Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 177

equilibrium is expressed by Euler's identity in the radial
direction and the pressure gradient

2

. The

∂ p ∂ r =−ρ u r

θ

( )

pressure distribution is in the

vortex , and outside of , where
is the pressure as . These relations can simply be
obtained by applying Bernoulli's equation along the circular
streamlines, and they show that the pressure does indeed
reach a local minimum on the axis of the circulatory
movement.

2 2 2

p ρ = p ρ − ω 2 a − r 8

∞ 0

∞ ρ − ω 2 a 4

8 r 2

< a

p ρ = p

> a

r r

p →∞ r

∞

Figure 3.27. Helical jet with canonical symmetry [JEO 95] (left). The
velocity field in spherical coordinates is
and , where ,
and , respectively, represent the radius, the axial and azimuthal angles.
The gray zones on the right are the contours . Zone 1 correctly defines
the vortex cone, but zone 2 is an artifact. The invariant is negative near
to the axis, and the criterion denotes a “hole” in the zone of rigid rotation
(bottom). This figure is adapted from [JEO 95]

()

v r =− Ψ ( x ) r , v

θ =−Ψ

( x ) r sin θ , v

φ =Γ

x r sin θ

= cos

′ x θ θ r

φ

Δ> 0

Q

Q