Page 42 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 25

(b) Potential flow theory
Many interesting inviscid flows (e.g. a uniform flow approaching a body) are initially
irrotational, i.e. the vorticity, w, is everywhere zero: o = V x V = 0. Hence, by Kelvin’s
theorem, such flows remain irrotational;’ the flow is then referred to as potential flow and
is associated with the velocity potential, 4, where V = V4. Euler’s equations in this case
simplify to the well known unsteady Bernoulli, or Bernoulli-Lagrange, equation

a4 P
- + $2 + - = 0, (2.67a)
at P
where p is measured relative to the stagnation pressure of the free stream.$ This form of
the equation applies if there are no body forces. If there are, for example due to gravity,
the following form may be more useful:

a4
- + ;v2+ - +gz =o, (2.67b)
P
at P
where z is the vertical height. There exists a highly developed mathematical treatment of
potential flow - see, e.g. Lamb (1957), Streeter (1948), Milne-Thomson (1949, 1958),
Karamcheti (1966), Batchelor (1 967).
(el Very low Reynolds number flows

In this case, when Re + 0, inertial effects become negligible, and the Navier-Stokes
equations reduce to the equations of creeping flow,

vp = pv2v. (2.68)

A number of well known solutions exist, e.g. for the plane Couette and Poiseuille
flows, classical lubrication theory (Lamb 1957), Stokes flow past a sphere and constant
pressure-gradient laminar flow through pipes; but, surprisingly perhaps, not for low-Re
two-dimensional cross-flow over a cylinder (Stokes’ paradox).
Id) Linearized flows

In some problems there is one dominant steady flow-velocity component, while all others
are perturbations thereof, say induced by structural motion, e.g. V = Ui + v, where llvll <<
i
U; is the unit vector in the x-direction. In such cases, the Navier-Stokes equations may
be linearized and simplified considerably. Thus, if U is steady, i.e. not time-dependent,
and spatially uniform, the Navier-Stokes equations reduce to
av av 1
- + u- = -- v p + v v2v. (2.69)
at ax P
‘Interestingly, this is not so if there is a density gradient to the fluid!
iThus, the integration constant that would otherwise appear on the right-hand side reduces to zero. This
constant, C(t), is generally a function of time if, unusually, the hydrostatic pressure varies with time.