Skin Friction Topology in a Region Enclosed by Penetrable ...

Skin Friction Topology in a Region

Enclosed by Penetrable Boundary

Tianshu Liu, S. A. Woodiga Department of Mechanical and Aerospace Engineering

Western Michigan University, Kalamazoo, USA

Tian Ma Department of Mathematics

Sichuan University, Chengdu, China

Objectives

To obtain high-resolution skin friction

fields in regions with penetrable

boundaries on surface in complex 3D

separated flows

To examine the skin friction topology

by application of the Poincare-Bendixson

index formula

Topological Features in Skin Friction Fields

Isolated Singular Points

Separation & Attachment Lines

Diamond

cylinder

Diamond

cylinder

Junction Flow

S

S

S

N

N

Separation line Attachment line

Ambiguity in Oil Flow Visualization

Oil Flow Visualization

(Hunt et al. 1978) Conjectured Topology

Topological Constraints

Lighthill (1963): Suggesting the Poincare-Hopf Index Theorem

as a topological constraint

Hunt et al. (1978): Applying the Poincare-Hopf Index Theorem

to several cases, such as junction flows

The topological rules have been widely used as a guideline

in interpreting flow visualizations in 3D separated flows

(Tobak & Peake 1982,Chapman & Yates 1991, Délery 2001).

Foss (2004, 2007) has utilized holes punched on a collapsed

sphere to explain the flow through a boundary

Previous Studies

Problem with Boundary Flow

Milestones

Oil Film Thickness (h) Luminescent Intensity (g)

Quantitative Global Skin Friction Diagnostics Based on Luminescent Oil Visualization

03

hg

X

p

2

h

Xt

h 3

i

i

2i

i

Thin-Oil Film Equation:

2

3

i

ii

2

i

i a3

gg

X

p

Xa2

g

Xt

g

)2,1i(

)2,1i(

Surface in Object Space Image Plane

Projection onto Image Plane

k

ki2

i

2

3

i

n

ni

k

kik

k

x

h

a2

g

a3

gg

x

ph

xhg

xt

g

)a2/(gˆkk

Equivalent Skin Friction:

Projective Skin Friction:

ikik hˆ 0|h| ki

)a2/(g

Equivalent Skin Friction:

)g,x,x(f 21Pressure Gradient and Gravity Terms:

)g,x,x(fgt/g 21

Liu & Shen, “Fluid Flow and Optical Flow”, JFM, 2008

Liu et al. “Global Luminescent Oil-Film Skin Friction Meter,”

AIAA Journal 2008

Liu et al. “Skin Friction Topology in a Region Enclosed by

Penetrable Boundary,” Exp. Fluids, 2011

Form of Physics-Based Optical Flow Equation

Inverse Problem:

To determine the equivalent skin friction field

21

2

2

2

1212

dxdxdxdxfgt/g)(J

Variational Formulation

The functional with a smoothness constraint:

0fgt

g

xg 1

2

1

0fgt

g

xg 2

2

2

The Euler-Lagrange equations:

0n /where the Neumann condition

From two successive images, a snapshot solution is obtained.

Fusion of Snapshot Solutions

Heuristic Fusion Methods:

(1) Superposition for Direct Fusion

(2) Wavelet-Based Fusion in Transform Domain

A sequence of snapshot solutions captures the major skin

friction signatures in different regions at different moments

in the whole process of oil film evolution.

This is due to different time scales in different regions .

Observation:

Snapshot solutions are not invariant in the process of oil

evolution unless the process is self-similar t/sh

Experimental Setup

Dow Corning silicone oil

(100 or 200 cs) with

oil-based UV dye

Skin Friction Magnitude Skin Friction Vectors

Example: Square Cylinder Junction Flow

(fusion of 300 snapshot solutions)

Square Cylinder Junction Flow

Skin Friction Lines

The Poincare-Bendixson Index Formula

A conservation law between the number of the singular points

in a region and the number of the switch points on the boundary:

In simple notation:

Negative Switch Point Positive Switch Point

k

k

k

k

k

k ZZ2

11I

2/)Z#Z#(1S#N#

k

kI)C(w

Direct Consequence of the Two Major Theorems

(1) The Poincare Index Theorem for a Disk:

(2) The Poincare-Bendixson Theorem:

2/)he(1)C(w

)C(w : the winding number

where e is the number of the elliptical sectors and

h is the number of the hyperbolic sectors

Sectors in Disk:

H – Hyperbolic sector

E – Elliptical sector

P – Parabolic sector

Classical Example 1: Hairy Sphere Theorem

A sphere with a small hole deformed into a disk:

The hairy sphere theorem is a direct consequence of

the Poincare-Bendixson index formula in this case:

2S#N#

Classical Example 2: Topological Rule of Hunt et al.

for Junction Flow

The topological rule of Hunt et al. is a direct consequence of

the Poincare-Bendixson index formula in this case:

0S#N#

Low-Aspect-Ratio Rectangular Wing

with NACA0012 Airfoil Section



oil-based UV dye

Low-Aspect-Ratio Rectangular Wing

with NACA0012 Airfoil Section

Topological Evolution as Angle of Attack is Changed

Luminescent Oil Image Skin Friction Vectors

Low-Aspect-Ratio Wing

The Upper Surface at AoA = 18 deg

12/)Z#Z#(1S#N#

Zoomed-in Views on the Upper Surface at AoA = 18 deg

The Poincare-Bendixson

index formula is satisfied in

these local regions

Normal Skin Friction Vector Magnitude on Boundaries of

Polygon ABCDEA on the Upper Surface at AoA = 18 deg

Switch Points Zero-Crossing Points

Leading and Trailing Edges

ABC & DE Side Edges CD & EA

The Side Surface at AoA = 18 deg

02/)Z#Z#(1S#N#



these local regions

Topological Change Caused by a Roughness Strip

along the Leading Edge at AoA = 18 deg

22/)Z#Z#(1S#N#

Zoomed-in Views on the Upper Surface at AoA = 18 deg



these local regions

Normal Skin Friction Vector Magnitude on Boundaries of

Polygon ABCDA on the Upper Surface at AoA = 18 deg

Switch Points Zero-Crossing Points

Leading and Trailing Edges

AB & CD Side Edges BC & DA


The Evolution of Skin Friction Topology

as AoA is Changed

AoA = 0 deg AoA = 5 deg



as AoA is Changed

AoA = 12 deg AoA = 13.5 deg



as AoA is Changed

AoA = 14 deg AoA = 16 deg

Double-Delta Wing with Yaw of 6 deg

AoA = 16 deg

02/)Z#Z#(1S#N#

Zoomed-in View

AoA = 24 deg

Double-Delta Wing with Yaw of 6 deg

02/)Z#Z#(1S#N#

Zoomed-in View

Junction Flows



oil-based UV dye

The Topological Rule of

Hunt et al.:

0S#N#

Junction Flow over a Circular Cylinder

Zoomed-in Views

02/)Z#Z#(1S#N#

Junction Flow over a Square Cylinder

Zoomed-in Views

02/)Z#Z#(1S#N#

Junction Flow over a Diamond Cylinder

Zoomed-in Views

02/)Z#Z#(1S#N#

Conclusions

● The high-resolution skin friction diagnostics allows clear

identification of the isolated singular points enclosed

by a penetrable boundary on a surface and the boundary

switch points

● The number of the isolated singular points and the number

of the boundary switch points in all the cases satisfy

a conservation law described by the Poincare-Bendixson

index formula

Skin Friction Topology in a Region Enclosed by Penetrable ...

Documents

Transcript of Skin Friction Topology in a Region Enclosed by Penetrable ...