Abstracts - KTH Mechanics

6
The dispersal, decay and instability of multiple trailing-line
vortices
Peter W. Duck ∗
The importance of the dispersal, decay and breakdown of trailing-line vortices is
now a key aerodynamic issue. The classical flow that has spawned much research
activity over many years has been the (viscous) solution of Batchelor 1 , describing
the downstream evolution of a solitary vortex generated behind one side of a wing.
Various aspects of the stability of this flow have been presented over the years (see
for example 2 3 4 ). The dominant mode of instability is generally of short wavelength
(and is therefore inviscid), and as a consequence growth rates are quite large (which
can lead to breakdown, which is advantageous from a practical point of view).
In this paper the approach adopted involves (i) the determination of the downstream
evolution of a multiple vortex system and (ii) an investigation of the shortwavelength
(inviscid) stability of this developing flow. The theory is mathematically
rational, being based on the assumption of large Reynolds numbers.
Consider first the downstream developing baseflow; assuming a uniform flow far
outside the trailing vortex system, it is then possible to reduce the Navier-Stokes
equations to the following nondimensional form:
UX + vy + wz =0, UUX + vUy + wUz = Uyy + Uzz,
UvX + vvy + wvz = vyy + vzz − py, UwX + vwy + wwz = wyy + wzz − pz.
Here (U, v, w) are the velocity components in the (X, y, z) directions respectively,
and p is pressure; the scaling in the downstream direction X is long, and includes a
Reynolds number factor (implying a slow downstream flow evolution, compared to the
vertical y and crossflow z directions). Correspondingly the velocity components in the
y and z directions are an order smaller in Reynolds number than the X component.
The system is parabolic in nature (the streamwise diffusion terms have been eliminated),
and can therefore be solved in a ‘marching’ fashion, downstream from some
prescribed initial conditions. A variety of such conditions have been implemented,
including two counter-rotating vortices and various systems involving four vortices.
When considering the stability of the base states (as described above), it is possible
to perform a 2D (inviscid) stability analysis, which reduces the (in)stability problem to
the following partial eigenvalue problem (in which either the streamwise wavenumber
α is specified and the wavespeed c is treated as an eigenvalue, or vice versa):
∂2 ˜p
∂y2 + ∂2 ˜p 2 ∂U ∂ ˜p 2 ∂U ∂ ˜p
− −
∂z2 U − c ∂y ∂y U − c ∂z ∂z − α2 ˜p =0,
where ˜p is the pressure eigenfunction. Using this type of approach, it is feasible to
rapidly and efficiently investigate different vortex configurations, in order to optimise
trailing-vortex characteristics with regard to dispersal, decay and breakdown.
∗ School of Mathematics, University of Manchester, England
1 Batchelor, J. Fluid Mech. 20, 645 (1964).
2 Lessen et al. J. Fluid Mech. 63, 743 (1974)
3 Lessen &Paillet J. Fluid Mech. 65, 769 (1974)
4 Duck &Foster, ZAMP 31, 524 (1980)