Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 190

The only specific situation where it is possible to establish an
exact relation between

( )

and is when the

λ 2 λ cR , λ cI

j , j and j in equation [3.40] form an

orthogonal basis, in which case

eigenvectors

R cR cI

⎡

2

⎤

⎛ ⎞ λ

2

⎢

cR

⎥ [3.68]

λλ = − 1

⎜ ⎟

2 cI

λ

⎢ ⎥

⎝ ⎠

⎣ ⎦

cI

Let us return to the example of the Burgers vortex
discussed by [CHA 05] to illustrate the differences that may
remain between the criteria , and detection by

compactness as shown in equation [3.43]. The Burgers vortex
is widely used in the literature in this field - among other
things, to model the turbulence scales. Consider the
potential flow

, λ < 0

Q > 0

2

in the cylindrical
coordinates [DRA 06]. This irrotational velocity field, where
γ

u

=− γ r and u = 2 γ z

z

represents the strain rate, formalizes the stagnation flow
arriving at a cylinder. Suppose that the cylinder is replaced
by a vortex, which causes a recirculation

at large Γ

distances. This configuration suggests the form

( ) ( )

of the tangential component of the velocity
field associated with the boundary condition

u =Γ 2 π r f r

θ

()

. By
substituting these relations into the Navier-Stokes equation
in cylindrical coordinates, we find

f ∞= 1

df

2

⎛ r γ

ν

2

⎞ df

r + − 1 = 0

⎜ ⎟

2

dr dr

⎝ ⎠

( )

which accepts the solution . The velocity

field associated with a Burgers vortex is, finally

2

f =− 1exp − γ r 2 ν

u =−

Γ

γ r

r

( )

( )

2

u = 1exp − − γ r 2 ν

θ

2
2

π
γ

r

u = z

z