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Exact Theory of Material Spike Formation in Flow Separation

The document discusses the exact theory of material spike formation in flow separation, highlighting the Lagrangian backbone's role in the development of material spikes. It presents findings on time-aperiodic flow and the connection of separation points to the wall, challenging traditional off-wall separation criteria. The document includes figures illustrating the evolution of separation and provides supplementary materials for further visualization.

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15 views1 page

Exact Theory of Material Spike Formation in Flow Separation

The document discusses the exact theory of material spike formation in flow separation, highlighting the Lagrangian backbone's role in the development of material spikes. It presents findings on time-aperiodic flow and the connection of separation points to the wall, challenging traditional off-wall separation criteria. The document includes figures illustrating the evolution of separation and provides supplementary materials for further visualization.

Uploaded by

asdafg_fewerwr
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Exact theory of material spike formation in flow separation 69

(a) (b) (c)


0.4
0.3
y 0.2
0.1
0
(d) (e) (f)
0.4
0.3
y 0.2
0.1
0
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
x x x

F IGURE 11. (Colour online) Lagrangian backbone B(t) of the separation spike (red)
extracted from κ 15
0 (z) (cf. figure 10d) at different times, along with the streamlines (blue)
and the material lines initially parallel to the wall (black). The full material evolution is
available as supplementary movie 5.

lines initially parallel to the wall (black). The full material evolution is available as
supplementary movie 5. The backbone curve that we locate acts as the theoretical
centrepiece of the material spike from its birth to its developed Lagrangian shape.
Figure 11 confirms also that the Prandtl point (i.e. where a singular streamline
connects to the wall) is unrelated to flow separation in unsteady flows.

6.2. Time-aperiodic flow


Here, we consider the general time-aperiodic rotating–translating cylinder flow with
the parameter values Ω = 3.5, U0 = 0.3, β = 0.5 and ωc = 2π/5, which appears
to create moving separation. From the classic asymptotic perspective, therefore,
separation is believed to be off-wall in this case. Therefore, prior separation criteria
seek an off-wall separation point in the flow.
Figure 12 shows the initial position B (t0 ) of the Lagrangian backbone of separation
https://doi.org/10.1017/jfm.2018.206 Published online by Cambridge University Press

for two different integration times. Panels (a,b) show the Lagrangian curvature change
fields κ tt00 +T (z) for t0 = 0, T = 7 and T = 8. Panels (c,d) show the contours of these
two scalar fields. The initial position B (0) of the backbone of separation is shown
in red and the zero set of κ tt00 +T (z) is shown in black. Remarkably, the Lagrangian
backbone of separation connects to the wall, consistent with the findings of Surana
& Haller (2008) in an appropriate frame, with the Lagrangian spiking point satisfying
the formulae in Proposition 2.
Figure 13 confirms that B (t) indeed acts as the evolving centrepiece of a forming
material spike. This is the case even though the upwelling in the lower material layers
is barely visible yet. The full material evolution is available as supplementary movie 6.

6.2.1. Off-wall perception of separation


Although we are primarily interested in the birth of separation, rather than its
long-term behaviour, we now analyse the same unsteady flow of § 6.2 for a longer
time interval. Similarly to figure 12, figure 14 shows the initial position of the
Lagrangian backbone of separation extracted from κ 10 0 (z). Figure 14(a) shows the
Lagrangian curvature change field κ 100 (z) and figure 14(b) shows the contours of the

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