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4.2 Continuous phase Lagrangian modelling
Helmholtz (1858; see Sarpkaya, 1989) showed that inviscid vortex
lines are also material lines, and therefore always composed of
the same fluid elements. This provided the basis for the
modelling of vortical flows represented as assemblages of
vortices of infinitesimal cross section embedded in a potential
flow. By tracking the vortices between time steps in a
Lagrangian manner the whole flow field is known in principle, so
long as the inviscid approximation is reliable. This method is
now termed the Discrete Vortex Method (DVM) , of which a brief
account and one specific application to two-plane flow are
discussed below. For detailed descriptions refer to Sarpkaya
(1989) and Leonard (1980) .
We begin with the vorticity equation describing advection,
rotation or, more strictly, redistribution, and diffusion of
vorticity u>
^ + (g-V) w-(o.Vg) =vV 2 o>
ot
Here q is the velocity vector and V 2 the Laplacian operator. In
two dimensional flows, the vortex rotation/stretching
contribution (last term LHS) is identically zero so the equation
reduces to a simple advective-dif fusive balance which, when nondimensionalised
on representative length and velocity scales (L' ,
W say) gives
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