Tim Jeter MAE 741 Final Presentation

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24 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Tim Jeter

MAE 741 Final Presentation

1

Classical Vorticity Criteria





Hill’s Spherical Vortex

Streamlines


Analytic solution of the Euler
equations that yield an
axi
-
symmetric
spherical vortex in a uniform flow.

Green, Rowley, Smits 2010

Q Criterion
(proposed by Hunt, Wray,
and
Moin
)

Rate of rotation
dominates over the strain
rate.

Q > 0

Green, Rowley, Smits 2010

2

Finite Time Lyapunov Exponent

-
Lagrangian analysis of the velocity field.

-
Scalar quantity representing the maximum rate of
separation between nearby particle trajectories.

-
Calculated at every point in space, large values represent
regions that distinctly diverge from one another.

-
FTLE of Hill’s Spherical vortex.


-
The FTLE has been resolved
enough to show the full structure
by combining the positive time
integration and the negative time
integration.

3

Lagrangian Coherent Structures


-

Sharp representations of high separation.

-

Small flux across the structures.

(
Shadden
)

-

Separatricies where particles will tend to fall away
from the ridge in a transverse fashion.


LCS can be categorized:


pLCS

as the positive time
integration of the FTLE


nLCS

as the negative time
integration of the FTLE

nLCS

pLCS

Shadden,2005

4

Eulerian Criteria


Q
-
criterion
(1988) Hunt, Wray, and
Moin


𝑄
=
1
2
𝜴
2

𝑺
2
>
0


Δ
-
criterion

(1990) Chong, Perry, and Cantwell



=
𝑄
3
3
+
det
𝛻
𝒗
2
2
>
0


λ
2
-
criterion

(1995)
Jeong

and
Hussain


𝜆
2
𝑺
2
+
𝜴
2
<
0

5

FTLE Strain Check


Haller(2010) exposes some of the vulnerabilities
associated with FTLE.


Not all LCSs are FTLE ridges and vise
-
versa.


Hyperbolic stretching near the FTLE ridges.







6

Experimental Setup

-
Fairing made of aluminum
and consisting of two
pieces cold welded
together.


-
Trailing edge pieces to
ensure smooth transitional
flow onto the panels
themselves.


-
Dye injection ports were
utilized to create more
means of flow visualization.

-


-
Panel and pitching shaft
are only pieces that
oscillate.

Motor

Aluminum

Mount

Fairing
Connector

Dye

Injection

NACA
-
0012

Fairing

Dye
Ports

10’’
Trap
-

Panel

Delrin

Trailing
Edge

Stabilizing
Bushing

End Plate and
Bushing

7

Green, Rowley, Smits 2011

Digital Particle Image
Velocimetry

(DPIV)


Seeded flow (glass
particulates).


Laser sheet to induce
reflections.


High
-
Speed camera to
capture staggered position
images.


Animation of vortex pair from a pump exit. (
Dantec
)

8

Multi
-
planar DPIV


121 planes uniformly spaced 2mm apart.


Normal to the z direction.


2mm thick laser sheet.


Green,
Rowley,
Smits 2011

-

3 sections were
obtained for use
with FTLE
calculations.

9

Dimensionless characterization.


Characterized by a couple dimensionless
numbers:


Strouhal number







Reynolds number




Relates the unsteadiness of the
panel to the fluid speed.

Relates the size of the panel to
the fluid’s viscous effects.

10

Wake Analysis


LCS analysis of vortex breakdown.


Varying Strouhal numbers.


Iso
-
surface interactions.


Trapezoidal Geometry effects.


3
-
D vorticity effects.

11


St = .17


St = .28

-
In the mid
-
span of the wake, we see that there is a clear correlation
between LCS interactivity and vortex break up.


-

Thrust producing formation of vortices occurs at mid
-
span, not at tips.


-
Force measurements needed for quantitative thrust analysis.

Green, Rowley, Smits 2011

Vortex Interactions

12

Vortex Interactions

-
Hyperbolic interactions of
LCS generated using FTLE.

-

Breakdown inferred around
2
times the chord length.

-

Clear breakdown near
1.4
times the chord length.

Green, Rowley, Smits 2011

13

Trapezoidal Geometry Effects


Additional vortex dynamics.


Span
-
wise, swept, edges of the panel.


Fluid rolling from the top surface toward the bottom.


Boundary layer is effected.


Occurs at all St.


At higher speeds, structures

became “trapped”, retaining

some coherence.

14

Error

-
Span wise velocity causing particles to
leave the plane.

-
Highest magnitude occurs at tips and in
one
Δ
t
approximately one half the original
particles are in the plane (2mm)

-
Assumption of zero span wise velocity
was made at the mid
-
span.


15

Conclusions


At the higher Strouhal number, a loss of the
characteristic LCS vortex core pattern was
observed to coincide with the structure
breakdown.


Saddle points that became merged at the
location of the vortex splitting.


Future Stereo
-
PIV required to fully understand
3
-
D flow.

16

References


Haller
, G. "An Objective Definition of a Vortex."
J. Fluid Mech.

525 (2005): 1
-
26.


Harris
, Miller, and Williamson. "Gallery
-

The Eye
-
catching Best of Fluid
Dynamics
-

Image 1
-

New Scientist."
Science News and Science Jobs from New
Scientist
. Web. 25 Apr. 2012. <http://www.newscientist.com/gallery/aps
-
physics
-
fluid
-
dynamics
>.


"Particle Image
Velocimetry

(PIV)."

Particle Image
Velocimetry
, PIV
.
N.p
.,
n.d
.
Web. 29 Apr. 2013.


Buchholz, James H. J.

The
Flowfield

and Performance of a Low Aspect Ratio
Unsteady
Propulsor
. Ann Arbor, MI: University Microfilms International, 2006.
Print.


Green, M.A., Rowley, C.W., and Smits, A.J.

The unsteady three
-
dimensional
wake produced by a trapezoidal panel
.

Journal of Fluid Mechanics.

685
. 117
-
145. (2011)


Green, M.A., Rowley, C.W., and Smits, A.J.

Using hyperbolic
Lagrangian

coherent structures to investigate vortices in
bioinspired

fluid
flows
.
Chaos
.

20
.
(017510) (2010)


Shadden
, Shawn C. "LCS Tutorial: Brief Overview."

LCS Tutorial: Brief Overview
.
Caltech, 15 Apr. 2005. Web. 15 Nov. 2012.
http://mmae.iit.edu/shadden/LCS
-
tutorial/overview.html
.


17