Mechanics of Fluids

358 Boundary layers, wakes and other shear layers
shape, and the sides of the elements can be straight or curved. The objective
of FEM is to determine the flow variables at selected points associated with
each element. These selected points are known as nodes, and may be positioned
at the corners, at the mid-side or at the centre of the element. Whereas
the FDM requires an orderly, structured grid, with the FEM the grid need
not be structured. The use of an unstructured grid allows complex geometrical
shapes to be handled without undue complication, and in this regard
provides FEM with a clear advantage over FDM. Another advantage of FEM
over FDM is in regard to its robust and rigorous mathematical foundations,
which embrace more precise definitions of accuracy than are inherent in
FDM. Perhaps the main drawback of FEM is that it is intellectually much
more demanding; in particular, with FDMs the relationship between the
derived algebraic equations and the partial differential equations they replace
are much more self-evident than is the case of FEM.
Boundary element The defining feature of the BEM for solving numerically the partial differmethod
ential equations is that all nodes are positioned on the boundary of the flow
field, and there are no interior elements.
Finite-volume method
The starting point for the FVM is the discretization of the integral forms of
the flow equations. This approach is particularly suited to the solution of
flow fields containing discontinuities, such as a shock wave in a compressible
flow field. The flow field is divided into cells, and the conservation equations
are solved numerically to determine the magnitude of the flow variables
at the nodes defined for each cell. The FVM shares the advantage of the
FEM of accommodating an unstructured grid. Advocates of FVM claim
that it combines the best feature of the FEM, namely its ability to handle
complex geometries readily, with the virtue of the FDM, the simple and
self-evident relationships between the finite-difference formulations and the
partial differentials they replace.
PROBLEMS
8.1 Determine the ratios of displacement and momentum thickness
to the boundary layer thickness when the velocity profile
is represented by u/um = sin(πη/2) where η = y/δ.
8.2 A smooth flat plate 2.4 m long and 900 mm wide moves
lengthways at 6 m · s −1 through still atmospheric air of density
1.21 kg · m −3 and kinematic viscosity 14.9 mm 2 · s −1 . Assuming
the boundary layer to be entirely laminar, calculate the
boundary layer thickness (i.e. the position at which the velocity
is 0.99 times the free-stream velocity) at the trailing edge of the
plate, the shear stress half-way along and the power required to
move the plate. What power would be required if the boundary
layer were made turbulent at the leading edge?
8.3 A smooth flat plate, 2.5 m long and 0.8 m wide, moves length
ways at 3 m · s −1 through still water. The plate is assumed to