation. The default filter size in commercial CFD software
is often related directly to the local grid size. Although less
computationally demanding than DNS, LES still requires
fine grids and small time steps and consequently comput-
ing resources that limit its use to research questions.
In view of the demands of DNS and LES, most tur-
bulence modelling still relies on the concept of Reynolds
averaging where the turbulent fluctuations are averaged
out and included as additional, modelled terms in the
Navier-Stokes equations.
Reynolds averaging considers the
in
stantaneous veloc-
ity,
u
, as consisting of an average,
u
, and a fluctuating
component,
u
as
6.2.5 The turbulence-closureproblemand
turbulencemodels
One of the fundamental phenomena of fluid dynamics is
turbulence. As the Reynolds number (the ratio of inertial
forces to viscous forces) of a flow increases, random
motions are generated that are not suppressed by viscous
forces. The resulting turbulence consists of a hierarchy of
eddies of differing sizes and orientations within the flow.
This hierarchy forms an energy cascade which extracts
energy from the mean flow into large eddies and in
turn smaller eddies extract energy from these which is
ultimately dissipated via viscous forces.
In environmental flows, turbulence is virtually ubiq-
uitous. The rough boundaries and complex topologies
usually found in surface water and atmospheric flows
generate significant turbulence. Turbulent flows are par-
ticularly good at mixing and dispersing pollutants in
environmental flow. Modelling turbulence accurately is
certainly the most important remaining challenge in
fluid dynamics.
In theory it is possible to predict all the eddy structures
from the large ones down to the smallest by solving
Equations 6.1 and 6.2 using the techniques described in
the preceding sections. This method is known as direct
numerical simulation (DNS) (Versteeg andMalalasekera,
2007) and relies on the finite volume mesh being able to
u
,
u
=
u
+
which is shown schematically in Figure 6.2. This substi-
tution can then be made in the Navier-Stokes equations.
A time-average can then be applied to the various terms
such as:
∂
u
∂
(
u
+
u
)
∂
u
x
=
=
x
,
(6.3)
∂
∂
x
∂
which is straightforward, but the nonlinear term is not so
straightforward:
(
u
i
u
j
)
∂
+
∂
∂
(
u
i
u
j
)
∂
∂
(
u
i
u
j
)
∂
=
.
(6.4)
x
x
x
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