Laser Velocimetry in Fluid Mechanics E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 533

Veröffentlichungsjahr: 2013

Beschreibung

In fluid mechanics, velocity measurement is fundamental in order to improve the behavior knowledge of the flow. Velocity maps help us to understand the mean flow structure and its fluctuations, in order to further validate codes.
Laser velocimetry is an optical technique for velocity measurements; it is based on light scattering by tiny particles assumed to follow the flow, which allows the local fluid flow velocity and its fluctuations to be determined. It is a widely used non-intrusive technique to measure velocities in fluid flows, either locally or in a map.
This book presents the various techniques of laser velocimetry, as well as their specific qualities: local measurements or in plane maps, mean or instantaneous values, 3D measurements. Flow seeding with particles is described with currently used products, as well as the appropriate aerosol generators. Post-processing of data allows us to extract synthetic information from measurements and to perform comparisons with results issued from CFD codes. The principles and characteristics of the different available techniques, all based on the scattering of light by tiny particles embedded in the flow, are described in detail; showing how they deliver different information, either locally or in a map, mean values and turbulence characteristics.

Leseprobe

Table of Contents

Preface

Introduction

Chapter 1: Measurement Needs in Fluid Mechanics

1.1. Navier–Stokes equations

1.2. Similarity parameters

1.3. Scale notion

1.4. Equations for turbulent flows and for Reynolds stress tensor

1.5. Spatial–temporal correlations

1.6. Turbulence models

1.7. Conclusion

1.8. Bibliography

Chapter 2: Classification of Laser Velocimetry Techniques

2.1. Generalities

2.2. Definitions and vocabulary

2.3. Specificities of LDV

2.4. Application domain of laser velocimeters (LDV, PIV, DGV)

2.5. Velocity measurements based on interactions with molecules

2.6. Bibliography

Chapter 3: Laser Doppler Velocimetry

3.1. Introduction

3.2. Basic idea: Doppler effect

3.3. Fringe velocimetry theory

3.4. Velocity sign measurement

3.5. Emitting and receiving optics

3.6. General organigram of a mono-dimensional fringe velocimeter

3.7. Necessity for simultaneous measurement of 2 or 3 velocity components

3.8. 2D laser velocimetry

3.9. 3D laser velocimetry

3.10. Electronic processing of Doppler signal

3.11. Measurement accuracy in laser velocimetry

3.12. Specific laser velocimeters for specific applications

3.13. Bibliography

Chapter 4: Optical Barrier Velocimetry

4.1. Laser two-focus velocimeter

4.2. Mosaic laser velocimeter

4.3. Bibliography

Chapter 5: Doppler Global Velocimetry

5.1. Overview of Doppler global velocimetry

5.2. Basic principles of DGV

5.3. Measurement uncertainties in DGV

5.4. Bibliography

Chapter 6: Particle Image Velocimetry

6.1. Introduction

6.2. Two-component PIV

6.3. Three-component PIV

6.4. Bibliography

Chapter 7: Seeding in Laser Velocimetry

7.1. Optical properties of tracers

7.2. Particle generators

7.3. Particle control

7.4. Particle behavior

7.5. Bibliography

Chapter 8: Post-Processing of LDV Data

8.1. The average values

8.2. Statistical notions

8.3. Estimation of autocorrelations and spectra

8.4. Temporal filtering: principle and application to white noise

8.5. Numerical calculations of FT

8.6. Summary and essential results

8.7. Detailed calculation of the FT and of the spectrum of fluctuations in velocity measured by laser velocimetry

8.8. Statistical bias

8.9. Spectral analysis on resampled signals

8.10. Bibliography

Chapter 9: Comparison of Different Techniques

9.1. Introduction

9.2. Comparison of signal intensities between DGV, PIV and LDV

9.3. Comparison of PIV and DGV capabilities

9.4. Conclusion

9.5. Bibliography

Conclusion

Nomenclature

List of Authors

Index

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Library of Congress Cataloging-in-Publication Data

Laser velocimetry in fluid mechanics / edited by Alain Boutier.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-397-5

1. Fluid dynamic measurements. 2. Fluid mechanics. 3. Laser Doppler velocimeter. I. Boutier, A. (Alain)

TA357.5.M43L385 2012

532-dc23

2012015529

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-397-5

Preface

This book has been elaborated from lectures given in the context of autumn schools organized since 1997 by AFVL – Association Francophone de Velocimetrie Laser (French-speaking Association of Laser Velocimetry).

AFVL activities are especially dedicated to foster and facilitate the transfer of knowledge in laser velocimetry and all techniques making use of lasers employed for metrology in fluid mechanics. Among the main objectives, a good use of laser techniques is investigated in order to fulfill requirements of potential applications in research and industry.

The authors of this book have thus shared their expertise with AFVL, which led them to write the various chapters within a teaching perspective, which allows the reader to learn and perfect both his theoretical and practical knowledge.

Alain Boutier

September 2012

Introduction1

In fluid mechanics, velocity measurement is fundamental to improve knowledge of flow behavior. Flow velocity maps are key to elucidating mean and fluctuating flow structure, which in turn enables code validation.

Laser velocimetry is an optical technique for velocity measurement: it is based on light scattering by tiny particles used as flow tracers, and enables the determination of local fluid flow velocity as well as its fluctuations. Particles, approximately 1 μm in size, are used because the light flux they scatter is about 104 more intense than this due to molecular diffusion. Nevertheless, these particles (which are the fundamental basis of this technique) have two main disadvantages: discontinuous information (because data sampling is randomly achieved) and inaccurate representation of the fluid velocity gradients.

For each technique, the basic principles, along with the optical devices and signal processors used, are described. Chapter 7 is specifically dedicated to flow seeding; it describes products currently used and appropriate aerosol generators. Data post-processing has been also extensively developed: it allows synthetic and phenomenological information to be extracted from the vast quantities of data coming from detailed measurements. As a result, a link can be established between flow physics and predictions from codes.

This book presents various laser velocimetry techniques together with their advantages and disdvantages and their specificities: local or planar, mean or instantaneous, 3D measurements.

Another book by the same authors, entitled Laser Metrology in Fluid Mechanics [Bou 12] describes velocity measurements by spectroscopic techniques, which are based on molecular diffusion and are better suited for very high-velocity flow characterization. In this other book, two chapters are specifically dedicated to light scattering and to particle granulometry by optical means, these measurement techniques being more dedicated to two-phase flow studies. The main recommendations concerning laser security are also recalled.

Bibliography

[BOU 12] BOUTIER A. (ed.), Laser Metrology in Fluid Mechanics, ISTE, London, John Wiley & Sons, New York, 2012.

1Introduction written by Alain BOUTIER.

Chapter 1

Measurement Needs in Fluid Mechanics1

Measurements provide useful information for the interpretation of physical phenomena and for code validation. Fluid mechanics is based on nonlinear Navier– Stokes equations, which are very difficult to solve directly; simplifying assumptions or numerical approximations are used in order to make calculation times reasonable. Sometimes empirical relations are established when theory is not available; in particular, turbulent regime analysis leads to the building of new theories that must be verified. All these processes require validation by experiments and accurate measurements.

The most famous names in physics are associated with knowledge evolution in fluid mechanics, from Newton to Euler, Navier and Stokes, and also Bernoulli, Lagrange, Leibniz and Cauchy.

Theoretical approaches consist of mathematical resolution of partial differential equations. When an analytical solution is not possible, numerical approaches are used, but must be verified by well–documented experiments. In fluid mechanics, more than elsewhere, the three approaches (theory, simulation, and experimentation) often cannot be separated.

Theoretical treatment is exact and universal, but requires good physical knowledge of the phenomena. Boundary conditions are often made ideal and solutions are not available for complex flow configurations.

Numerical simulation provides complete flow information, with conditions that can be easily modified. Nevertheless, the process is often very expensive to put into operation, is limited by the computer power, and as turbulence models are not universal, a certain ability is required for correct employment.

Experimental investigations make parametric studies possible, in order to recognize which parameters are influent; sometimes it is the only way to obtain information. Yet they may appear rather complicated and expensive to implement; not all the variables can be measured and the intrusive character of the measuring method must be minimized.

1.1. Navier–Stokes equations

General equations in fluid mechanics are based on mass and energy conservation, as well as on movement quantity equations. These equations, called Navier–Stokes equations, make use of spatial and temporal partial derivatives of velocity and temperature, at first and second order. Even if exact solutions exist for simple laminar flows, for real flows, which are turbulent and 3D, calculations become much too complex to be solved by current computers within acceptable timescales. Therefore, numerical solutions are not exact and generate errors that must be evaluated by experiments and appropriate measurements.

The continuity equation (mass conservation) is expressed by:

[1.1]

where ρ is the volume mass and the velocity vector, with (u, v, w) coordinates in the frame (x, y, z) or (u1, u2, u3) in the frame (x1, x2, x3).

[1.2]

The movement quantity equation expresses the fact that the system movement quantity derivative is equal to the sum of the forces acting on the system. Using some assumptions, mainly that of Newtonian flow, this vector equation is written:

[1.3]

is the constraint tensor, which makes pressure P and dynamic viscosity µ appear. represents the unity tensor.

In incompressible conditions, movement quantity equation along x is reduced to:

[1.4]

The energy conservation equation interprets that total energy variation E of the fluid contained inside a volume is equal to the summation of the mechanical and thermal energies introduced into this volume. It is written as:

[1.5]

is the conduction heat flux, expressed by Fourier’s law; in this expression, λc is thermal conductivity. We can also derive similar equations for internal energy, enthalpy, total enthalpy or entropy. These equations are deduced from one another using the definitions of considered quantities.

The velocity gradient tensor describes deformation kinematics of a volume element:

[1.6]

It is decomposed into a symmetric tensor (deformation) and an anti–symmetric tensor (rotation):

[1.7]

The rotational part of the velocity field is called the vorticity:

[1.8]

Flows having a velocity potential are characterized by , a condition that is not valid for turbulent flows. A transport equation for vorticity is generally obtained when combining Navier–Stokes equations.

1.2. Similarity parameters

Dynamic and geometric similarity between two flows can be established using general adimensional equations. The following adimensional variables are generally used:

[1.9]

where ρ∞, V∞, p∞ and µ∞ are reference values and L a length characteristic scale, for instance, a wing or model chord.

When introducing these adimensional variables into movement quantity equation [1.3], written for a stationary flow (for instance), it becomes:

[1.10]

is the viscous part of the constraint tensor (terms in µ). γ is the ratio between specific heats at constant pressure (Cp) and at constant volume (Cv).

The Reynolds number (Re∞) and Mach number (M∞) are adimensional numbers. If two flows with the same boundary conditions provide identical values for Re∞ and M∞, then general equations of both flows are identical, as are their solutions.

The Reynolds number and Mach number are not the only similarity parameters. When taking into account various effects such as compressibility, instationarity, gravity, etc. other adimensional numbers appear in equations. The following table summarizes the main adimensional numbers used in fluid mechanics.

Geometric similarity is obtained when a geometric homothety allows passage from reality to a model. Thermal and dynamic similarities impose conservation of the adimensional parameters previously defined. Generally, all these conditions cannot be simultaneously satisfied.

Coupling experimental and numerical methods is indispensable for a better handling of phenomena in fluid mechanics. Good validation is achieved only if these two approaches are coupled in a complementary way. Validation requires using appropriate equations and boundary conditions; the nature of numerical solutions must be checked before analyzing the experimental results. For laminar flows, validation does not raise any specific problems. Description of flows with shock waves remains a problem.

1.3. Scale notion

Turbulent flows are also treated by simulation means, but the problem is caused by the fact that these flows present a wide spectrum of space and timescales. In order to obtain an exact solution for a turbulent flow, small and large scales (time and space) contained in the flow must be solved. The ratio between length scales (according to Kolmogorov, small η over large δ) is given by the following relationship:

[1.11]

where Re is the Reynolds number formed with characteristic scales (velocity, length) of large structures. It appears that ranges of large and small scales deviate more with increasing Reynolds numbers, which induces increasing difficulties for the resolution of all scales at higher Reynolds numbers. The computation of turbulent flow inside a volume of 1 m3 would take too much time (depending upon the Reynolds number, velocity and viscosity). Methods that avoid solving all scales make use of models in order to reduce prohibitive Direct Numerical Simulation (DNS) calculation times: these use Large Eddy Simulation (LES) and Reynolds Averaged Navier–Stokes (RANS) methods.

1.4. Equations for turbulent flows and for Reynolds stress tensor

The classic statistical description of turbulent flows is based on velocity and instantaneous pressure decomposition into a mean part (which is time independent) and a fluctuating part (which is time dependent). For the velocity component, ui becomes:

[1.12]

The mean temporal value is:

[1.13]

The resulting mean Navier–Stokes equations then include additional terms, called Reynolds stresses. For instance, the movement quantity equation along x for a 2D and stationary flow takes the following form:

[1.14]

Taking into account these new terms and , closure of this equation is one of the main objectives for turbulence modeling.

The fluctuating part of velocity leads to the definition of turbulence intensity. It can be related to a velocity component or to the vector modulus:

[1.15]

Kinetic turbulent energy is defined by the relation:

[1.16]

[1.17]

From Navier–Stokes equations, specific equations for Reynolds stresses , can be deduced. Generally, these equations are as follows:

[1.18]

For small flow scales, turbulent kinetic energy is dissipated as internal energy (heat) by the action of viscosity. Considering a kinematic viscosity v[m2s–1] and a dissipation rate by a mass unit  ε[m2s–3], the smallest movement scales fixed by viscosity are characterized by the length scale n previously introduced (Kolmogorov scale):

[1.19]

When developing the tensor form of dissipation, the said "true" (slightly different from this identified in equation [1.20]) dissipation rate, ε, per unit mass of turbulent kinetic energy k is obtained:

[1.20]

If the turbulence field is locally isotropic, the dissipation rate divided by v is given by:

[1.21]

where only one gradient has to be determined. In several cases, this gradient may be indirectly obtained using Taylor’s hypothesis, which is based on the following arguments.

If then the characteristic time l/u′1 of turbulent large scale movement becomes large compared to time , which characterizes their convection, so that the scales are “frozen” during their observation. Then it is settled:

[1.22]

A temporal gradient is easier to obtain than a spatial gradient by experimentation. Only one point measurement is required and it may be directly computed using a temporal analysis of velocity evolution.

1.5. Spatial–temporal correlations

Spatio–temporal correlation function, Rij(xk,t,rk,τ) is defined by:

[1.23]

Fluctuation u′i is measured at a point A with coordinates xk at a time t. Fluctuation u′j – is measured at a point B separated from A by rk, with a time delay τ (which is considered to be zero in this case). Spatial correlations contain complete information about the structure of the velocity field. A Fourier transform of these functions provides the second–order spectral tensor, depending on wavelength where kk is the wave number in direction k:

[1.24]

The half sum of diagonal components of 1/2 Φij, i.e. Φ11 + Φ22 + Φ33 represents the total kinetic energy per unit mass, at a given wavelength.

The dissipation rate per unit mass can also be obtained from the wavelength spectrum by:

[1.25]

It is not generally possible to measure the three wavelength spectral components in 3D, because it would involve the determination of spatial correlations for all components separately in all directions. Therefore, only one velocity component and one spatial correlation are acquired (or a component associated with a temporal correlation): using Taylor’s assumption, spatial correlation is deduced along the main flow direction.

[1.26]

Measurement of frequency spectral density deduced from the fluctuation measurement, u′1, provides, for instance, Φ11(k1) via Taylor’ s assumption:

1.6. Turbulence models

The objective for using turbulence models is to determine a temporal mean value of the Reynolds stress and scalar transport term , where  φ represents, for instance, a temperature fluctuation (or a fluctuation of chemical species concentration, etc.).

These algebraic or differential models contain empirical constants, which must be experimentally determined; otherwise direct numerical simulations must be used. Empirical information is introduced into models via two methods:

– by derivation of exact equations (transport equations) for , and then is deduced (which is very complex);

– by giving an expression to and which is deduced from semi–empirical transport equations.

The most widely used modeling strategies are based on the assumption of turbulent viscosity, which mimics Newtonian behavior law: the turbulent constraint is linearly linked to the mean deformation rate tensor:

with (turbulent kinetic energy).

δij is Kronecker’s symbol.

Turbulence models are classified as a function of the number of transport equations used to determine µt. Several forms of models exist; four should be noted.

1.6.1. Zero equation model

The velocity scale and length scale are given algebraically (for instance, the Prandtl mixing length, a model used since 1925):

1.6.2. One equation model

One semi–empirical transport equation is used to determine velocity scales. Length scales are given algebraically. For instance:

ε being the dissipation rate:

Then the exact equation of kinetic energy k is:

It is modeled along the following form:

[1.27]

Constant σk ≈ 1 is determined by comparing experimental and numerical results on selected flows.

1.6.3. Two equations model

[1.28]

1.6.4. Reynolds stress models (RSM, ARSM)

In this case, the notion of turbulent viscosity disappears and partial differential equations of are solved:

[1.29]

Closing relations are obviously required in order to solve the system.

1.7. Conclusion

Experiments are essential for the development and implementation of turbulence models. New models will be proposed if high–order correlation measurements and measurements giving access to the dissipation rate are developed, because the quantities to be resolved are described in equation [1.20].

Validation of LES models requires a volume mean value of velocity fluctuations, with high spatial resolution. Validation of global flow parameters requires that spatial structure measurements are temporally solved.

Nowadays, DNS are validated using data issued from simple flows; in the future, more complex quantities would be needed, as acceleration or two–point correlations. Dissipation measurement would be the optimum.

Detailed developments of the subjects described in this chapter (in particular mathematical demonstrations) can be found in [BON 89, COU 88, COU 89].

1.8. Bibliography

[BON 89] BONNET A., LUNEAU J., Théories de la dynamique des fluides, Cepaduès, Paris, 1989.

[COU 88] COUSTEIX J., Couche limite laminaire, Cepaduès, Paris, 1988.

[COU 89] COUSTEIX J., Turbulence et couche limite, Cepaduès, Paris, 1989.

1Chapter written by Daniel ARNAL and Pierre MILLAN.

Chapter 2

Classification of Laser Velocimetry Techniques1

Velocity measurement techniques may be classified into two categories depending on whether they are based on the diffusion of molecules or particles. The purpose of the molecules or particles is to track flow fluctuations so that these can be studied [BOU 93]. In each category, methods exist to enable either local or planar measurements; they can be instantaneous or averaged. Methods are based on two main principles: time-of-flight measurement (dx or dt are fixed) or determination of the double Doppler shift due to the chain fixed light source/mobile scattering tracer/fixed observer.

Measuring systems based on molecules are less well developed than those based on particles, essentially because they have lower accuracy and their domain of application is more limited to reacting flows and very high velocity flows.

Measuring systems based on particles are more frequently used, and are classified according to their method and data. The main difficulty of all devices based on particle scattering is that particle inertia must be taken into account; flow seeding is a key problem for measurement quality and the degree of confidence that may be applied to the results.

The different methods to determine velocity using laser velocimetry will be introduced, as well as the concepts leading to various optical set-ups. The vocabulary associated with the different apparatus will be provided, and the advantages and drawbacks associated with each method will be listed.

2.1. Generalities

Velocity determination in fluid mechanics is fundamental to elucidate flow behavior. Maps of the velocity flow field enable the validation of flow structure and turbulence properties, and thus, the validation of codes.

Laser velocimetry is an optical technique, which uses tiny particles as flow tracers in order to determine local velocities and their fluctuations; many optical schemes have been developed since the first paper published by Yeh and Cummins in 1964 [YEH 64]. We shall distinguish the various techniques related to laser velocimetry. During velocity determination, the information delivered by each instrument must be well clarified.

Setting up any laser velocimeter involves the following subjects:

– the purpose of velocity measurements in a well-defined flow generally enables the determination of the optical scheme of the apparatus;

– as measurements are generally local, the mechanics supporting and moving optics must be studied in order to cope with optical access in facilities;

– as all measurements rely upon the presence of scattering particles inside the flow, flow seeding and particle characterization are the two main issues that must be carefully checked;

– data signal processing is one of the most important features of any apparatus; depending upon processor efficiency, measurement quality may vary enormously: sensitivity to different particle sizes, time required to obtain convenient statistics, etc.;

– a computer connected to the laser velocimeter is fundamental, because of its multiple functions: data acquisition from processors; displacement management of the measuring point; data reduction (taking into account initial flow conditions) in order to display results along understandable curves, maps, or organized data banks.

In the following chapters, optical devices will be described in detail because their choice is crucial: the general arrangement defines the way velocity is measured and thus further possible data interpretation. The quality of the optical set-up must be carefully checked in order to provide optimized signals to the electronic processors; in fact these processors will never improve or correct previous mishandlings in optics. The main topics covered are:

– processing and post-processing of signals;

– flow seeding and particle size measurement;

– measurement accuracy.

Laser velocimetry has become an operational measuring tool; this is evident by the periodic international conferences that are held dealing with the advances and applications of laser velocimetry; the proceedings of the main conferences are listed in [THO 72, THO 74, LDA 75, THO 78, ISL 80, LAS 85, CLV 87, LAA 85, 87, 89, 91, 97, ALA 82, 84, 86, 88, ALT 90, 92, 94, 96, 98, 00, 02, 04, 06, 08, 10]. National French conferences are listed in [FAL 88, FVL 90, 92, 94, 96, 98, 00, 02, 04, FTL 06, 08, 10]. VKI (Von Karman Institute) lectures are listed in [VKI 88, 90, 91, 96, 99, 00].

2.2. Definitions and vocabulary

Two terms are currently used in velocity measurement in fluid mechanics: laser velocimetry and laser anemometry. The first comes from the Latin “velox”, which means a measurement of the speed of rapid objects; the second comes from the Greek “anemos”, which means wind measurement. Therefore, as fluid dynamics includes aerodynamics as well as hydrodynamics and combustion, the expression “laser velocimetry” appears more general. The same technique is either called laser Doppler velocimetry or laser Doppler anemometry, Doppler being used for historical (and obviously physical) reasons linked to the first developments of the technique.

Velocity is by definition the first derivative of the trajectory of a mobile object; it is a vector determined by three components. The usual terminology for the measuring apparatus is as follows:

– One dimensional (1D): apparatus giving access to only one velocity component;

– Two dimensional (2D): apparatus giving access simultaneously to two velocity components (velocity vector projection on a plane): which is the area of interest in 2D flows;

– Three dimensional (3D): apparatus giving access simultaneously to the three velocity components, i.e. the instantaneous velocity vector, which further enables the determination of all the turbulence properties of the flow using the appropriate calculations and statistics.

A velocity component (i.e. the projection of the velocity vector on the x-axis for instance) of any mobile object (which is a particle in laser velocimetry) is given by this basic relationship:

This means that the object has traveled (on this projection axis) along a distance dX during a time interval dT. It immediately appears that any local measurement requires a short dX, which induces a small dT. This feature is a perpetually important constraint in laser velocimetry, because improvements in accuracy are obtained by increasing either dX or dT, which induces poorer localization of the measurement, which is generally prohibited in flow regions where important velocity gradients exist. This is the reason why compromises are necessary in any laser velocimeter set-up due to this fundamental discrepancy between good knowledge of the position and good knowledge of the velocity of any mobile object.

The various types of laser velocimeters may be classified into three categories:

1) dX is known and dT is measured. This means that the velocimeter is measuring the time interval, dT, during which a particle travels over a fixed distance dX. This kind of laser velocimeter is called an optical barrier velocimeter because distance dX is materialized by a given geometry of laser beams inside the probe volume. This technique is currently called laser transit anemometry (LTA), laser transit velocimetry (LTV), or laser two-focus (L2F; laser beams are focused along two spots or two dashes inside the probe volume). The main advantage of these systems is due to a high signal-to-noise ratio, which makes them very efficient in turbomachinery experiments where flows are confined in very narrow channels (stray light scattered by walls close to the probe volume is thus very high); their main drawback is the difficulty in performing measurements in turbulent flows.

In a new concept for apparatus, the optical barrier is set in the receiving part where the image of the particle trajectory is analyzed. This device, which looks like a probe, makes use of optical fibers and is called a mosaic laser velocimeter; it enables 3D measurements in highly turbulent flows.

3) The last idea may be expressed in two ways, which a priori appear completely different, but are hopefully physically the same; we shall emphasize this feature in Chapter 3, in order to establish a clear coherent statement:

- when an object is moving, we must think of the Doppler effect: the frequency it receives from a fixed source is changed, as well as that it emits when received by a fixed observer. These frequency shifts are directly related to the velocity vector of this moving object. This idea was first published in 1964 [YEH 64], taking advantage of the interesting characteristics of a new light source, called laser, which provided an intense light beam, with small divergence and a well-defined optical frequency (with narrow spectral bandwidth).

These systems are currently called LDV or LDA; we shall see that there are four possible optical set-ups leading to the following set-ups:

– spectrometer: essentially used in transient flows at high velocities;

– reference beam: historically the first, and still used in airborne devices;

– one beam: practically never employed;

– dual-beam or fringe system: the most widely used and able to provide the instantaneous velocity vector for each particle.

Since 1992, a new type of instrument has been developed, which transforms the Doppler shift into a variation of light intensity (the Doppler shifted scattered light is filtered through a steep slope of an iodine absorption line). A one-component velocity map is then obtained in real time as a result of this technique, which is called Doppler global velocimetry (DGV); presently this technique gives access to the three velocity components in a plane.

2.3. Specificities of LDV

2.3.1. Advantages

Laser velocimetry (LDV) is considered as a non-intrusive technique, because it is an optical method with a probe volume made of laser beams. This is particularly important in transonic and supersonic flows or in combustion phenomena where introducing probes frequently modifies the flow structure. Nevertheless as measurements rely on the presence of particles inside the flow, flow seeding must be very carefully achieved, with an injection point situated far upstream of the probe volume, in order that the injection device does not disturb the flow. It should be remembered that in physics any measurement modifies the measured object to a certain degree. Seeding is generally so diluted that the flow cannot be considered as a two-phase flow; pressure radiation on particles and flow heating due to laser beams inside the probe volume are two very weak effects, essentially due to the finite and short transit time of particles (and fluid) through the probe volume.

Instrument response is a linear function of the velocity parameter, which is an important feature compared to hot-wire anemometry, which is sensitive to a combination of velocity and temperature.

Owing to special optical devices used in most fringe laser velocimeters, the sign of each measured component is determined, allowing confident results in complex flows (such as recirculation zones, vortices, highly turbulent flows) in which the instantaneous velocity vector may have any direction in space as a function of time. PIV, DGV, and mosaic laser velocimeters also have this capability. Therefore, laser velocimetry techniques are the sole techniques able to provide valid measurements in highly turbulent flows or in reverse flows.

A laser velocimeter performs local and instantaneous measurements of the velocity vector (or at least of one or two of its components). Typically, the size of the probe (or spatial resolution in global measurements) is approximately a few hundred microns and each velocity sample is obtained within a few microseconds (or hundred milliseconds in low-speed flows). The size of the probe is smaller than the smallest significant flow turbulence scale, and the duration of a measurement is so short that it enables measurement of the highest frequencies of the turbulence spectrum (if the particle response to high frequencies is adequate, i.e. if the particles are small and light enough and if the data rate is high enough). It is assumed that the velocity is constant during the time a particle crosses the probe volume.

2.3.2. Use limitations

Unfortunately no method is perfect. The main problem in laser velocimetry comes from its basic principle: the fluid velocity is not directly measured but instead the velocity of tiny particles (which are assumed to accurately track the flow) is measured. This is due to the scattering cross-section of particles, which is much larger than that of the molecules. The main limitations are thus the following:

– as the particle arrival rate across the probe volume is random, statistical bias may occur, which must be carefully checked, and velocity fluctuation spectral analysis is difficult to undertake;

– very high temperature flows are difficult to investigate because refractory particles must be used; the present limitation is imposed by the use of zirconium dioxide (ZrO2) aerosols resisting up to 2,700 K (but they are heavy);

– the wall approach for boundary layer exploration remains difficult due to problems caused by stray light; when laser beams are parallel to the wall, the probe volume can be set tangent to the wall (allowing measurements down to a few hundred microns from the wall), but when they are perpendicular to the wall the minimum distance is usually in the range of a few millimeters (unless special systems are designed: L2F velocimeter, fluorescent paints, reflecting or glass wall, etc.);

– very low turbulence levels, i.e. below 0.1%, are not obtained (contrary to hot-wire anemometry) and the reasons for this still require further investigation.

The volume occupied by an operational laser velocimeter, laser implementation, and consumption make this technique rather difficult to use initially, more so as a scientist will have to deal with optics, mechanics, signal processing, data analysis, flow seeding, and finally, fluid dynamics. The cost of operational apparatus is relatively high, because high-quality components must be used.

2.4. Application domain of laser velocimeters (LDV, PIV, DGV)

Application of laser velocimetry in fluid mechanics covers many domains:

– hydrodynamics with velocities ranging from centimeters per second to a few meters per second; if signals are generally large, the implementation difficulties are due to the air-glass-water (or any liquid) interfaces;

– aerodynamics in subsonic, transonic, supersonic, and even hypersonic regimes, i.e. in a velocity range from 1 m/s up to 3,900 m/s;

– convective flows and flows in medical research are in the micron per second or millimeter per second velocity range, which are time consuming regarding data acquisition and generally require non-conventional electronics;

– combustion or flame studies: seeding and laser beam distortions are the main difficulties;

– turbomachinery experiments: flows are periodic and rapidly changing; optical access is reduced and stray light scattered by walls close to the probe volume is important;

– two-phase flows: velocity and particle measurements are simultaneously achieved (for particles typically larger than 5–10 μm);

– airborne systems for measuring aircraft speed.

Because the application domain of laser velocimetry is so wide, there are many types of apparatus set-ups. The most appropriate set-up will be chosen as a function of flow characteristics, of its surroundings, and of the data required. New trends in LDV development are focused on the use of optical fibers (which leads to more flexible set-ups), improvements in signal processing, reduction in instrument size by employment of diode lasers, and data post-processing. PIV has become the most widely used technique because it provides images of instantaneous velocity fields, allowing visualization and understanding of flow instabilities. DGV is more dedicated to high-velocity flows or large-dimension facilities. Laser granulometry is more concerned with two-phase flows.

2.5. Velocity measurements based on interactions with molecules

Four main techniques have been developed to measure velocities in hypersonic flows: electron beam fluorescence, laser fluorescence, absorption spectroscopy making use of tunable infrared diode lasers and coherent anti-Stokes Raman scattering (CARS). Depending upon the means employed for signal detection, these techniques perform a time-of-flight measurement or are based on the Doppler effect. They are detailed in the forthcoming book by the same authors entitled Laser Metrology for Fluid Mechanics [BOU 12].

2.5.1. Excitation by electron beams

A pulsed electron beam has been used for velocity determination in a hypersonic flow, linked to a Langmuir probe situated downstream [HIR 93]. The electron beam creates ions having a long life span; their time of flight is measured during the known distance they travel downstream from the electron beam, among neutral molecules. Beam pulses last 1 µs, with a repetition rate of 2 to 8 kHz, and the Langmuir probe behaves as an ion collector. The time interval between electron beam pulse and pulse delivered by the probe is measured and provides the velocity. The axial velocity component is obtained, because the reference distance separates the ion collection point on the probe from the vertical line defined by the electron beam. Depending upon the gas used in the flow, velocities measured by this technique range from 500 m/s up to 4,500 m/s.

A similar device has been designed for a low-density airflow (4×10–4 kg/m3) at Mach 10 [MOH 95a]. A pulsed electron beam excites the gas molecules along a vertical line. The movement of the plasma columns is made visible owing to the post-luminescence produced inside them by secondary electron excitation; these secondary electrons have enough energy to induce fluorescence from N2+1N and N22P nitrogen states. These secondary electrons have a long life span over a large distance downstream of the electron beam. The radiative lifetime of the excited molecules is very short (typically 60 ns), which means that fluorescence is emitted at the excitation point in columns for velocities less than 104 m/s. The velocity is obtained on measuring the distance traveled by these luminous plasma columns during a known time interval. For a given column, a camera shutter is open at different known instants in order to follow column displacement: the duration of opening is very short in order to fix column displacement at each picture. In practical terms, a series of columns is created when pulsing the electron gun at regular time intervals: images are added (assuming that flow velocity is constant) by the operating camera and electron gun at the same rate. An image intensifier is used (with a gain of several thousands) in order to adequately capture the weak post-luminescence of nitrogen. Using the integrated image obtained over 1 ms, distances between columns are measured, which provides the mean axial velocity component, which was 1,450 m/s in this application.

2.5.2. Laser fluorescence

RELIEF (Raman excitation plus laser-induced electronic fluorescence) [MIL 93] measures velocity using a time-of-flight method. Airflow is tagged by a vibrational excitation of oxygen molecules, and is later observed by laser-induced fluorescence. Lines are thus instantaneously written inside the flow (10 ns); their displacement (after an accurately measured time interval) provides the velocity. When a cross is placed inside the flow by means of two exciting crossed laser beams, this cross displacement provides an accurate measurement of the local velocity; cross rotation and line distortion can be used to evaluate the vorticity and turbulence. Tests have been performed up to Mach 4; even if results appear promising, a shorter wavelength laser source would deliver higher fluorescence intensities in low-density hypersonic flows.

In stable, non-reactive flows, velocity measurements in a plane can be achieved by taking advantage of the fluorescence induced by an argon laser on iodine molecules (the line width is refined by adding a Fabry-Pérot interferometer in the cavity); plane fluorescence is recorded using a charge-coupled device (CCD) camera, with integration times of 30 seconds [MCD 93]. The laser scans the iodine absorption line profile, which provides an image series from which the Doppler shift is deduced. These tests being relatively long, they require a stable flow and a uniform iodine vapor seeding; the major drawback of this technique comes from the high degree of corrosion induced by the iodine vapor, which makes this method difficult to apply. In reacting flows, the same technique is implemented on OH (a naturally present radical), using a tunable UV laser with a narrow line width.

2.5.3. Spectroscopy with a tunable laser diode in the infrared

A beam emitted by an infrared laser diode (around 5 μm) crosses a hypersonic flow containing nitrogen oxide (NO) molecules; absorption along the whole path is recorded. This is an integrated non-local measurement [MOH 95b]. Nominal laser diode frequency is monitored by temperature adjustment, via an electric current. This frequency rapidly scans the non-resolved absorption NO doublet at 1,924.457/cm and a saw-tooth modulation (period 2 ms) is added to the continuous current supplying the diode. The beam issued from the diode is collimated owing to a concave mirror, and only one laser mode is selected using a grating and a slit. This beam is then re-collimated and sent towards the test section. It is divided into two beams: beam A crosses the flow perpendicularly to its main axis, the other beam B crosses the same flow (owing to a mirror system) but along a direction making an angle of 63° with the flow axis (which creates a Doppler effect). These two beams are separately focused on two infrared detectors HgCdTe, cooled with liquid nitrogen. The signals are thus digitized and recorded. Doppler shift, and thus flow velocity, is deduced from the position difference of doublet centers in signals issued from the A and B paths. Velocities ranging from 2,000 m/s up to 4,500 m/s have been measured, with a ±5% uncertainty, mainly due to angle measurement between the laser beam and main flow axis.

2.5.4. Coherent anti-Stokes Raman scattering technique

New developments in coherent anti-Stokes Raman scattering (CARS) in the temporal domain, achieved at ONERA [LEF 96], have allowed instantaneous velocity measurements to be performed in hypersonic flows at low densities. This measurement technique is based on temporal analysis of anti-Stokes signals created by mixing inside flow a pump laser beam (30 ns) and a Stokes radiation of short duration (1 ns). The anti-Stokes signal is detected by a quadratic detector (quick photomultiplier having a bandwidth of several hundred megahertz); it is characterized by a decaying oscillation, the frequency of which provides instantaneous velocity flow. This method enables the measurement of two velocity components of nitrogen molecules, with a transverse spatial resolution of a few hundred microns. Nevertheless, in order to obtain a convenient signal-to-noise ratio, it is necessary to integrate the anti-Stokes response over several pulses, which leads to a temporal integration of several seconds. This CARS technique is mainly dedicated to the characterization of low-pressure flows at high velocities (supersonic or hypersonic regimes). For high-pressure conditions and low velocities, Brillouin scattering can be coupled to classical Raman scattering in order to simultaneously obtain velocity and temperature measurements.

2.5.5. Tagging techniques

A velocity measurement using a photo-thermal effect is proposed in [NAK 97]. A line is marked in a nitrogen and ethylene jet by a carbon dioxide laser having a pulse duration of 50 ns; absorption of laser energy heats the medium, so that the refraction index decreases. This index modification is detected by laser beam deflection. In order to improve detection sensitivity and speed, a differential interferometer measures the phase difference observed downstream of the tagged region. This method provides information on the deformation of a line traced inside the flow as it propagates downstream; it is not very accurate.

A similar method, called ozone tagging velocimetry (OTV), is described in [DEB 98]. This method uses photochemical tagging: an ozone line is created in the airflow using a pulsed ultraviolet (UV) laser of narrow bandwidth, emitting at 193 nm. Ozone is produced inside a reaction including molecular oxygen and atomic oxygen; this initial position is recorded by laser-induced fluorescence or by Rayleigh scattering. Ozone is created within 20 µs and resists more than 100 ms. Downstream, a second UV laser (reading laser) emits radiation at 248 nm, the light dissociates the ozone and creates atomic oxygen and molecular oxygen, which is excited in vibration: its fluorescence is recorded on an intensified CCD camera. This method measures time of flight and observes line deformation; it is more accurate than methods based on the Doppler effect, which often have a limited range. Nevertheless, temperatures must not exceed 500 to 600 K, because the ozone concentration decreases beyond these temperatures, but if the pressure increases the proportion of ozone is higher.

2.5.6. Summary

Table 2.1.Main information delivered by the different velocimetric devices

Table 2.1.(Continued) Main information delivered by the different velocimetric devices

Table 2.1.(Continued) Main information delivered by the different velocimetric devices

2.6. Bibliography

[ALA 82] 1st International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, 5–7 July 1982.

[ALA 84] 2nd International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, 2–5 July 1984.

[ALA 86] 3rd International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, 7–9 July 1986.

[ALA 88] 4th International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, 11–14 July 1988.

[ALT 90] 5th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 9–12 July 1990.

[ALT 92] 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 20–23 July 1992.

[ALT 94] 7th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 11–14 July 1994.

[ALT 96] 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.

[ALT 98] 9th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 13–16 July 1998.

[ALT 00] 10th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 10–13 July 2000.

[ALT 02] 11th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 2002.

[ALT 04] 12th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 12–14 July 2004.

[ALT 06] 13th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 26–29 June 2006.

[ALT 08] 14th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 7–10 July 2008.

[ALT 10] 15th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 5–18 July 2010.

[BOU 93] BOUTIER A., YANTA W.J., SMEETS G., “Velocity measurements in hypersonic flows: a review”, New Trends in Instrumentation for Hypersonic Research, NATO ASI Series, vol. 224, p. 593–602, 1993.

[BOU 12] BOUTIER A. (ed.), Laser Metrology in Fluid Mechanics, ISTE, London, John Wiley & Sons, New York, 2012.

[CLV 87] International Specialists Meeting on the Use of Computers in Laser Velocimetry, ISL Report no. 105/87, Saint-Louis, France, 18–20 May 1987.

[DEB 98] De Barber P.A., RIBAROV L.A., WEHRMEYER J.A., BATLIWALA F., PITZ R.W., “Quantitative unseed molecular velocimetry imaging”, 9th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 13–16 July 1998, paper 34–1.

[FAL 88] 1es journées françaises d’anémométrie laser, Marseille, France, 21–23 September 1988.

[FTL 06] 10e congrès francophone de techniques laser, Toulouse, France, 19–22 September 2006.

[FTL 08] 11e congrès francophone de techniques laser, Poitiers, France, 16–19 September 2008.

[FTL 10] 12e congrès francophone de techniques laser, Vandoeuvre-les-Nancy, France, 14–17 September 2010.

[FVL 90] 2e congrès francophone de vélocimétrie laser, Meudon, France, 25–27 September 1990.

[FVL 92] 3e congrès francophone de vélocimétrie laser, Toulouse, France, 21–24 September 1992.

[FVL 94] 4e congrès francophone de vélocimétrie laser, Poitiers, France, 26–29 September 1994.

[FVL 96] 5e congrès francophone de vélocimétrie laser, Rouen, France, 24–27 September 1996.

[FVL 98] 6e congrès francophone de vélocimétrie laser, Saint-Louis, France, 22–25 September 1998.

[FVL 00] 7e congrès francophone de vélocimétrie laser, Marseille, France, 19–22 September 2000.

[FVL 02] 8e congrès francophone de vélocimétrie laser, Orsay, France, 17–20 September 2002.

[FVL 04] 9e congrès francophone de vélocimétrie laser, Brussels, Belgium, 14–17 September 2004.

[HIR 93] HIRAï E., BüTEFISCH K., DANKERT C., “Velocity and density determination by electron beam technique”, New Trends in Instrumentation for Hypersonic Research, NATO ASI Series, vol. 224, p. 361–371, 1993.

[ISL 80] Symposium on Long Range and Short Range Optical Velocity Measurements, ISL Report 117/80, Saint-Louis, France, 15–18 September 1980.

[LAA 85] International Conference on Laser Anemometry, Advances and Applications, Manchester, UK, 16–18 December 1985.

[LAA 87] 2nd International Conference on Laser Anemometry, Advances and Applications, Glasgow, UK, 21–23 September 1987.

[LAA 89] 3rd International Conference on Laser Anemometry, Advances and Applications,. Swansea, UK, 26–29 September 1989.

[LAA 91] 4th International Conference on Laser Anemometry, Advances and Applications, Cleveland, USA, 5–9 August 1991.

[LAA 97] 7th International Conference on Laser Anemometry, Advances and Applications, Karlsruhe, Germany, 8–11 September 1997.

[LAS 85] International Laser Anemometry Symposium, ASME-FED, vol. 33, Miami, USA, 17–22 November 1985.

[LDA 75] “The accuracy of flow measurements by laser doppler methods”, Proceedings of the LDA Symposium, Copenhagen, Denmark, August 1975.

[LEF 96] LEFEBVRE M., SCHERRER B., BOUCHARDY P., POT T., “Transient grating induced by single-shot time-domain CARS; application to velocity measurements in supersonic flows”, JOSA B, vol. 13, 1996.

[MCD 93] MCDANIEL J.C., HOLLO S.D., KLAVUHN K.G., “Planar velocimetry in high speed aerodynamic and propulsion flowfields”, New Trends in Instrumentation for Hypersonic Research, NATO ASI Series, vol. 224, p. 381–390, 1993.

[MIL 93] MILES R., Lempert W., FORKEY J., ZHANG B., ZHOU D., “Filtered Rayleigh and RELIEF imaging of velocity, temperature and density in hypersonic flows for the study of boundary layers, shock structures, mixing phenomena and the acquisition of in-flight data”, New Trends in Instrumentation for Hypersonic Research, NATO ASI Series, vol. 224, p. 391–398,1993.

[MOH 95a] MOHAMED A.K., POT T. “Diagnostics by electron beam fluorescence in hypersonics”, ICIASF ’95, Dayton, OH, USA, 18–21 July 1995.

[MOH 95b] MOHAMED A.K., ROSIER B., HENRY D., LOUVET Y., VARGHESE P.L., “Tunable diode laser measurements on nitric oxide in a hypersonic wind tunnel”, AIAA 95–0428, 33rd Aerospace Sciences Meeting, Reno, NV, USA, January 1995.

[NAK 97] NAKATANI N., “Measurement of the flow velocity distribution of gases by the laser photothermal effect with a differential interferometer”, 7th International Conference on Laser Anemometry, Advances and Applications, p. 27–34, Karlsruhe, Germany, 8–11 September 1997.

[THO 72] THOMPSON H.D., STEVENSON W.H., “The use of laser doppler velocimeter for flow measurements”, Proceedings of a Workshop, Purdue University, Hammond, IN, USA, 9–10 March 1972.

[THO 74] THOMPSON H.D., STEVENSON W.H., Proceedings of the 2nd International Project Squid Workshop on the Use of Laser Velocimeter for Flow Measurements, Purdue University, Hammond, IN, USA, 27–29 March 1974.

[THO 78] THOMPSON H.D., STEVENSON W.H., “Laser velocimetry and particle sizing”, Purdue Workshop, 11–13 July 1978, Hemisphere Publishing Corporation, Washington, DC, 2005.

[VKI 00] Particle Image Velocimetry and associated Techniques, VKI Lecture series 200003, 17–21 January 2000.

[VKI 88] Particle Image Displacement Velocimetry, VKI Lecture Series 1988-06, 21–25 March 1988.

[VKI 90] Measurement Techniques for Hypersonic Flows, VKI lecture series 1990-05, 28 May to 1 June, 1990.

[VKI 91] Laser Velocimetry, VKI Lecture Series 1991-05, 10–14 June 1991.

[VKI 96] Particle Image Velocimetry, VKI Lecture Series 1996-03, 3–7 June 1996.

[VKI 99] Optical Diagnostics of Particles and Droplets, VKI lecture series 1999-01, 25–29 January 1999.

[YEH 64] YEH Y., CUMMINS H.Z., “Localised fluid flow measurement with a He-Ne laser spectrometer”, Applied Physics Letters, vol. 4, p. 1964.

1Chapter written by Alain BOUTIER.

Chapter 3

Laser Doppler Velocimetry1

In this chapter, four different velocimetry set-ups to measure one velocity component, deduced from Doppler effect theory, are described. The most widely used fringe laser velocimeter (laser velocimeter) is described in more detail: signal-to-noise ratio (SNR) evaluation, velocity sign determination, general structure of the emitting and receiving optics, signal processors.

Progressively two velocity components were simultaneously measured (for two-dimensional (2D) flow investigations), then three components simultaneously (which gives access to the local velocity vector). Currently, complex three-dimensional (3D) flows can now be studied in detail.

The great variety of systems able to simultaneously provide three velocity components is reviewed; among these set-ups, the most operational 3D laser velocimeter is based on a probe volume formed by three fringe patterns, with different directions in space and with three colors. Nevertheless, this optimal concept of a 3D laser velocimeter may comprise several mechanical and optical structures, the merits of which are discussed.

3.1. Introduction

The basis of laser velocimetry is that a moving particle changes the frequency of incoming light due to the Doppler effect, so that the velocity of the particle can be deduced from this frequency shift. We shall recall this theory and apply it to optical waves; in fact their frequencies are so high that special optical devices, which are described below, had to be designed in order to extract velocity information.

Then, having shown how one velocity component is determined, different means to measure the local instantaneous velocity vectors are described.

3.2. Basic idea: Doppler effect

3.2.1. Double Doppler effect

We assume that the illumination source is a laser: we shall justify this choice later when discussing formulas. The laser beam (emitting an optical frequency f0) and the observer (which will be a photodetector) are stationary in the laboratory.

In a first step the moving particle is a mobile observer relative to a stationary laser beam source. Due to the Doppler effect it receives a frequency fp slightly different from f0.

In a second step, the moving particle, which is thus a mobile source, emits the frequency fp towards the stationary observer, which receives another frequency f due to a second Doppler effect. We compute f as a function of f0, the velocity vector, , and the geometry of the system defined by two unit vectors: along the direction of laser beam and along the direction of observation.

3.2.1.1. Fixed source, mobile observer

A laser beam of frequency f0 in a vacuum propagates along a unit vector in a medium having a refraction index n. The wavelength in the medium is:

Figure 3.1.Doppler effect: fixed source, mobile observer

Therefore, the apparent frequency fp of light radiation received by the moving particle is:

3.2.1.2. Mobile source, fixed observer

The moving particle emits a frequency fp and the observer (photodetector) receives a frequency f along observation vector that we shall compute (see Figure 3.2).

Figure 3.2.Doppler effect: mobile source, fixed observer

The second wavefront emitted by S″ a time Tp later than that emitted by S′, is at O″ after time interval t. Then it becomes:

The apparent wavelength received by the observer is O″O:

In conclusion, the detected frequency f has the following expression:

This equation can be simplified because the ratio V/c is always very small, even at the highest velocities found in hypersonic systems (i.e. 3 × 103 m/s, for instance); so V/c is always smaller than 10–5. Therefore, it becomes (with Figure 3.3 notations):

[3.1]

Figure 3.3.Basic formula diagram of laser Doppler velocimetry

3.2.2. Four optical set-ups

In equation [3.1], as the ratio V/c is very small, f has the same order of magnitude as f0, i.e. about 1014 Hz. To extract velocity information, two approaches are possible: either direct measurement of f by optical means or by interference techniques (or beating techniques), the objective of which is to eliminate term 1 in equation [3.1], in order to obtain frequencies to which the detectors used are sensitive (typically 108 Hz or less).

Figure 3.4.The four optical set-ups for laser Doppler velocimetry

3.2.2.1. Spectrometer velocimeter