Statistical Nature of Spark Ignition

```Shadowgraph and Schlieren Techniques
Sally Bane
Explosion Dynamics Laboratory
Directed by Professor Joseph Shepherd
Ae104b Lecture
February 9, 2010
Schlieren Visualization
• optical techniques have been used for decades to study
inhomogeneous media
• Robert Hooke (1635-1703) – “Father of the optics of inhomogeneous
media”, invented the schlieren method
• many different optical
techniques for studying fluid flow
• will focus on the classic
schlieren technique
Schlieren image of explosion in hydrogen-air
2
Basic Concepts:
Light Propagation Through Inhomogeneous Media
allow us to see the
phase differences in light
Q: Why do stars “twinkle”?
A: atmosphere is inhomogeneous –
disturbances due to turbulence etc. change
the air density
→ change in the refractive index
→ rays of starlight bend, wave front of the
light is wrinkled
→ star not a point, but fluctuates (“twinkles”)
on the time scale of the atmospheric
disturbances
3
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Refractive Index: describes how the speed of light changes upon
interacting with matter
n : refractive index  1
c0
n
c
e.g. nair  1.000292
c0 : speed of light in a vacuum  3x108 m/s
c : speed of light in the medium
Gases: linear relationship between n and the gas density
n  1 : refractivi ty
n 1  k
 : gas density
k : Gladstone - Dale coefficien t
e.g. k air  0.23 cm 3 /g
4
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Refractive index only very weakly dependent on density
→ k = 0.23 cm3/g = 2.3 x 10-4 m3/kg
Increase air density by
two orders of magnitue
→ 2.3% increase in n!
1.0252
1.0005
1.0202
1.0152
n
n
1.0004
1.0003
1.0102
1.0052
1.0002
1.0002
1
1.5
2
 (kg/m3)
Require very sensitive optics!
1
10
100
 (kg/m3)
5
Basic Concepts:
Light Propagation Through Inhomogeneous Media
What does “schlieren” mean?
Schliere (singular of schlieren):
 German for “streak,” “striation,” or “cord”
transparent media
 object that has a gradient in the index of
refraction, i.e.
n
n
or
x
y
6
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Example schliere:
n
x
Laminar Candle Plume
2
The gas in the
plume is hotter and
less dense than the
surrounding gas, so
n
x
2  1
and therefore
1
1
n2  n1
y
x
producing a
7
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Planar wave front
Increasing n
Dz = cDt = (c0/n)Dt
y1 Dz
1
Rays (normal
to wave front)
y2 Dz
2
Dz1 = (c0/n1)Dt
and
Dz2 = (c0/n2)Dt
Since n2 &gt; n1, c2 &lt; c1 so
t=0
y
x
z
t = Dt
Negative vertical
refractive-index
Dz2 &lt; Dz1
8
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Increasing n
RESULT:
Refracted wave front
Huygen’s Principle:
Light rays, always normal to the
local speed of light, are bent
toward the zone of higher
refractive index (zones of higher
density in gases).
t = Dt
y
x
z
Negative vertical
refractive-index
9
Basic Concepts:
Light Propagation Through Inhomogeneous Media
Distance wave front moves in time Dt:
dn/dy &lt; 0
y2
(c0/n2)Dt
Dz
De
Refraction angle:
y
Dy
c0
 cDt  Dt
n
z2
z1
De
y1

c0 / n2 Dt  (c0 / n1 )Dt
tan De   De 
Dy
Also:
n
Dt  D z
c0
n n1  n2 
De 
Dz
n1n2 Dy
y
x
z
10
Basic Concepts:
Light Propagation Through Inhomogeneous Media
dn/dy &lt; 0
y2
Dy  0 , Dz  0
(c0/n1)Dt
y
Dy
de 1 dn

dz n dy
Dz
De
z2
z1
De
y1
Because e is a very small angle, it is
approximately equivalent to dy/dz, the
slope of the refracted ray.
 2 y 1 n

2
z
n y
y
x
and
 2 x 1 n

2
z
n x
z
Curvature of refracted ray
11
Basic Concepts:
Light Propagation Through Inhomogeneous Media
So the angular ray deflection in the x and y
directions are:
dn/dy &lt; 0
y2
(c0/n1)Dt
y
Dy
1 n
e y   dz
n y
Dz
De
1 n
dz
and e x  
n x
z2
z1
De
y1
For a 2D schliere of length L along the
optical axis (z):
L n
ey 
n0 y
y
x
z
L n
and e x 
n0 x
Refraction caused by gradients of n, not
overall level of n!
12
Only need a light source, a schlieren object, and screen on which
screen
extra illumination
e
*
less illumination
schliere
point light source
lens
e
*
point light
source
Denser sphere (i.e. a bubble)
Screen
screen
13
Dark circle due to light
refracted from outline of
sphere
Light circle due to
refracted light from the
outline illuminating this
part of the screen
Screen
background illumination
due to non-uniform
refraction of rays as the
light wave travels down
the optical axis (x)
14
don’t get outline of the
schliere
y
Uniform
shift of
illumination
 as move down optical
path (z-direction),
n y  constant
so all rays shift the same!
z
 as move down optical
path (z-direction),
y
Nonuniform
illumination
 2 n y 2  constant
so rays shift nonuniformly
z
15
He/N2 mixing layer (Settles 2001)
Sphere flying at M=1.7 (Merzkirch 1987)
Oil globs in water (Settles 2001)
Shock wave diffraction
around wedge (Settles 2001)
16
Schlieren Imaging
Focused optical image
formed by a lens
Not an image but a shadow
Requires cutoff of the
refracted light
No cutoff of refracted light
Illuminance level responds to
∂n/∂x and ∂n/∂y
Schlieren image displays the
deflection angle e
Responds to second spatial
derivative, ∂2n/∂x2 and ∂2n/∂y2
ray displacement
More sensitive in general
Less sensitive except for special
cases (e.g. shock waves)
More difficult to set up –
use lamps, mirrors, lenses
Extremely easy to setup, occurs
naturally
17
Schlieren System – Point Light Source
screen
lens
lens
focused back
to same point
on screen
*
point light
source
schliere in
test section
deflected
rays miss
the focus
• merely a projector, imaging opaque objects in the test section
18
Schlieren System – Point Light Source
screen
lens
lens
*
point light
source
schliere in
test section
knifeedge
Brighter
point on
screen
• translating phase difference causing a vertical gradient ∂n/∂y to amplitude of
light on the screen
• refracts many rays in many directions – all downward deflected rays get
blocked, painting at least a partial picture
• gives black and white image
19
Schlieren System – Extended Light Source
screen
lens
extended
light source
lens
knifeedge
• the light source is first collimated by a lens then refocused by the second lens
• an inverted image of the light source is formed at the knife-edge
• the extended light source can be considered as an array of point sources – each
“point source” forms a schlieren beam that focuses to a corresponding point in the
light source image (extreme rays shown in cartoon above)
• knife-edge blocks a portion of the image of the extended light source
• another lens focuses an inverted image of the test area on the screen
20
Schlieren System – Extended Light Source
screen
lens
extended
light source
lens
knifeedge
• each “point source” in the extended light sources illuminates every point in the
test section → each point in test section is illuminated by rays from the entire
extended source
• when focused to knife-edge, each point in test section produces an entire
“elemental” source image to the “composite” image at the knife-edge
• e.g. if insert a pinhole in the test section, would still see an image of the extended
source, but much weaker in intensity than the “composite” image
21
Schlieren System – Extended Light Source
screen
lens
extended
light source
lens
knifeedge
IMPORTANT POINT:
• with no schliere present, if we advance the knife-edge to block more the
“composite” image of the extended light source → block each “elemental” source
image equally
therefore blocking equal amount of light from every point in
the test area
Screen darkens uniformly! This is how you know your
alignment is good and that you are at the true focus!
22
Schlieren System – Extended Light Source
screen
lens
lens
extended
light source
knifeedge
NOW PLACE A SCHLIERE IN THE TEST AREA
• consider one point in the test area to be subject to refraction by the schliere
• since all of the “point sources” on the extended light source contribute a ray to this
point, a group of rays from all “point sources” is deflected (dashed lines in cartoon)
• this group of rays are focused to produce an “elemental” image of the light source
at the knife-edge
but the image is displaced due to the refraction
• the group of rays is returned to the same relative position on the
screen by the third lens → true image of the schliere at the screen
23
Schlieren System – Extended Light Source
• the displacement of the “elemental” source image separates the rays refracted by
the schliere from the rays that provide the background illuminance
• because the refracted light is separated, can have a different amount of cut-off by
the knife edge → recombined in the schlieren image at the screen → variations in
the illumination with respect to the background
schlieren image that
shows the shape
and strength of the
schliere
Many points of
varying illuminance
Da
Weak source
image
displaced by
schlieren
object
a
Undisturbed
composite source
image
Note: using an
extended light sources
gives continuous grayscale schlieren images!
Knifeedge
24
Schlieren System – Extended Light Source
Sensitivity:
Constrast:
Sensitivity:
DE :
DE f se y
C

E
a
S
d output  dC

d input  de
fs
S
a
Larger focal length =
better sensitivity
More obstruction of source
image = better sensitity
differential illuminance
at an image point
E:
background illuminance
fs :
focal length of the
schlieren lens
ey :
refraction angle
Da
Weak source
image displaced
by schlieren
object
a
Undisturbed
composite source
image
Knifeedge
25
Z-Type Schlieren Arrangement
condenser
lens
parabolic mirror
light source
pinhole
or slit
parabolic mirror
test area
knife-edge
camera
Most common arrangement: easy to set-up, allows for a schlieren
mirror with long focal length (high sensitivity) and large field-of-views
26
Cool Schlieren Images
Bullet and
candle flame
(Settles 2001)
Glass fibers
(Settles 2001)
Projectile fired at
Mach 4.75 in reactive
H2/air mixture –
cyclic detonation
behind the shock
(Settles 2001)
27
Cool Schlieren Images
Removing
frozen pizza
from case
(Settles 2001)
Blackjack dealer
and players
(Settles 2001)
Space heater
(Settles 2001)
28
Cool Schlieren Images
Image of a T-38 at Mach 1.1 (Leonard M. Weinstein, NASA
Langley Research Center) – taken using a telescope, the sun,
and a cutoff, field of view of 80 m!
29
Cool Schlieren Images
Color schlieren
of the space
shuttle orbiter in
supersonic wind
tunnel test
(Settles 2001)
3D schlieren of a glass
figurine (Settles 2001)
Color schlieren of a
gun firing 0.22
caliber bullet
(Settles 2001)
30
Important Equations
Object
(FOV)
Lens 1
Knifeedge
y0
x1
Lens 2
yI1
f1
x2
x3
Equations:
Gaussian Lens
Equation:
Magnification:
Total
Magnification
Object image
Inverted
object image
yI2
f2
x4
Constraints:
1 1
1
1 1
1
 
and
 
x1 x2 f1
x3 x4 f 2
y I 1  x2
y
 x4

and I 2 
y0
x1
yI1
x3
M
y I 2 x4 x2

y0
x3 x1
For Real Image:
x2 , x3  0
x1  f1 and x3  f 2
Table Size:
x1  x2  x3  x4  L
where L is limited by the size
of the optics table
31
Important Equations
Object
(FOV)
Lens 1
Knifeedge
y0
x1
Lens 2
yI2
yI1
f2
f1
x2
x3
Must Satisfy:
SUMMARY
Object image
Inverted
object image
1 1 1
 
x1 x2 f1
1 1
1
 
x3 x4 f 2
x4 x2
M
x3 x1
x4
Under the Constraints:
x1  f1
x3  f 2
4
x
i 1
i
L
32
How My Schlieren Setup Works
Flat mirror
Flat mirror
Baffle
(to block
stray light)
Aperture
(to make 1”
&Oslash; beam)
Concave mirror
(schlieren lens)
f = 1000 mm
Test section
(1” &Oslash; field-of-view)
Flat mirror
Vertical slit
Achromatic lens (to
collimate the light)
f = 200 mm
Optical
assembly
Light source
(Xe arc lamp)
Knife-edges
High-speed camera
focusing lens used)
Flat mirrors
33
How My Schlieren Setup Works
Camera Side:
Schlieren “Lens”
(concave mirror)
Object
(1” &Oslash; field-of-view)
Knifeedges
Inverted object
image on
camera CCD
yI
yO
f
f
x1
Equations:
1
2
Unknowns:
1 1 1
 
x1 x2 f
y
x
M I  2
yO x1
x1 , x2
x2
Knowns:
M
f  1000 mm
Size of Camera CCD
Beam Diameter
5 / 8 in.

 0.625
1 in.
34
How My Schlieren Setup Works
1
x2  x1 1

x1 x2
f
2 into
x2  Mx1
2
Invert 3
1
Mx1  x1 M  1 1


x1 Mx1 
x1M
f
3
Mx1
 f
M 1
Solve for x1:
1 M 
 1  0.625 
x1  
f 
1000 mm 
 M 
 0.625 
x1  2600 mm
x2  Mx1  0.6252600 mm 
x2  1625 mm
Then from
2
:
Remember: Sensitivity is proportional to the focal length
so f should be as large as possible!
35
Setting Up a Schlieren System:
Step-by-Step (1)
Step 1: Calculate the required distances between he object, schlieren lens,
focusing lens, and camera based on the equations on the previous slide and the
Step 2: Set up the light source, any flat mirrors, and test section with windows in
place if applicable
Step 3: Set up a laser in the place where the camera will go
Step 4: Turn on the laser and ensure that the beam is straight in both the
vertical and horizontal directions along the optical axis (line to next mirror)
Side View
y
Top View
optical
axis (z)
optical
axis (z)
laser
laser
x
z
ruler or
height
gauge
right angle
ruler
36
Setting Up a Schlieren System:
Step-by-Step (2)
Step 5: Adjust any mirrors on this side of the set-up to direct the laser to the test
section, ensuring that the beam stays the same height the whole way (use a
ruler or a height gauge to test this at every mirror)
Tip 1: Try to keep the laser dot as close to the center of the mirrors
as possible
Tip 2: The laser light corresponds to approximately the center of the
ultimate light beam, so locate the laser beam through the test section
where you want the center of the light beam
Step 6: If there are windows on the test section, check for reflections to ensure
the laser is perpendicular to the windows
Incident laser beam
window
Tip 1: Use a piece of paper to probe all around
the incident beam – any reflections will show up
on the paper
Tip 2: When it is properly aligned, when you look
through the windows all the laser dots will
appear in a straight line through the glass
37
piece of
paper
reflection
Setting Up a Schlieren System:
Step-by-Step (3)
Step 7: Adjust any mirrors on the light-source side to direct the laser beam to
the light source, ensuring the beam stays the same height and is centered on the
mirrors
Step 8: Adjust the height of the light source so that it is at the same height as
the laser beam
Tip 1: The two most common types of light sources are filament and arc
light sources, and there are often lenses mounted on the front
Tip 2: First, adjust the height of the light source so that the laser beam is
centered on the lens on front of light source if present
Tip 3: Check for reflections from the lens using the method described
before – adjust light source orientation to minimize relfections
Step 9: Remove the cover of the light source (make sure it is unplugged and
cold!) so you can see the filament or arc bulb.
Step 10: Use the controls on the light source to move the filament or bulb until
the laser light hits the center of the filament or bulb.
38
Check for reflections.
Setting Up a Schlieren System:
Step-by-Step (4)
Step 11: Once alignment of the laser, mirrors, and light source is complete, be
sure to secure all the optics in place.
Step 12: One-by-one, add the lenses to the setup.
Tip 1: The laser light should go through the center of the lens.
Tip 2: Check for reflections using the method described before (probe
around the beam with a piece of paper between the incident laser beam
and the lens). Get rid of reflections by adjusting the height of the lens and
angle of the lens with respect to the laser.
Step 13: Once the alignment is complete, secure well all of the optical
components.
Step 14: Replace the laser with the camera, place the knife-edge at the
approximate location of the focus of the schlieren lens, and turn on the light
source.
Now the REAL work begins! Remember, the best tool
for setting up a good schlieren system is PATIENCE!
39
Setting Up a Schlieren System:
Step-by-Step (5)
Step 15: Starting at the light source, very carefully make slight changes to the
focusing lens (if one is not included on the light source) to focus the light source
down onto the pin-hole or slit.
Step 16: Using a precision translation stage, adjust the distance between the
pin-hole or slit and the collimating lens until the beam is collimated. Use an
aperture if desired to define the size of the beam
lens 1
lens 2
light
source
f1
f2
aperture
Tip 1: Position the collimating lens
(lens 2) one focal length (f2) from the
pin-hole or slit first.
Tip 2: Put up a piece of paper a good
distance from the lens, then carefully
adjust the distance between lens 2
and the pin-hole/slit until the beam on
the paper is the same size as at the
aperture – then the light is collimated!
40
Setting Up a Schlieren System:
Step-by-Step (5)
Step 17: After the beam has been collimated, if it is not in the location where
you want it in the test area, make adjustments to move the beam.
Tip 1: Make horizontal adjustments by moving the mirrors – NOT
tilting the mirrors, but actually moving them horizontally. It is a good
idea to mount the mirrors on translation stages to allow for this.
Tip 2: Make vertical adjustments by changing the aperture (if you are
using one) if possible; if not, change the height of both the lenses and
the light source.
Step 18: Follow the same procedure to position the image correctly on the
camera, heeding the Tips 1 and 2.
Step 19: Find the approximate location of the focus of the schlieren lens, and
place the knife-edge there on translation stages.
Step 20: Step the knife-edge in/up to block part of the light – if you are at the
focus, the background will become dimmer uniformly. Adjust the location of the
knife-edge using the translation stages until you find the focus.
41
References &amp; Where to Buy Optics
Reference Books on Schlieren Methods:
G. S. Settles. Schlieren and Shadowgraph Techniques. Springer-Verlag, 2001.
W. Merzkirch. Flow Visualization. 2nd Ed. Academic Press, Inc., 1987.
Where to purchase optical components:
Thorlabs, Inc
Newport
Edmund Optics
CVI Melles Griot
http://www.thorlabs.com
http://www.newport.com
http://www.edmundoptics.com
http://www.cvimellesgriot.com
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