Page 82 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 67

4.1 Helmholtz Instability

Now we will study the stability of an inviscid, incompressible, parallel
fluid flow, containing a velocity discontinuity, following [22]. Precisely,

we will suppose that, above the Oz axis, the fluid moves with a uniform
velocity U in the positive sense and, below, it moves with a uniform
velocity of equal magnitude but in the opposite sense. In this case, the
Oz axis represents a discontinuity surface for the velocity and it is the
site of a vortex sheet of uniform circulation 2U per unit of width. We
remember that the circulation is

= [veds

where V is the magnitude of the velocity of the fluid and ds is the arc
element along a closed curve encircling the vortex.
Such a vortex sheet is unstable 1e., if a displacement happens the
sheet will go away and will not return to its initial position. This could be
shown by analytical studies, considering small sinusoidal perturbations.

Here we will numerically analyze the time evolution of such perturba-
tions.
We divide the vortex sheet into segments of equal length A on Ox and
each segment will be divided into m equispaced discrete vortices. As
the total circulation per unit length is 2U, each discrete vortex has the

circulation 2UA/m. We will suppose that at the initial moment these
vortices are displaced from their initial positions y, = 0 to the positions

2
ne = asin (| nee k=---—2,-1,0,1,::- (2.2)

Let us consider the row of vortices containing the vortices k, k +m,
k + 2m,... The complex potential generated by this row is
_
oo )
=e)
wr(z) = » a mn log(z — z, — nA) = i—- log sin m (2 ;
= 24)

DN
Thus the complex potential generated by all the m rows which compose
the sheet is

w(2) = Dol )= Di tg sin EE)

Replacing this potential in the relation

dw ;
— =u-—iv,
dz