Page 41 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

24 SLENDER STRUCTURES AND AXIAL FLOW
Equations (2.62) and (2.63) together with appropriate boundary conditions, including
equations matching the motion of a moving boundary (which could be part of the structure
of interest), should in principle be sufficient to solve problems involving incompressible
fluids. Similarly for compressible fluids, but the equations in this case are more complex
and will not be presented here. Possible boundary conditions for a body surface moving
with velocity v, in the fluid are

V - n = v,,,. n and V x n = v,, x n, (2.66)

the first matching the normal components of fluid and solid-surface velocities, and the
second being a form of the no-slip boundary condition, matching fluid and body velocities
parallel to the surface; n is the unit normal to the surface.
By ‘solution’ of the fluid equations we mean the determination of the velocity and
pressure fields, V and p. For fluid-structure interaction problems in which the forces
induced by the fluid on the structure are the only concern, most of the information on V
and p is ‘thrown away’. This is because the forces on the structure may be determined by
the pressure and viscous stresses on the body surjuce, cf. equations (2.64) and (2.65). This
allows for approximate treatment of some classes of problems, which will be discussed
in what follows. Indeed, the rest of this preamble will introduce, in general terms, some
of the broad classes of admissible simplifications and hopefully guide the reader towards
other ones.
The topic of tcrrbiifent fIows [subsection (f)] is treated at considerably greater length
than the other classes of flows. The reasons for this anomaly are that turbulence is more
complex and generally less well remembered than the rest, at least by those not in constant
touch with it. Nevertheless, the concepts and some of the relations to be recalled will be
needed later on, e.g. in treating turbulence-induced vibrations of pipes and cylinders in
axial flow; see Chapters 8 and 9 in Volume 2.

fa) High Reynolds number flows; ideal flow theory
If U is a characteristic flow velocity (e.g. a mean flow velocity in the system) and D
a characteristic dimension, the Reynolds number is Re = UD/u. If equation (2.63) is
written in dimensionless form, the last term is divided by Re; hence, for sufficiently
high Re this term is negligible, and the Navier-Stokes equations reduce to the so-called
Euler equations. Thus, away from any solid boundaries, the fluid is considered to be
essentially inviscid. Close to a boundary, in the boundary layer, the effects of viscosity
are predominant, but, they may be treated separately. In such cases, precluding situations
of large-scale turbulence and separated flow regions, the pressure field is determined as
if the flow were inviscid and then the shear stresses on the body are determined by
boundary layer theory or via empirical information.’ This is the treatment adopted for
slender cylindrical structures in axial flow in Chapters 8 and 9. Strictly, this approach
constitutes but a first approximation; in general, the boundary-layer and inviscid-flow
calculations should be matched iteratively.
For sufficiently high Re, the flow becomes turbulent and, if the effects of turbulence
cannot be ignored, this introduces new complexity [see subsection (01.

‘The key idea making this possible is that of a constant pressure across the boundary layer.