laser diagnostics in turbulent combustion research · laser diagnostics in turbulent combustion...
TRANSCRIPT
-
Laser Diagnostics in Turbulent Combustion Research
Jeffrey A. Sutton Department of Mechanical and Aerospace EngineeringOhio State University
Princeton-Combustion Institute Summer School on Combustion, 2019
Lecture 1 – Introduction and Overview
Turbulence and Combustion Research Laboratory
-
Turbulence and Combustion Research Laboratory
I direct the Turbulence and Combustion Research Laboratory (TCRL) at OSU (https://tcrl.osu.edu/)
My Research Group
http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=PMTZGQIccyqV6M&tbnid=7AB395gJd3J1jM:&ved=0CAUQjRw&url=http://www.eng.warwick.ac.uk/%7Eespbc/courses/turbine/pivset.htm&ei=3HEFU9icM9O5qQGp04GwCA&psig=AFQjCNFygG-AyK4pr222x6_zvHJLYN8vkg&ust=1392952021090204http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=PMTZGQIccyqV6M&tbnid=7AB395gJd3J1jM:&ved=0CAUQjRw&url=http://www.eng.warwick.ac.uk/~espbc/courses/turbine/pivset.htm&ei=3HEFU9icM9O5qQGp04GwCA&psig=AFQjCNFygG-AyK4pr222x6_zvHJLYN8vkg&ust=1392952021090204
-
Turbulence and Combustion Research Laboratory
Goal: Setting the stage for the remaining lectures
Motivation – Why combustion and Why Laser Diagnostics?
Interaction of Light and Matter – the Basics for Combustion Scientists
Overview of Popular (and Useful) Laser Diagnostic Approaches
General Challenges – From a Measurement Point-of-View
Overview and Outline of Lecture
-
Turbulence and Combustion Research Laboratory
Why Study Combustion?
= 68%
= 85%
-
Turbulence and Combustion Research Laboratory
Laser Diagnostics?
image courtesy of AFRL
How do lasers and photons help with the energy problem?
How do lasers and photons help understand combustion/engines?
How do lasers and photons help improve combustion/engines?
?
http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=EcurPuAvhWresM&tbnid=WJvDiaB1f6RFwM:&ved=0CAUQjRw&url=http://www.lbl.gov/Science-Articles/Archive/EETD-LSI.html&ei=BUgGU7CGBtPbqwG_hoGYBg&bvm=bv.61725948,d.aWc&psig=AFQjCNEyIifX4rTzj32-75B-EiY1ksasTw&ust=1393006785757370
-
Turbulence and Combustion Research Laboratory
Why Laser Diagnostics ?Laser-based measurements are an important tool for studying
combustion processes in detail
-
Turbulence and Combustion Research Laboratory
Why Laser Diagnostics ?Laser-based measurements are an important tool for studying
combustion processes in detail
Selectivity (you can pick from various techniques to measure the quantity of interest – species, temperature, pressure, velocity, particle characteristics)
Typically non-intrusive (photons do not interfere with fluid mechanics or chemistry like physical probes)
Good spatial resolution (laser beams can be focused to small probe volumes; typically limited by diffraction)
Sensitivity (laser-based measurements can measure minor species down to ppb levels at times)
Survivability under harsh conditions (photons do not “melt” at high temperatures)
In situ (direct interaction of light and matter)
http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=PMTZGQIccyqV6M&tbnid=7AB395gJd3J1jM:&ved=0CAUQjRw&url=http://www.eng.warwick.ac.uk/%7Eespbc/courses/turbine/pivset.htm&ei=3HEFU9icM9O5qQGp04GwCA&psig=AFQjCNFygG-AyK4pr222x6_zvHJLYN8vkg&ust=1392952021090204http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=PMTZGQIccyqV6M&tbnid=7AB395gJd3J1jM:&ved=0CAUQjRw&url=http://www.eng.warwick.ac.uk/~espbc/courses/turbine/pivset.htm&ei=3HEFU9icM9O5qQGp04GwCA&psig=AFQjCNFygG-AyK4pr222x6_zvHJLYN8vkg&ust=1392952021090204
-
Turbulence and Combustion Research Laboratory
Why (or When) NOT Laser Diagnostics?
If you do not need to!!!
You need optical access (“real” engines don’t have windows)
Environment may not be “friendly” (windows, soot, and flow field particulate interfere with many really good laboratory diagnostics)
Qualitative vs. Quantitative (interpreting measured signals can be challenging)
Expensive (thermocouple $20 vs. laser/camera $100K)
Complexity (large footprint; system alignment/repeatability; sensitivity to surroundings such as vibrations)
?
-
Turbulence and Combustion Research Laboratory
Role of Laser Diagnostics in Energy/Propulsion
Laser Diagnostics
Remote Sensing and
Control
Characterize Ground Test
Facilities
Understand Fundamentals
of Reactive Flows
Assess Models (Physical and
Chemical)
Assess and Develop New Technologies
-
Turbulence and Combustion Research Laboratory
Combustion Diagnostics (A Family Tree)
Combustion Diagnostics
Physical probes Optical Diagnostics
Laser-Based Non-Laser-Based
“Spectroscopic” “Non-Spectroscopic”
Absorption spectroscopyLaser-induced fluorescenceScattering processes
Particle imaging velocimetryLaser Doppler velocimetryHolography
-
Turbulence and Combustion Research Laboratory
Categorization of Laser Diagnostics
“Resonant” or “non-resonant” techniques (do you have to tune a laser to a specific wavelength?)
“Line-of-sight” (path integrated), single point (“0D”), line (“1D”), planar (“2D”), volumetric (“3D”), or volumetric and time resolved (“4D”)
Linear vs. non-linear techniques (signal vs input laser intensity)
Lasers can be continuous wavelength, long pulse (ns – µs), short pulse (fs – ps)
Lasers can be spectrally narrow or broad in spectral bandwidth
Lasers can range from infrared (IR)→ visible → ultraviolet (UV)
-
Turbulence and Combustion Research Laboratory
Quantities of InterestFlow field
Mean velocities, RMS fluctuations, Reynolds stresses
Gradients (strain rate, vorticity, dilatation)
Integral scales, spectra
Scalar Fields
Mean and RMS fluctuations of temperature and species fields
Topology from imaging
Gradients and dissipation rates
Boundary Conditions for both!
Acoustics, unsteadiness, etc.
-
Turbulence and Combustion Research Laboratory
Suggested Reading
K. Kohse-Höinghous, R.S. Barlow, M. Aldén, J. Wolfrum, “Combustion at the Focus: Laser Diagnostics and Control”, Proc. Combust. Inst., 30 (2005), 89-123.
R.S. Barlow, “Laser Diagnostics and Their Interplay with Computations to Understand Turbulent Combustion”, Proc. Combust. Inst., 31 (2007), 49-75.
-
Turbulence and Combustion Research Laboratory
Instrumentation - A Generic Configuration?There is no possible way to draw a schematic that covers all possible measurement scenarios that are actively used today
However, the majority of laser measurements share common instrumentation (let’s spend a slide familiarizing ourselves)
Laser
Test SectionUVVisibleNear-IR/IR
Focusing optics Beam dump or detector
Light collection optics(camera lens)
Detector (i.e., camera)
Optical filter
Notes:Lasers take many forms (cw, pulsed, broadband, single-frequency, high-rep, tunable, etc.)
Detectors take many forms (CCD, ICCD, CMOS, PMT, photodiode, spectrometer, etc.)
-
Turbulence and Combustion Research Laboratory
Instrumentation - Lasers?
Not what we will talk about today - typically we want non-intrusive methods for measurements
-
Turbulence and Combustion Research Laboratory
Instrumentation - Lasers
Lasers are (typically)…quasi-monochromatic; coherent; directional (i.e., beam)
Lasers can be…tunable, CW or pulsed (10-18 < τ (s) < 10-6); low or high average power; high pulse energies; high instantaneous power
Light Amplification by Stimulated Emission of Radiation
He – Ne Laser
Argon-ion Laser
Tunable diode laser
Nd:YAG laser
Excimer laser
Dye Laser
Optical Parametric Oscillator
} Continuous wavelength lasers
“pulsed” lasers}} Broadly tunable; wavelength extension
-
Turbulence and Combustion Research Laboratory
Instrumentation - DetectorsCharge-coupled device (CCD) camera: Incident photons create electron-hole pair in a silicon portion of a pixel (biased to a potential); generates photoelectrons which migrate to “potential well” of CCD.Essentially a capacitor (stores charge); charge is proportional to incident photons
Pixels are O (10) µm; arrays are typically > 106 pixels
Yields 2D array of information, i.e., “images”
Very linear and uniform; low noise; high dynamic range
-
Turbulence and Combustion Research Laboratory
Instrumentation - DetectorsIntensified CCD camera: photocathode converts incident photons to photoelectrons. The multichannel plate (MCP) multiplies the photoelectrons. The phosphor screen converts photoelectrons back to photons. The photons are captured by the coupled CCD camera.
Intensifies signal (high gain); intensifies and adds noise
Works as a very short time gate – enables low signal collection
Increases wavelength detection range (i.e., UV for PLIF)
-
Turbulence and Combustion Research Laboratory
Instrumentation - DetectorsComplimentary Metal Oxide Sensor (CMOS) camera: “alternative” technology to CCD camera. CMOS is an “active pixel” sensor. Each pixel has own A/D converter, amplifier, noise correction, and digitization circuits
Very fast framing rates (~10,000 full sensor)
Higher noise, lower uniformity, lower dynamic range as compared to CCD
-
Turbulence and Combustion Research Laboratory
An Introduction to the Interaction of Light and Matter
What happens when a laser and gas molecules meet? (Photons are destroyed, created , or “re-routed” – energy loss/gain manifests itself in various phenomena that we use to understand the gaseous medium)
Absorption
Fluorescence/Phosphorescence
Elastic Scattering (Rayleigh scattering)
Inelastic Scattering (Raman scattering)
Incandescence
Others?
Gas
-
Turbulence and Combustion Research Laboratory
Brief Overview of Laser Diagnostics
I will now give a brief overview of the more useful (and common) laser diagnostic techniques (unfortunately I can not present them all)
We will get into details about the diagnostics, equipment, analysis in subsequent lectures
I will introduce and discuss concepts (some specific and some broad), but we WILL come back to these same concepts throughout the week with more depth
If you don’t understand something, please ask me a question – I will try my best to answer them (or at least I will go off and think about it)
Throughout the week I will present the diagnostics from a practitioner’s point-of-view, i.e., an engineer that wants to use these diagnostics to understand physics!
-
Turbulence and Combustion Research Laboratory
Particle Imaging Velocimetry
“PIV” is the most common method for measuring flow field properties
First, you seed the flow with tracer particles (small enough such that particles accurately follow the flow)
The flow is illuminated with two laser pulses (typically 532-nm light from Nd:YAG lasers), where the laser pulses have been formed into laser sheets using a particular set of optics. The two pulses are separated by a user-selected ∆t.
“Mie” scattered light is collected or “imaged” onto a dual-frame camera. The two exposures are bracketed around the two laser pulses.
-
Turbulence and Combustion Research Laboratory
Light Source, Optics, and Camera
-
Turbulence and Combustion Research Laboratory
Particle Imaging Velocimetry
Gas-phase velocity is determined from the motion of the tracer particles (we’ll discuss the potential problems with this later!)
We can’t really track the motion of the individual particles (we can –but that is particle tracking velocimetry), so we divide the images into somewhat coarse regions, called interrogation windows.
The interrogation windows will be a large number of pixels, typically 16 x 16 to 64 x 64 squared pixels
Each interrogation window from frame 1 is correlated with the interrogation window from frame 2
The peak correlation between the two sets of particle images determines the “average” particle displacement between the two sets of images
With the known ∆t, a velocity is determined.
-
Turbulence and Combustion Research Laboratory
Particle Imaging Velocimetry
Stohr et al, CNF, 2012
-
Turbulence and Combustion Research Laboratory
Particle Imaging Velocimetry
Summary and Review
PIV uses scattering from tracer particles (non-resonant technique)
It is spatially resolved and can be operated with laser sheet or laser “slab” for 2D or 3D imaging
Uses two pulses of light and a dual-frame camera
Linear technique (signal scales with I, although this is not really important)
Velocity data is indirectly inferred
Experimental challenges are with seeding and correlation processing
Other challenges include limits in spatial resolution and measurement dynamic range
-
Turbulence and Combustion Research Laboratory
Laser-Induced Fluorescence
LIF is a common technique for probing or visualizing species concentration distributions
The flow is illuminated with a tunable laser that overlaps an allowed electronic-vibrational-rotational transition in an atom or molecule
What wavelength do we need? Example: OH (1,0) - excitation from theground state (v’’ = 0) to excited state(v’ = 1)
From the figure: ∆E ≈ 35500 cm-1λ = 1/∆E*107 ≈ 282 nm
Laser light is absorbed (“a”) and theabsorbing atoms/molecules occupythe A-state (are in the “excited” state)
Depiction of lower electronic state (X-state) and first electronic state (A-state) for a diatomic molecule
a
-
Turbulence and Combustion Research Laboratory
Laser-Induced Fluorescence
The molecule can stay in the excited (A-) state for a short time (10-8 sec) b/c that state has a natural lifetime
During its lifetime in the upper state, the energy is re-distributed across other vibrational and rotational levels in the A-state
Then the molecule starts to “fall” backdown….what happens?
The molecule can “collide” with anothermolecule and give up its energy (b)(“collisional quenching”)
Molecules can relax back to the groundstate and emit photons as an energyrelease (“fluorescence”, c)
Signal is proportional to quantum yieldΦ = fluorescence photons/incident laserphotons (~0.1% or so) Depiction of lower electronic state (X-state) and first
electronic state (A-state) for a diatomic molecule
abc hν
-
Turbulence and Combustion Research Laboratory
Laser-Induced Fluorescence
LIF can be performed at a single point, line, or with a laser sheet (PLIF)
For PLIF imaging, a camera is setup normal to the direction of the propagating light sheet
PLIF allows visualization of the structure of a species
Typically intensified CCD cameras are used b/c of low light levels
Majority of excitation/collection strategies are in the UV
OH
CH2O
-
Turbulence and Combustion Research Laboratory
Laser-Induced Fluorescence
Summary and Review
PLIF allows detection of minor species; selection is limited (OH, NO, CH, tracers, O, H, CO, N, and a few others)
It is spatially resolved and can be operated with laser sheet for 2D imaging
A resonant technique; single-photon LIF is linear; multi-photon LIF is non-linear
Predominantly a qualitative visualization technique; quantification is difficult
Signal levels can be low, which necessitates the use of an ICCD and leads to higher noise levels
-
Turbulence and Combustion Research Laboratory
Previously we talked about absorption and LIF. Resonant processes require that the laser is “tuned” to a certain wavelength/frequency
Scattering does not have this restriction. It is a non-resonant process
Laser scattering from particles is commonly referred to as “Mie scattering” – recall PIV
Here, we are interested in scattering from gas-phase molecules (see picture below)
Spontaneous Scattering Processes
MieRayleigh
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
We will discuss two types of spontaneous scattering processes: (i) Rayleigh scattering and (ii) Raman scattering
Miles et al, MST, 2001
Rayleigh-Brillouin
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
Rayleigh scattering is the quasi-elastic scattering from particles with diameter
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
Molecular scattering processes are relatively weak, so they require high-energy, pulsed lasers for “single-shot” measurements in turbulent flows (cw lasers can be used, but one must integrate for large time periods). Most common is frequency-doubled Nd:YAG at 532 nm
Rayleigh scattering is proportional to total number density
Measurement is not species specific
If one knows the species present (measurement or assumption), then number density can be measured
'RAY laser mixS CI Nσ=
' 'N
mix i ii
Xσ σ= ∑2 2
2 4
4 ( 1)' iin
Nπσ
λ−
=
N = Number density
n = index of refraction
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
Most common application of Rayleigh scattering is to determine temperature in turbulent flames
Application of ideal gas law in an isobaric process yields
We will discuss how to handle the mixture-average cross section in subsequent lectures.For now, let’s just assume it is known…
1 'RAY mixS Tσ∝
Kaiser and Frank, PCI, 2010
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
The exact wavelength shift of the Raman-scattered light is bond-specific, so it can be “species-specific” and thus we can write that Raman scattering is proportional to number density of each species
It is about 1000 times weaker than Rayleigh scattering, so measurements are largely confined to point or line measurements
As you can imagine, there will be various amounts of Raman-shifted light at various frequencies depending on the species present in the laser probe volume. This leads to Raman-scattered spectra
We will discuss how to collect and interpret the signal in a subsequentlecture
, ,RAM i laser i RAM iS CI N σ=
Wehr et al. (2007)
-
Turbulence and Combustion Research Laboratory
Spontaneous Scattering Processes
Most common application of Raman scattering (in conjunction with Rayleigh scattering) is to determine major species in turbulent flames
http://crf.sandia.gov/combustion-research-facility/reacting-flow/flow-experiments/turbulent-combustion/
Sandia piloted partially premixed flame
-
Turbulence and Combustion Research Laboratory
Rayleigh and Raman Scattering
Summary and Review
Rayleigh and Raman scattering are non-resonant processes that can be used with any laser source
Signals are relatively weak and thus require high-energy pulsed lasers for turbulent combustion applications. Signals are collected with CCD, ICCD, or EMCCD (depending on application and laser energy)
The techniques are linear; signal scales with laser intensity
Rayleigh scattering is a measure of number density (if species are known).
Raman scattering can be a measured of individual species number density (if temperature and pressure are known → mole fractions)
Raman signal interpretation is complex; requires lots of calibration
-
Turbulence and Combustion Research Laboratory
Coherent Anti-Stokes Raman Scattering
Coherent Anti-Stokes Raman Scattering (CARS) uses several laser beams (at different wavelengths) that cross at a point within a flame
There is a ‘pump’ beam at frequency ωp, a Stokes beam at frequency ωs, and a probe beam at frequency ωprThe pump and Stokes beams (actually ωp- ωs) excite ‘Raman coherences’ and the probe beam (typically the same as the pump beam) reads them out, generating a coherent ‘laser-like’ beam at frequency ωp- ωs+ωpr (energy conservation)
Example: pump = 532 nm; Stokes = 607 nm (dye laser)What is the CARS signal beam wavelength?
ωCARS = 2ωp - ωS = [2(1/532) – (1/607)]-1 = 473 nm
-
Turbulence and Combustion Research Laboratory
Coherent Anti-Stokes Raman Scattering
The difference in the frequency between the pump and Stokes beams (ωp- ωs) is matched to the energy level difference in a vibrational-rotational line of a molecule
The three beams (pump, Stokes, probe)have to be arranged at specific anglesrelative to one another to conservemomentum, known as “phase matching”
CARS signal comes out at a known angle – known location makes signal collection straightforward!
CARS
X-state
A-state
“virtual” state
CARS p pr sk k k k= + +
CARSk
pk
prk
sk
-
Turbulence and Combustion Research Laboratory
Coherent Anti-Stokes Raman Scattering
Many experimental setups exist for satisfying the phase matching condition, one popular one is the “BOXCARS” configuration
How to generate a CARS spectrum?
The pump/probe typically is narrowband (i.e., Nd:YAG)
Anderson et al. (1986)
PUMP STOKES CARS spectra
Broadband output
-
Turbulence and Combustion Research Laboratory
Coherent Anti-Stokes Raman Scattering
Most common application of CARS is thermometry
Temperature is determined by fitting experimental spectra to theoretical (modeled) spectra
Zheng et al. (1984)
N2 ro-vibrational spectra
-
Turbulence and Combustion Research Laboratory
Coherent Anti-Stokes Raman Scattering
Summary and Review
CARS is a non-resonant technique; it is a non-linear technique (signal scales as I2)
It is spatially resolved (multi-beam crossing), but the resolution is limited by the crossing angle; it is a “single-point” technique
It is the most accurate gas-phase thermometry approach (~2%)
It can be used to determine species concentrations as well
The setups are complex (multiple lasers and sensitive spectrometer)
Recent advances with short-pulse (ps and fs) lasers have shown very high signal levels. This has allowed extensions to 1D and 2D imaging
Short-pulse approaches are insensitive to pressure – opens many new possibilities in realistic environments
-
Turbulence and Combustion Research Laboratory
Lecture Summary and Looking Forward
There are a large number of laser diagnostics used in combustion environments
In this course we will focus on the approaches applicable to “single-shot” measurements in turbulent flames
In turbulent flames we are interested in characterizing the flow field, temperature, species concentrations, reaction rates, etc.
We need to use different diagnostics to gather the information that we want – each with different challenges, accuracies, resolution
In this lecture, we reviewed several common approaches, highlighting the major aspects of the diagnostics
In the next lecture, we will “start at the beginning”, discussing Maxwell’s equations, light as an E-M wave, and classic and physical optics
We will then discuss instrumentation – lasers, cameras, and signal collection
-
Laser Diagnostics in Turbulent Combustion Research
Jeffrey A. Sutton Department of Mechanical and Aerospace EngineeringOhio State University
Princeton-Combustion Institute Summer School on Combustion, 2019
Lecture 2 – Light and OpticsTurbulence and Combustion Research Laboratory
-
Turbulence and Combustion Research Laboratory
Goal: Providing a Background for Optics, Wave Propagation, and Instrumentation
Geometric and Physical Optics
Ray Tracing, Thin Lens Formula
Maxwell’s Equations – A General Framework for Light Propagation
Background and Fundamentals of Lasers
Detection Equipment – Fundamentals of Signal Collection
Overview and Outline of Lecture
-
Turbulence and Combustion Research Laboratory
Light is an electromagnetic (EM) wave; EM waves are described by their position in the EM spectrum
Light and Electromagnetic Waves
-
Turbulence and Combustion Research Laboratory
Definitions that will be used throughout these lectures:
co = speed of light in a vacuum (m/s)c = speed of light in a material (m/s)c = co /n, where n = refractive indexλo = wavelength of light in a vacuum (nm)λ = wavelength of light in a medium (nm)f = c/ λ = frequency of light (1/s or Hz)ω = 2πf = angular frequency (rad/sec)ν = f / co = spectroscopic frequency (cm-1)
Light has a bandwidth (no such thing as monochromatic light) – frequency and wavelength bandwidth are related via
dν = -(1/nλ2)dλ
Light and Electromagnetic Waves
-
Turbulence and Combustion Research Laboratory
Light and light propagation can be described in two ways: (i) “classical” optics and (ii) modern optics
“Classical optics” is further divided into two branches:(i) geometric optics and (ii) physical optics
Geometric optics treats light as a collection of rays
Light rays travel in straight lines in homogenous media and bend at the interface of two media with different refractive indices.
Light rays are governed by the laws of reflection and refraction at interfaces
The energy levels of the light is not important; the primary interest is in the re-direction of light (i.e., ray tracing, focusing, magnification)
Light
-
Turbulence and Combustion Research Laboratory
Physical optics treats light propagation as a wave andthe solution is in the form of a ‘wave equation’
This is necessary to describe effects such as interference, diffraction, laser propagation, etc.
The solution is described by Maxwell’s equations
Geometric and physical optics are dually applicable when the wavelength of light
-
Turbulence and Combustion Research Laboratory
Fermat’s principle: Of all of the possible paths connecting two points, light travels along the path that requires the least time of travel
This principle underlies the laws of reflection, refraction, and optical lenses
Easy example: reflection
Ray starting at P interacts with OThe path with minimum time going through O would occur if the ray were allowed to pass through O on a straight line to image Q’ beneath the surfaceLocation of Q’ is along the extension to the normalfrom Q to O that equals a distance between Q andthe mirrorOQ = OQ’ and thus the time for passage along PQ’ and POQ are identicalThus, θi = θr , which is the law of reflection.
Geometric Optics
Q’
-
Turbulence and Combustion Research Laboratory
Fermat’s principle: Of all of the possible paths connecting two points, light travels along the path that requires the least time of travel
This principle also can be used to describe refraction
Let t be the time for light to travel from Q to P
Application of Fermat’s principle implies dt/dx = 0 (extrema or minima point)
Geometric Optics
Q
P
O
a
b
d
x
2 22 2
1 2
(d )b xa xtc c
+ −+= +
2 2 2 21 2
(d )(d )
dt x xdx c a x c b x
−= −
+ + −
1 2 2sin x
a xϑ =
+2 2 2
(d )sin(d )
xb x
ϑ −=+ −
1 1 2
2 2 1
sinsin
c nc n
ϑϑ
= = Snell’s Law (1)
-
Turbulence and Combustion Research Laboratory
Lenses are refracting elements (so are waveplates and prisms)
To analyze refracting elements we must introduce the “optical distance”
Lo is the distance that light would propagate in free space during the same time that it would take to propagate through a refracting medium (with refractive index n) of length L
Propagation time is the same for all rays traveling through the same optical distance
Geometric Optics
oo
cL L L nc
= = ⋅ (2)
-
Turbulence and Combustion Research Laboratory
The goal of a lens is to collect radiation emitted at O (object) and transfer to point I (image)
Rays emitted at O are in a cone with solid angle Ωo and imaged at a point I with angle Ωi
According to Fermat’s principle rays emerging at O can only intersect at I if their propagation times are identical
Clearly rays propagating along the edge of envelope travelfarther than the rays along the optical axis
The only way for the rays to end up atI at the same time is for the optical distances to vary within the cone
This leads to the lens shape – thickeralong axis compared to the edges
Optical Lenses
Laufer, 1998
-
Turbulence and Combustion Research Laboratory
Detailed analysis/characterization of lens parameters are needed for manufacturing, but a fairly simple analysis can be used to describe the lens operation in terms of one parameter – focal length
so = object distance (object to center plane of lens)si = image distance (image to center plane of lens)f = focal length of lens
Paraxial approximation is invoked that assumes that rays make a small angle to the optical axis (so >> ho) of the system and lies close to the axis throughout the system
Optical Lenses
2 2oh s+
2 2ih s+
os is
hO Iθ
V
Laufer, 1998
os is
OIθ
-
Turbulence and Combustion Research Laboratory
Using an analysis of similar trianglesand
Combine these results
Eliminate ho and divide by si
Expand out LHS
Optical Lenses
tan( ) o io i
h hs s
θ = = o ii
h hf s f
=−
( )o oi io
h hs f sf s
− =
1 1( )ii o
s ffs s
− =
1 1 1
o if s s= − Thin Lens Formula
Laufer, 1998
os is
OI
(3)
θ
-
Turbulence and Combustion Research Laboratory
Define (using similar triangles) the magnification, m
m > 1 microscope (small object on large frame)
m < 1 telescope (large object on small frame)
Magnification (or de-magnification) is an important parameter in imaging
Combining the thin lens equation with the definition of “m” yields
OR
Optical Lenses
i o
o i
h smh s
= =
1 1
o
mms f
+=
1 1
i
ms f+
=
Laufer, 1998
os is
OI
(4)
θ
-
Turbulence and Combustion Research Laboratory
Portions of analysis follows that presented in Clemens, N.T., “Flow Imaging”, 2000
Cameras have pixels of size δxδy
The relationship between spatial regions in the flow and the individual pixels is
The solid angle subtend by the lens is
The camera f-number (FN) isdefined as f/D, thus
Imaging Example
;xxmδ
∆ =
Clemens, 2000
;yym
δ∆ = i i
o o
y smy s
= =
2
2 ;4 o
Ds
π∆Ω = D = lens diameter
2
2 24 ( ) ( 1)m
FN mπ
∆Ω =+
(5)
-
Turbulence and Combustion Research Laboratory
Let’s assume we have a field-of-view illuminated with a laser of height yL, thickness ∆z, and laser fluence FL, where FL = energy/area.
The total number of photons collected by each pixel can be written as:
h = Planck’s constant
ν = laser spectroscopic frequency
∆V = ∆x∆y∆z = volume
= arbitrary scattering cross section (process dependent)
n = number density of scattering medium
ηt = transmission efficiency of lenses
Imaging Example
Clemens, 2000
Lpp t
FS V nh
σ ην
∂= ∆ ∆Ω
∂Ω
𝜕𝜕𝜎𝜎𝜕𝜕𝛺𝛺
(6)
-
Turbulence and Combustion Research Laboratory
Let’s assume the laser sheet has a uniform energy distribution such that FL is approximated as EL/yL∆z, EL is the laser pulse energy
The total number of photons is now written as:
This equation is useful in the context that the laser sheet does not change size no matter what you make the field-of-view
This equation shows that as m→0, Spp increases; that is, more collected photons per pixel as the camera is moved further away from object plane!
Is this counter-intuitive?
Remember, as m→0, ∆Ω→0 …
… but ∆x and ∆y increase; that is, each pixel collects more light from a larger region in the flow
Imaging Example
Clemens, 2000
2 2
14 ( ) ( 1)
Lpp t
L
E x yS nh y FN m
δ δ σ π ην
∂= ∂Ω +
(7)
-
Turbulence and Combustion Research Laboratory
Now lets assume that as the camera is moved further away from the object plane (m→0) the laser sheet size is increased to accommodate field-of-view
Assume yL = yo= Npδy/m; Np =yi/δy = # pixels in one column array
The total number of photons is now written as:
Some useful observations:
(1) signal depends on EL/hν (total number of incident photons)
(2) signal is independent of laser sheet thickness(tighter focus → FL ↑, # scatterers ↓)
(3) as Np ↑ (smaller pixel size for fixed sensor),one achieves better spatial resolution, butSpp ↓ since pixels collect less light
(4) Spp ∝ m/(m+1)2 (signal maximized at m = 1)
(5) Spp ∝1/FN2 (low FN lenses lead to large increases in signal)
Imaging Example
Clemens, 2000
2 2 24 ( ) ( 1)iL
pp tp
yE mS nh N FN m
σ π ην
∂= ∂Ω +
(8)
-
Turbulence and Combustion Research Laboratory
Physical Optics considers the propagation of electromagnetic (EM) waves and the interaction of these EM waves with matter
Light is an electromagnetic wave
We need some definitions:
ε (electric permittivity) and µ (magnetic permeability) allow 𝐸𝐸 and 𝐵𝐵 to sustain one another, leading to EM-wave propagation
Physical Optics
:E
the electric field [V/m or N/C]
:B
the magnetic induction field:H
the magnetic field [A/m];B Hµ=
µ is the magnetic permeability:j
electric current density field:D
electric displacement field;D Eε=
ε is the electric permittivity;j Eσ=
σ is the specific conductivity:ρ charge density [C/m3]
-
Turbulence and Combustion Research Laboratory
In the late 19th century, James Maxwell compiled the known laws of electromagnetism (Maxwell’s equations)
He also was responsible for “connecting” the fact that light propagation is governed by EM principles
For propagation through free space:
Electromagnetism
o
E ρε
∇ ⋅ =
(9)D ρ∇ ⋅ =
or Guass’s theorem
( ) 0Hµ∇ ⋅ = or 0B∇ ⋅ = (10) “no magnetic monopole”BEt
∂∇× = −
∂
(11) Faraday’s law
o oEB jt
µ ε ∂
∇× = + ∂
(12) Ampere’s law + modification
Displacement current – added by Maxwell
-
Turbulence and Combustion Research Laboratory
We now want to get a wave equation
We will assume that all light/optical interactions are electronic such that we seek a solution for 𝐸𝐸 that is independent of the magnetic field.
Feynman, Leighton and Sands (1963) go over this solution in detail. Applying vector calculus (and assuming ε is constant and ρ = σ = 0)
These are just wave equations with wave speed c = (µoεo)-1/2
Electromagnetism
22
2o oEE
tµ ε ∂∇ =
∂
(13)
22
2o oBB
tµ ε ∂∇ =
∂
(14)
( )
-
Turbulence and Combustion Research Laboratory
Let’s get a solution to the 𝐸𝐸 field equation:
The solution must have a form of 𝐸𝐸(𝑟𝑟,t) = 𝐸𝐸(𝑟𝑟)f(t), where 𝑟𝑟 is the position vector given by 𝑟𝑟 = x�̂�𝑒𝑥𝑥 + y�̂�𝑒𝑦𝑦 +z�̂�𝑒𝑧𝑧
ω is the oscillation frequency of the electric field (ω = 2πf ; rad/sec)𝐸𝐸o is the amplitude of the electric field𝑘𝑘 = kx�̂�𝑒𝑥𝑥 + ky �̂�𝑒𝑦𝑦 + kz �̂�𝑒𝑧𝑧 is the propagation vector ; 𝑘𝑘 = 2π/λφ is the phase angle (often unimportant for our applications)
Describes the motion of a set of plane waves
Wave equation is linear; complex solns can be formed by combining solutions for various values of 𝐸𝐸o , 𝑘𝑘, and ω. Foundation for physical optics solutions
Electromagnetism
( )( , ) Re i t k roE r t E eω φ− ± ⋅ + =
(15)
-
Turbulence and Combustion Research Laboratory
Let’s substitute Eq. (15) into Eq. (9)
𝐸𝐸 ⊥ 𝑘𝑘
We also can show that 𝐵𝐵 ⊥ 𝑘𝑘 and 𝐸𝐸 ⊥ 𝐵𝐵 implyingthat all three vectors are mutually perpendicular
Thus 𝐸𝐸 and 𝐵𝐵 are restricted to a plane ⊥ to the direction of propagation and are denoted as transverse electromagnetic (TEM) waves
Electromagnetism
0E ik E∇ ⋅ = ⋅ =
-
Turbulence and Combustion Research Laboratory
How is the propagation of the 𝐸𝐸 field different from something like an acoustic wave?
Acoustic wave = amplitude is a scalar; propagation is a vector
The vectorial amplitude of 𝐸𝐸o is quite important and distinguishes this solution of the wave equation from others
The components of 𝐸𝐸o represent the optical polarization; that is, the direction of the electric field is called optical polarization.
𝐸𝐸 oscillates normal to 𝑘𝑘, but can oscillate at any angle in the plane wave
Thus, any polarization state can be decomposed into two states
e.g., 1D propagation along z-axis
Electromagnetism
( )( , ) Re ;oi t k zoE z t E eω φ− ± + =
ˆo o o zk eω µ ε=
ˆ ˆo ox x oy yE E e E e= +
(x polarization) (y polarization)
-
Turbulence and Combustion Research Laboratory
𝐸𝐸 is the outcome of the superposition of two independent fields (polarizations)
For each field, the frequency, ω and the propagation vector 𝑘𝑘 are the same, but the amplitude nor phase angle need to be the same, 𝐸𝐸𝑜𝑜𝑥𝑥 ≠ 𝐸𝐸𝑜𝑜𝑦𝑦; φ1 ≠ φ2Energy carried by one component is independent of the other!
The two fields (polarization states) are spectrally indistinguishable, but can be separated by optics that are sensitive to polarization (birefringent materials)
Optical Polarization
1 2 ( )ˆ ˆ( , ) oi t k zi iox x oy yE z t E e e E e e eωφ φ − ±− − = +
-
Turbulence and Combustion Research Laboratory
Optical Polarization
In general the vector 𝐸𝐸 describes an ellipse on a plane perpendicular to the direction of propagation
Define a phase shift as ∆ = φ1 - φ2
Linear and circular polarization are special cases of elliptical polarization
Ey = 0 Ex = Ey∆ = 0
Ey /Ex = M0< ∆ < π/2
Ex = Ey∆ = π/2
-
Turbulence and Combustion Research Laboratory
Energy of EM WavesWe will consider an analogy to the 1st law of thermodynamics
Applying divergence theorem:
We seek an analogous description of EM energy in a CVRate-of-work done by an electric field on the charges
Energy stored within a unit volume (u) → stored EM energy density
VCV CS
Q W ud uVdAt
∂− = +
∂ ∫ ∫
Rates of heat and work transferred across boundaries
Rate-of-change of internal energy w/n CV
Energy flux across boundaries carried by flow at velocity 𝑉𝑉
( )uq w uVt∂
− = + ∇ ⋅∂
w j E→ ⋅
2 2
2 2ou E Hµε→ +
(16)
(17)
-
Turbulence and Combustion Research Laboratory
Energy of EM WavesThe last term in the 1st law of thermo represents an energy flux across the boundary of a volume. This is the energy carried by the EM wave as it propagates. This energy flux is represented by Poynting’s vector
The units of 𝑆𝑆 are identical to irradiance (W/m2). This flux may be interpreted as the irradiance or “light intensity”.
Combining all of the energy terms into an equation similar to the 1st law of thermodynamics we obtain (assuming optical medium is adiabatic)
With no electrostatic charge (𝚥𝚥=0) and negligible absorption (internal energy remains unchanged):
S E H= ×
(18)
2 2
2 2oj E E H S
tµε∂ ⋅ = + + ∇ ⋅ ∂
(19)
0S∇ ⋅ = Irradiance is constant as it
propagates through medium
-
Turbulence and Combustion Research Laboratory
Energy of EM WavesExamining Eqs. (10), (13) - (15), the magnitude of magnetic field vector relative to the electric field vector can be determined as
Substituting Eq. (20) into Eq. (18) and integrating over a time period which is long relative to an electric field cycle, but short relative to characteristic time of illumination
For a TEM wave, the direction of the light intensity is coincident with the propagation vector
Note the quadratic dependence of the intensity of light on the electric field components
( , ) ( , )oo
H r t E r tεµ
=
(20)
22 *
0
1 1 1ˆ ˆ ˆRe2 2
To o o
s o o s o so o o
S E dt e E E e E eT
ε ε εµ µ µ
= = =
∫
(21)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionTo date, we have discussed light (EM) propagation in a vacuum
Now, we need to understand what happens when light propagates through another medium (i.e., “interacts”) as this is the basis for any laser diagnostic
We will now introduce the “material polarization”, also called the “polarization vector” in some texts. The material polarization is NOT the optical polarization
The material polarization is collection of active polar molecules within a unit volume
where N is the total number density of the active dipoles and 𝑞𝑞 ⋅ 𝑑𝑑 is an average dipole moment
(dipole moments)volume
P N q d= = ⋅ ⋅∑ (21)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionWhen an electric field transmits through a medium (i.e., optical wave propagating through a gas), material polarization can be induced. We can describe this through the relation
where χ is the electric susceptibility of the dielectric.
The susceptibility (χ) is a material parameter representing the ability of the dipoles to respond to the ‘polarizing’ electric field
Recall the displacement vector 𝐷𝐷 = 𝜀𝜀𝐸𝐸 .
In the absence of a polarizing electric field, this must be 𝐷𝐷 = 𝜀𝜀𝑜𝑜𝐸𝐸
Thus, we can write 𝐷𝐷 = 𝜀𝜀𝑜𝑜𝐸𝐸 + 𝑃𝑃 = 𝜀𝜀𝑜𝑜𝐸𝐸+𝜀𝜀𝑜𝑜𝜒𝜒𝐸𝐸 = 𝜀𝜀𝑜𝑜(1+ 𝜒𝜒) 𝐸𝐸
Now the electric permittivity in a medium (or dielectric constant) is
oP Eε χ=
(22)
(1 )oε ε χ= + (23)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionWe re-write Maxwell’s equations (now with EM wave-matter interactions)
This leads to a new wave equation
and a new wave speed (speed of light in the medium), c = (µoε)-1/2
This also allows us to calculate the refractive index within a medium
0,E∇ ⋅ =
0,B∇ ⋅ =
,BEt
∂∇× = −
∂
,oEBt
µ ε ∂∇× =∂
(1 )oε ε χ= +
22
2oEE
tµ ε ∂∇ =
∂
(24)
1/2(1 )oo
cnc
ε χε
= = = + (25)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionThe wave equation can be re-written (in a neat way)
The LHS side is the wave equation in vacuum; the RHS is a source term due to material polarization.
But…the material polarization comes from the interaction of the EM wave and the material…what does this mean?
In between the molecules, there is free space and the wave propagates. When it encounters dipoles (molecules), it interacts and sends a wave, which then propagates in free space until it encounters more dipoles, and so forth…
22
2 0oEE
tµ ε ∂∇ − =
∂
22
0 2(1 ) 0oEE
tµ ε χ ∂∇ − + =
∂
2 22
0 02 2o oE EE
t tµ ε µ ε χ∂ ∂∇ − =
∂ ∂
2 22
0 2 2o oE PE
t tµ ε µ∂ ∂∇ − =
∂ ∂
(26)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionNow, lets assume our propagating light wave has a very strong electric field. In this manner, a non-linear material interaction can occur:
The first term on the RHS is the linear term given by Eq. (22). The remaining terms each have a non-linear susceptibility (χ(n>1)), which are material constants describing various non-linear optical interactions
This requires introducing the non-linear polarization into the wave equation
𝑃𝑃𝑁𝑁𝐿𝐿 is a source term for generating a new wave (perhaps different wavelength, direction, etc.)
(1) (2) 2 (3) 3 (m)... mNL oP E E E Eε χ χ χ χ = + + + +
(27)
222
0 2 2NL
o oPEE
t tµ ε µ ∂∂∇ − =
∂ ∂
(28)
-
Turbulence and Combustion Research Laboratory
Light-Matter InteractionAbsorption and single-photon fluorescence are 𝜒𝜒 1 processes,SHG, SFG, DFG, OPO are 𝜒𝜒 2 processes,THG, two-photon absorption and fluorescence and CARS are 𝜒𝜒 3processes.
Note: do not confuse the relationship between optical signal and intensity of the incident radiation with the light-matter relationship.
Techniques are characterized as ‘linear’ or ‘non-linear’ based on the relationship between optical signal and incident intensity. Techniques that exhibit a quadratic or higher-order dependence on the input radiation are ‘non-linear’
For example, spontaneous Raman scattering is ‘linear’ (signal scales with I), but the light-matter interaction is non-linear.
-
Laser Diagnostics in Turbulent Combustion Research
Jeffrey A. Sutton Department of Mechanical and Aerospace EngineeringOhio State University
Princeton-Combustion Institute Summer School on Combustion, 2019
Lecture 3 – Measurement Resolution and Challenges
Turbulence and Combustion Research Laboratory
-
Turbulence and Combustion Research Laboratory
Goal: Providing a Background for Combustion, Turbulence, and Turbulent Combustion Measurements
Quick description of combustion and turbulent combustion environments
Experimental resolution and limitations
Turbulence length and time scales
Requirements for “good” measurements
Overview and Outline of Lecture
-
Turbulence and Combustion Research Laboratory
Many of you already are quite familiar with this…(research, classes at university, etc.)
Fuel + Oxidizer + heat → reactions (energy release + light) → species formation
Many times we will write something like this:
What is Combustion?
4 2 2 22 2CH O CO H O+ → +
-
Turbulence and Combustion Research Laboratory
…but really, it is something like this…
Simple fuels such as CH4 are described with ~ 53 species, 325 reactions (GRI-mech 3.0)
What is Combustion?
Low temperature High temperature NOx formation
-
Turbulence and Combustion Research Laboratory
Simple fuels such as CH4 are described with ~ 53 species, 325 reactions (GRI-mech 3.0) – and their structure looks like this (for a laminar flame)
Many more species than we can measure…
Spatial distributions of the species and heat release rate may be confined to very small regions (more on this later in terms of resolution)
What is Combustion?
Turns: An Introduction to Combustion
-
Turbulence and Combustion Research Laboratory
Turbulent flows are highly transient and three-dimensional andwhen coupled to a set of chemical reactions create a complexsystem occurring on multiple length and time scales.
Scientific Challenge
Length Scales
Time Scales
Device
10010-210-310-410-510-6
µm mm10-1
m
OuterIntegralKolmogorovMolecule (x 104)
δFδF
δF
µs
ms
-
Turbulence and Combustion Research Laboratory
Short time scales
Scientific Challenge
δ
Reaction zones O(100) µm at 1 atm; smaller at higher pressures!
Human hair
δ
SMALL
FAST
Image courtesy of Photron
http://en.wikipedia.org/wiki/File:Menschenhaar_200_fach.jpg
-
Turbulence and Combustion Research Laboratory
The focusing properties of laser beams are related to the mode structure of the beam
For an infinite plane wave, only one TEM mode (intensity distribution) appears. A single-mode (TEM00) beam has a Gaussian intensity distribution and can be considered diffraction limited.
The transverse intensity distribution of a TEM00 or simple Gaussian beam is given by
where Io is the maximum intensity and w is the beam radius (also called “waist”). w is defined as the point where I/Io has dropped to its 1/e2 value
Optical Resolution
2 2 22( )/x y woI I e
− += (27)
Silvfast: Laser Fundamentals
-
Turbulence and Combustion Research Laboratory
If the beam is focused, a minimum waist wo will appear.
At any position z along the beam, the beam waist w(z) is given by
where zR is called the “Rayleigh range” and is defined as the distance along the beam from the focus to the point where w is 2 times wo
Optical Resolution
1/22
2( ) 1oR
zw z wz
= +
(28)
2o
Rwz πλ
= (29)
-
Turbulence and Combustion Research Laboratory
wo is determined by first examining the angular spread of a Gaussian beam for z > zR (invoking small angle approximation):
The spreading angle also can be approximated from geometrical optics as the inverse of the “optical” f-number:
where d is the beam diameter illuminated on the focusing lens
Combining (30) and (31):
Optical Resolution
2 ( ) 2( ) limz o
w zzz w
λπ→∞
Θ = = (30)
( ) dzf
Θ ≈ (31)
02 fw
dλ
π= (32)OR 0
4 fdd
λπ
=
-
Turbulence and Combustion Research Laboratory
While we typically model a laser as a plane wave with infinite lateral extent, this is not possible in a real cavity (a laser is “almost” a plane wave)
The finite lateral size of the beam is limited by the size of cavity mirrors, amplifier, etc. This, along with imperfections in the laser cavity resonator lead to beam diffraction, distortion and losses within the laser cavity.
These generate higher-order transverse (TEMpl or TEMnm) modes.
Optical Resolution
Cylindrical transverse mode patterns Rectangular transverse mode patterns
2 2 2 2
2 2
/ /2 2x w y wo m n
x xI I H e H ew w
− −
=
-
Turbulence and Combustion Research Laboratory
The existence of higher-order TEM modes (i.e., multi-mode beams) lead to higher beam divergence and poorer focusing
Multimode beams are characterized with an M2 value, where M2 increases for an increasing number of modes (M2 = 1 for Gaussian beams)
This leads to new definitions of beam parameters
Scientific lasers such as Q-switched Nd:YAGs are said to have M2 ≈ 1
Other lasers have M2 as high as 10 – 100
Typically dye laser output and corresponding UV generation have M2 > 1
Equations show that a smaller focal spot can be achieved by using a shorter focal length lens and/or increasing d; smaller focal spot = decreased Rayleigh range. As M2 increases → decreased Rayleigh range
Optical Resolution
2
04 fMd
dλπ
=2
2o
RwzM
πλ
= (33, 34)
-
Turbulence and Combustion Research Laboratory
In planar imaging, the laser beam is formed into a sheet using optics
Since the optics only expand the beam in one direction, the laser sheet thickness is approximately the same as the spot size given by Eq. (33)
An example: Let’s say we have a pulsed, Nd:YAG laser at 532 nm; d = 10 mm; f = 500 mm
for M2 = 1 (Gaussian beam), do ≈ 34 µm
This seems pretty good, but we have neglected the fact that the incident beam has divergence and thus M2 > 1
Optical Resolution
http://www.ophiropt.com/photonics
-
Turbulence and Combustion Research Laboratory
If we knew M2, then we can use Eq. (33), but normally we only know the divergence (Θ, mrad) of the beam (given in technical specs)
The actual beam diameter measured at a distance f from the lens is related to the divergence angle through
With the small angle approximation, the focused beam diameter or laser sheet thickness is calculated as
Back to our example: a typical pulsed, Q-switched Nd:YAG has a divergence < 0.5 mrad: df = 250 µmIn this manner, we can determine and “effective” M2 from:
Optical Resolution
1tan fmdf
− Θ =
(35)
fd f= Θ (36)
22( 1)f
effo
dM
d M=
=(37)
-
Turbulence and Combustion Research Laboratory
Let’s go back to our schematic of a camera array…
Unfortunately it is common to find in the literature that the spatial resolution is theprojected area onto the pixel, (∆x, ∆y)
This is NOT necessarily the spatial resolution of the measurement – the spatial resolution is can be much worse!
Spatial resolution depends on the point-spread function (PSF) as images are the convolution of the PSF and emitted intensity distribution
The PSF describes the response of an imaging system to an infinitesimally small point object
The “best” that a system can do is the ‘diffraction limit’; that is, at a minimum, diffraction will blur out any point-like object to a certain minimal size and shape (real systems are worse as discussed on the next slides)
Camera Resolution
Clemens, 2000
-
Turbulence and Combustion Research Laboratory
The ideal PSF is a 3D diffraction pattern, where the size or “blur spot” is given by an Airy disk
Example: (532 nm laser; 7 µm camera pixel)
In practice, aberrations in the optical system lead to spot sizes that are much larger than the diffraction-limited blur size
Camera Resolution
, 2.44(1 )blur diffd m FNλ= + (38)
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1
blur
spot
size
(µm
)
Magnification (-)
FN = 22FN = 16FN = 8FN = 2.8FN = 1.4Projected Pixel Area
-
Turbulence and Combustion Research Laboratory
So how do we measure the PSF or other suitable metric of the actual camera/lens system?
You can directly image it on a camera…
However, quantization of the image makes itdifficult to determine the PSF.
Higher resolution imaging in the biological sciences fitthese types of images to models (Gaussian, etc.) to determine the PSF
Perhaps it is easier to consider the line spread function (LSF):
Camera Resolution
( ) ( , )LSF x PSF x y dy∞
−∞
= ∫ (39)
https://carlesmitja.net/2011/02/06/image-quality-of-photographic-cameras/
-
Turbulence and Combustion Research Laboratory
The LSF is essentially a “1D” surrogate for the PSF and will give you the width of the blur spot in one dimension
To determine the LSF, we must first determine the step response function (SRF), also called the edge spread function (ESF)
To measure the ESF, Clemens (2000) offers a suitable experimental setup
The LSF is easily determined from the ESF:
Camera Resolution
( ) (x)dLSF x ESFdx
= (40)
-
Turbulence and Combustion Research Laboratory
Camera Resolution
https://carlesmitja.net/2011/02/06/image-quality-of-photographic-cameras/
-
Turbulence and Combustion Research Laboratory
Out-of-plane resolution most likely is determined by the laser sheet thickness (LST)
However, the LST can influence the actual in-plane spatial resolution IF the LST is greater than the depth-of-field (DOF)
We need to make sure the depth-of-field is larger than the LST:
Example: m = 0.5; FN = 1.4:
DOF, diff = 30 µm DOF = 420 µm (if dblur = 50 µm)
When DOF < LST, the “wings” of the laser sheet are not in focus and thus it degrades the measurements since the 2D image is integrated in the direction of the laser sheet thickness
There is a careful balance between choosing f, m and FN and the resulting LST, DOF, dblur
Camera Resolution
1DOF 2 blurmd FN
m+
= (41)
-
Turbulence and Combustion Research Laboratory
Two important imaging parameters are signal-to-noise ratio (SNR) and dynamic range (DR)
Earlier, we introduced an equation for signal collection, in terms of photons per pixel:
The collected signal (photoelectrons in units e-) can be written as
where ηq is the camera quantum efficiency and G is the gain from the photocathode to the CCD (for CCD, G = 1)
We will only consider two camera noise sources here: (i) shot noise and (ii) read noise
Other Camera Parameters
Lpp t
FS V nh
σ ην
∂= ∆ ∆Ω
∂Ω(6)
q ppS S Gη= (42)
-
Turbulence and Combustion Research Laboratory
Shot noise refers to statistical fluctuations in the number of photoelectrons generated at each pixel for a certain number of incident photons, i.e., it is due to the random arrival of photons
The statistical fluctuations of the photoelectrons exhibit Poisson statistics, but in the limit of a large number of photons, the statistics can be approximated as Gaussian
The shot noise (units of e- RMS) is given as
where K is a noise factor that quantifies the noise generated through the gain process ( K = 1 for CCD; K > 1 for ICCD)
Read noise (NR) occurs during the process of converting pixel charge to voltage that is read by an A-D converter. This is an amplification and digitization stage
Other Camera Parameters
( )s q ppN S K Gη= (43)
-
Turbulence and Combustion Research Laboratory
If it is assumed that the noise sources are uncorrelated, then the signal-to-noise ratio (SNR) is given by:
For the majority of imaging cases, “shot-noise-limited” operation occurs; that is, NR
-
Turbulence and Combustion Research Laboratory
The best definition (and measurement) of SNR is /σS, where is the mean signal and σS is the standard deviation of the signal measured in a uniform region.
SNR is a metric of measurement precision, not accuracy. It tells you how precise your single-shot measurement will be in a turbulent combustion environment (assuming same laser and camera settings)
Other Camera Parameters
-
Turbulence and Combustion Research Laboratory
Camera dynamic range (DR) is defined as the ratio between the maximum and minimum useable signals.
The maximum signal (Ssat) typically is referred to as the “saturation level” and is limited by the total number of photoelectrons that can be stored in a CCD pixel (“well depth”)
The minimum useable signal is limited by camera noise. Hence
where SDC is the integrated dark charge, which typically is small for short-gate experiments
Example: Lets say a camera has a well depth of 105 e-, a conversion of 24.4 e-/count and read noise of 40 e- RMS. What is the DR?
DR = 2500 (assuming DC is negligible) or 11.3 bits! This camera would have been listed as a 12-bit camera due to its well depth, but it actually loses 0.7 bits of DR due to noise
Other Camera Parameters
sat DC
R
S SDRN−
= (45)
-
Turbulence and Combustion Research Laboratory
We have now discussed “hardware” spatial resolution in terms of what you can achieve with a laser and signal collection (lens/camera)
General rule: better spatial resolution lower SNR lower accuracy/precision
Experimental temporal resolution* (e.g., acquisition rate) is determined by hardware only
So now knowing some of the issues that limit experimental resolution, we can state that we have two levels of resolution: desired resolution and actual resolution (due to combined effects discussed above)
So what resolution do we desire to achieve in turbulent combustion research?
We need to look a bit at turbulent flows and understand what sets the length/time scales…
Resolution Requirements
* For laser-based measurements, temporal pulse widths
-
Turbulence and Combustion Research Laboratory
Energy is supplied at outer scales (δ)Integral scale (lo) characterizes the large-scale “eddy” motionKolmogorov scale (λk) describes smallest velocity length scale; velocity fluctuations damped by viscosity (ν) and kinetic energy is dissipated (ε is the kinetic energy dissipation rate)
Batchelor scale (λB) describes smallest scalar fluctuation length scale; fluctuations damped by diffusion (D is the diffusivity; Sc = ν /D = Schmidt number
Length Scales (Flow Turbulence)
Outer scales
log (k)
log E(k)
Inertial subrange
Dissipation range
Energy input
δ-1 λΚ-1
Increases as Re ↑
Integral scales
lο-1 ( )1/43 /Kλ ν ε= < >
( )1/42 1/2/B KD Scλ ν ε λ −= < > =
“classic” turbulence view
(46)
(47)
-
Turbulence and Combustion Research Laboratory
Within the inertial subrange (“energy cascade”), is constant
Net energy input rate = net kinetic energy dissipation rate
There is a continuous spectrum of scales in turbulent flows
Length Scales (Flow Turbulence)
Outer scales
log (k)
log E(k)
Inertial subrange
Dissipation range
Energy input
δ-1 λΚ-1
Increases as Re ↑
Integral scales
lο-1
, δ
Outer scales
log (k)
T(k)
Inertial subrange
Dissipation range
Energy input
δ-1 λΚ-1
Integral scales
lο-1
un, λn
T(k) = 0
Uk, λk
= constant
2 3
3n n n
n n n
u u λετ λ τ
< >∝ ∝ ∝
“classic” turbulence view
33' Ko K
uul λ
∝1/4 3 3/4
3/43 3/4 3/4
( ) Re' '
KT
o ol u u lλ ν ε ν −< >∝ ∝ ∝
3/4 3/43/4
3/4 1/4 3/4 ReK
Uδ
δ δ δλ ν ε ν−
< >∝ ∝ ∝
< >
3/4ReK C δλδ
−= (48)
-
Turbulence and Combustion Research Laboratory
In a shear flow, δ is the full width (5 – 95%) of the velocity profile (i.e., jet or wake width)
Measurements in jets have show that C ≈ 2.3
There have been a number of studies that have targeted the “required” resolution for turbulence measurements
Recent temperature measurements in non-premixed flames by Wang et al. (2008) suggested that in order to resolve 90% of the scalar variance, a measurement should resolve > 37 λB and to resolve 90% of the dissipation energy, a measurement should resolve > 9 λB (or λK)
Keep in mind that this recommendation is not “law”. Other researchers recommend much more stringent requirements of 2-3 λB (or λK) for resolving gradients.
Also keep in mind that “resolving” or “not resolving” is not so simple. As the scales get finer and finer, they will be affected more and more by the imaging system (remember the MTF that I showed)
Length Scales (Flow Turbulence)
-
Turbulence and Combustion Research Laboratory
Physical/chemical processes of combustion generate additional scales
Let’s first consider turbulent premixed combustion first (flame is a propagating wave with velocity scale = SL)
Length Scales (w/ Combustion)
“corrugated flamelets” (δF < λK)
Eddies w/ a turnover velocity = SLwrinkle the flameDefine Gibson scale as
“thin reaction zones” (δF > λK)
Eddies interact with preheat zone; do not penetrate primary reaction layer Define mixing scale as
δF δF
3L
GSlε
=< >
3 3 3 4
3 3 3 3'L L L K
n o Kn K F
S S Slu u u
λλ λδ
= = = = 3 1/2( )m Fl tε=1/2 1/23 33
3 3n FF
L L
US S
δεδλ
= =
lo ≥ lm ≥ λKlo ≥ lG ≥ λK
-
Turbulence and Combustion Research Laboratory
Now let’s consider non-premixed combustion
No characteristic velocity; diffusion controlled
Responsive to turbulent fluid dynamics
Interaction between flow and chemistry over broad range of scales (λK/δ ~ Reδ-3/4)
Smallest interaction scale is ~ λK (or λB)
Length Scales (w/ Combustion)
Outer scales
log (k)
log E(k)
Inertial subrange
Dissipation range
Energy input
δ-1 λΚ-1
Increases as Re ↑
Integral scales
lο-1
Terascale Direct Numerical Simulations of Turbulent Combustion using S3DJ H Chen, et al., Computational Science & Discovery Volume 2, January-March, 2009
OH
HO2
-
Turbulence and Combustion Research Laboratory
High-Reynolds number flows?
What resolution can be achieved (today) with well-designed experiments? Velocity (PIV); > 500 µm Temperature (Rayleigh scattering) ; ~ 100 µm Temperature (CARS); ~100 µm Major species (Raman scattering); ~ 200 µm Minor species (laser-induced fluorescence); 200 to 500 µm
Estimating Length Scales
1
10
100
1000
100 1000 10000 100000 1000000
λ K(µ
m)
Reynolds number, Reδ
δ = 5 cmδ = 1 cmδ = 0.5 cm
1
10
100
1000
0.1 1 10 100 1000
Kol
mog
orov
Sca
le, λ
K(µ
m)
Bulk Velocity (m/s)
T = 300 KT = 700 KT = 1500 KT = 2000 K
δ = 2 cm
Depends heavily on hardware (i.e., ICCD vs. CCD)
-
Turbulence and Combustion Research Laboratory
From turbulent flow theory, the highest spatial frequency of wavenumber κ for turbulent fluctuations is estimated as κ λK = 1 (Pope, 2000)
Corresponds to < 2% of the total mean dissipation
To resolve a spatial frequency of wavenumber κ (rad/mm) =1/ λK, the physical wavelength to be resolved is ~ 2πλK; Nyquist criteria ~3λK
Spatial Resolution
1
10
100
1000
100 1000 10000 100000 1000000
λ K(µ
m)
Reynolds number, Reδ
δ = 5 cmδ = 1 cmδ = 0.5 cm
1
10
100
1000
0.1 1 10 100 1000
Kol
mog
orov
Sca
le, λ
K(µ
m)
Bulk Velocity (m/s)
T = 300 KT = 700 KT = 1500 KT = 2000 K
δ = 2 cm
measurements measurements
-
Turbulence and Combustion Research Laboratory
Can we measure the smallest length scales instead of estimating them?
Yes – if the measurements are well resolved, then the smallest length scales can be determined
Back to the concept introduced by Pope…measure the power spectral density (PSD) of a fluctuating quantity
Measuring Length Scales?
2( ) (2 / ) ( )sE N DFT xκ κ=2( ) 2 ( )sD Eκ κ κ≈
(49)
(50)
-
Turbulence and Combustion Research Laboratory
Turbulent flows show a broad range of time scales similar to the range of length scales
Recent explosion of “high-speed” imaging diagnostics dictates that temporal (or sampling) resolution needs discussing.
What is the shortest time scale that should be resolved, τK?
Majority of measurements are in the Eulerian frame, thus the turbulent flow is convecting past the measurement volume
Thus, a convective time scale, τc, should be considered
Temporal Resolution
;Uδδτ ∝
< >;
'oo
llu
τ ∝2
;K KKKu
λ λτν
∝ ∝1/2Re
K
δδ
ττ
∝Large TDR but more manageable than SDR
Kc U
λτ =< >
δ = 2 cm0.01
0.1
1
0.1 1 10 100 1000
τ c/τ
K
Bulk Velocity (m/s)
T = 300 KT = 1000 KT = 2000 K
δ = 2 cmτc/τK dependent on δ;δ ↑, τc/τK ↓
-
Turbulence and Combustion Research Laboratory
For time-resolved measurements, the data acquisition rate (required frequency response, f ) should be fast enough to resolve velocity/scalar fluctuations as they are convected over a particular scale (wavenumber).
Temporal Resolution
0.1
1
10
100
1000
0.1 1 10 100 1000
Con
vect
ive
Velo
city
(m/s
)
Resolved Wavenumber, k (1/mm)
110 0.1 0.01Length Scale (mm)
Typical commercial kHz lasers
Typical spatial resolution
-
Turbulence and Combustion Research Laboratory
Lecture Summary
We examined the various aspects that determine optical resolution including the ability to focus a laser (form a “thin” sheet) and limitations in hardware (lens/camera)
The laser spot size (laser sheet thickness) is primarily limited by the divergence of the laser beam for high-quality, scientific lasers
The projected area onto a pixel is NOT the spatial resolution of a camera/lens system
The spatial resolution is a complex function of lens FN, magnification, and other optical transfer functions that may exist in the system
In general, the resolution of a camera system is much worse than the pixel projected area and the diffraction-limited blur size
The hardware resolution limits (LST and camera resolution) must be compared to the smallest turbulent/flame length scales.
For high-Reynolds number flows (even lab scale), it may be difficult to resolve the smallest length scales
Lecture 1 - IntroductionLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44
Lecture 2 - Light and OpticsLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34
Lecture 3 - Measurement ResolutionLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37
Lecture 4 - Velocimetry TechniquesLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29
Lecture 5 - PIV basicsLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33
Lecture 6 - PIV analysis and examplesLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30
Lecture 7 - Emerging and Altnerative VelocimetryLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2
Lecture 8 - Basic spectroscopyLaser Diagnostics in Turbulent �Combustion ResearchSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33