62 M. Serra, J. Vétel and G.

Haller

∇ · f = 0. Lagrangian spiking point: (sp , 0)

Steady Time-periodic: f (z, t + Tp ) = f (z, t) Temporally aperiodic
T = nTp , n ∈ N+
Z t0 +Tp Z t0 +T
vsssηη (sp , 0, t) dt = 0 vsssηη (sp , 0, t) dt = 0

 

 
 

vsssηη (sp , 0) = 0 Zt0t0 +Tp Zt0t0 +T

 

  
vssssηη (sp , 0) > 0 vssssηη (sp , 0, t) dt > 0 vssssηη (sp , 0, t) dt > 0
v (s , 0) < 0
 
 t0 
 t0
ssηη p 
 Z t0 +Tp

 Z t0 +T
vssηη (sp , 0, t) dt < 0 vssηη (sp , 0, t) dt < 0

 


 

t0 t0

TABLE 4. The same as the right column of table 2, but expressed in terms of averaged
Eulerian quantities.

P ROPOSITION 2. Over the finite time interval [t0 , t0 + T], we have the following.
(i) The initial position B (t0 ) of the Lagrangian backbone of separation can be
computed as the set of points s ∈ [s1 , s2 ], η ∈ [0, η1 ],

∂s κ t0 (s, η) = 0, η ∈ (0, η1 ],
 t0 +T

∂ κ t0 +T (s, η) < 0, η ∈ (0, η ],

ss t0 1
B (t0 ) := (4.6)
 κ
 t0

t0 +T
(s, η) > 0, η ∈ (0, η 1 ],
(s , η),

η = 0.
p

The Lagrangian spiking point (sp , 0) can be computed in terms of Lagrangian

quantities using the formulae in table 2, in terms of averaged Eulerian quantities
using the formulae in tables 3 and 4, or as the intersection of B (t0 ) with the
no-slip boundary (cf. (4.5)).
(ii) Later positions B (t) of the Lagrangian backbone of separation can be computed
as B (t) = Ftt0 (B (t0 )), t ∈ [t0 , t0 + T].
(iii) The Lagrangian spiking point (sp , 0)
https://doi.org/10.1017/jfm.2018.206 Published online by Cambridge University Press

Steady flow Time-periodic flow: f(z, t + Tp ) = f(z, t) Aperiodic flow

is fixed if T = nTp , n ∈ N+ , is fixed moves
and independent of t0 , T and independent of t0 , n depending on t0 , T.

By Proposition 1, the Lagrangian backbone of separation is objective. Although the

analytic formulae in tables 3 and 4 involve higher derivatives of the velocity field, the
spiking point can also be identified as the intersection of B (t0 ) with the wall (cf. (4.5))
with low numerical effort.
While our main interest is to capture the birth of material spikes, in appendix D.3,
we also derive explicit formulae for the asymptotic (T → ∞) mean location of the
Lagrangian spiking point in the case of general unsteady velocity fields that admit a
finite asymptotic mean, such as periodic and quasiperiodic velocity fields. The latter
is relevant for several separation problems, such as separation behind a cylinder or a
backward facing step, which admit a finite number of dominant frequencies in their
Fourier spectra. In these flows, we find that the asymptotic Lagrangian spiking point
oscillates with respect to a well-defined mean position. We summarize these formulae