Laser Diagnostics in Turbulent Combustion Research · Laser Diagnostics in Turbulent Combustion Research Jeffrey A. Sutton Department of Mechanical and Aerospace Engineering ... broadband,

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


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


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


Why Study Combustion?

= 68%

= 85%


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


Why Laser Diagnostics ?Laser-based measurements are an important tool for studying

combustion processes in detail


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


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)

?


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


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


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)


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.


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.


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.)


Instrumentation - Lasers?

Not what we will talk about today - typically we want non-intrusive methods for measurements


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


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


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)


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


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


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!


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.


Light Source, Optics, and Camera



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.



Stohr et al, CNF, 2012



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


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



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



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



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


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



We will discuss two types of spontaneous scattering processes: (i) Rayleigh scattering and (ii) Raman scattering

Miles et al, MST, 2001

Rayleigh-Brillouin



Rayleigh scattering is the quasi-elastic scattering from particles with diameter



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



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



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)



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


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


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



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



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



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



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


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




Lecture 2 – Light and OpticsTurbulence 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



Light is an electromagnetic (EM) wave; EM waves are described by their position in the EM spectrum

Light and Electromagnetic Waves


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


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


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


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’


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)


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)


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


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


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


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


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)


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)


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)


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)


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]


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


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)

( )


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)


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∇ ⋅ = ⋅ =


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)


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 − ±− − = +


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


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)


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


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)


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)


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)


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)


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)


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)


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.




Lecture 3 – Measurement Resolution and Challenges



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



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+ → +


…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


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


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


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


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


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)


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

=


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

− −

=


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)


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


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)


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


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


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/


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)


Camera Resolution

https://carlesmitja.net/2011/02/06/image-quality-of-photographic-cameras/


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)


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)


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


( )s q ppN S K Gη= (43)


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


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)



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


sat DC

R

S SDRN−

= (45)


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


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)


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


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)


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)



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


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


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)


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


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)


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 ↓


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


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

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Laser Diagnostics in Turbulent Combustion Research · Laser Diagnostics in Turbulent Combustion Research Jeffrey A. Sutton Department of Mechanical and Aerospace Engineering ... broadband,