Global rainbow thermometry assessed by Airy andLorenz–Mie theories and compared with phaseDoppler anemometry

Jeronimus Petrus Antonius Johannes van Beeck, Thomas Grosges, andMaria Grazia De Giorgi

Global rainbow thermometry �GRT� measures the mean size and temperature of an ensemble of spraydroplets. The domain of validity of the Airy theory for this technique is established through comparisonwith Lorenz–Mie theory. The temperature derivation from the inflection points of the Airy rainbowpattern appears to be independent of the type of spray dispersion. Measurements in a water spray arereported. The mean diameter obtained from the rainbow pattern lies between the arithmetic and theSauter mean diameters measured by phase Doppler anemometry. The temperature measurement byGRT is shown to be accurate within a few degrees Celsius. © 2003 Optical Society of America

OCIS codes: 120.0120, 290.0290, 010.1110, 120.6780, 120.5820.

1. Introduction

Rainbow thermometry has been investigated sincethe late 1980s.1–3 The rainbow technique has beenused to measure the temperature and the size of in-dividual droplets. Sankar et al.4 used the techniqueto measure spray combustion but encountered majorproblems. These problems were related to the tem-perature gradient inside the droplet,5–7 to droplet as-phericity,8 and to a ripple structure that stronglyperturbed the rainbow interference pattern fromwhich one deduces the droplet’s parameters.5 Thelast two problems have been solved by global rainbowthermometry. In particular, the fact that there is asolution for the asphericity problem is important be-cause until now a droplet asphericity of 1% could leadto an error in temperature measurement of as muchas 40 °C. Near the end of the 20th century, twoalgorithms for detection of spherical droplets wereproposed, but neither of them was sufficiently gener-al.4,9 Global rainbow thermometry �GRT�, however,

J. P. A. J. van Beeck �vanbeeck@vki.ac.be�, Th. Grosges, andM. G. De Giorgi are with the Von Karman Institute for FluidDynamics, 72, Chaussee de Waterloo, B-1640 Rhode-Saint-Genese,Belgium. Th. Grosges is also with the Laboratoire de PhysiqueAtomique et Moleculaire, Universite Catholique de Louvain, 2,Chemin du Cyclotron B-1348 Louvain-la-Neuve, Belgium.

Received 29 January 2003; revised manuscript received 29 Jan-uary 2003.

0003-6935�03�194016-07$15.00�0© 2003 Optical Society of America

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is applicable to research in flashing and combustingsprays, spray dryers, and raindrops as well as toliquid jets and to the manufacture of fiber bundles.

2. Principles of Global Rainbow Thermometry

GRT is aimed primarily at eliminating the aspheric-ity effects that have marred the rainbow techniquesince its invention in 1988. The basic principles ofGRT were published by van Beeck et al. in 1999.10

Figure 1 shows a picture of the GRT setup. A cwargon-ion laser beam illuminates a flat fan waterspray. The typical laser power for this spray is 100mW, and the beam is expanded to a thickness of 15mm. A transparent screen is installed in the focalplane of a lens system containing a spatial filter thatselects a probe volume of �1 cm3. This volume issmall compared to the spray dimensions. All thedroplets that cross this probe volume will contributeto the angular scattered-light distribution, which isvisible on the transparent screen. This distributionis called global rainbow pattern, and that is recordedfrom the other side of the transparent screen by adigital video camera. The use of a transparentscreen makes alignment of the video camera a minorissue. To calibrate the magnification factor of thecamera, one records a graph paper that is attached tothe screen; then the relationship between pixel num-ber and scattering angle is found by means of thefocal length of the lens system.

A typical global rainbow interference pattern isshown in Fig. 2. The pattern is recorded while the

probe volume is positioned in the core of an isother-mal water spray. One can observe several Airyfringes, which are also called supernumerary bows.These fringes are formed through the interference ofrays that have experienced one internal reflectioninside the droplets. The pattern of Fig. 2 is stable;i.e., the Airy fringes are not moving in time, whichshould imply that the pattern is formed by construc-tive interference of the spherical droplets because forall the droplets the geometrical rainbow position isidentical. Destructive interference occurs for the as-pherical droplets and liquid ligaments because theirrespective rainbow patterns are randomly orientedand thus yield a uniform background. Conse-quently there is no need for one to use complex se-lection criteria to look for rainbow patterns fromspherical droplets. The selection of spherical drop-

lets is done automatically, just as occurs in a rainbowin the sky, which is a static phenomenon even thoughnumerous raindrops are not spherical.

Besides the fact that the global rainbow pattern�Fig. 2� is formed by spherical droplets only, it isinteresting to note that the visibility of this pattern isless than that formed by a single droplet �Fig. 3�. Aripple structure such as that in Fig. 3 cannot be ob-served at all in the global rainbow pattern of Fig. 2.This ripple structure contains no temperature infor-mation and thus can only deteriorate the accuracy ofthe temperature measurement; for a single dropletsmaller than 30 �m the uncertainty in the tempera-ture measurement already exceeds �6 °C when stan-dard rainbow thermometry is used. Therefore thefact that a ripple structure does not appear in theglobal rainbow pattern is favorable for accurate de-tection of the Airy fringes from which droplet size andtemperature are derived. The disappearance of theripple structure has already been discussed by Rothet al.11,12 Those authors studied the mean scatter-ing diagram of a polydispersed burning dropletstream: A variation in droplet size of less than 1 �mmade the ripple structure vanish completely. How-ever, they made no comment on the natural selectionof spherical droplets by means of this global interfer-ence pattern.

For the pattern of Fig. 2, phase Doppler anemom-etry �PDA� measures an arithmetic mean diameter of45 �m and a Sauter mean diameter of 107 �m. Thequestion arises: What mean diameter can be de-duced from the global rainbow pattern? This ques-tion is answered below.

3. Numerical Simulation of the Global Rainbow

A proper numerical model for the interference pat-tern will help in selection of the optimum data inver-sion scheme for GRT.

Fig. 1. Setup for GRT. The water spray, the receiving lens sys-tem, a transparent screen, and a video camera can be seen. Thelaser beam is added during postprocessing of the photograph anddoes not represent the real laser beam thickness.

Fig. 2. Typical global rainbow pattern in a water spray recordedby a video camera. Only the so-called Airy fringes are visible.The horizontal axis is proportional to the scattering angle.

Fig. 3. Typical rainbow pattern coming from a single droplet in awater spray. Note the high-frequency ripple structure superim-posed upon the low-frequency Airy fringes.

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A. Model

In the present case, the rainbow interference patternis simulated by a spray of spherical particles thatobey a statistically perfect log-normal droplet-sizedistribution at uniform temperature. The densityfunction is given by

f �d� �1

�d�2��1�2 exp��12 �ln�d���

� �2� , (1)

where the integral from zero to infinity over densityfunction f �d� yields unity. � and � are parametersthat represent the mean diameter of the spray andthe dispersion about it, respectively. Because weconsider a finite number of droplets, Ntot, the distri-bution must be discretized. di � di�2 d di �di�2 is the diameter range that contains only onedroplet with diameter di. From this, it follows that

f �di�di � 1�Ntot. (2)

The arithmetic mean diameter D10 and the com-monly used Sauter mean diameter D32 are given by

D10 � �i 1

Ntot

�dif �di�di� �1

Ntot�i 1

Ntot

di, (3)

D32 � �i 1

Ntot

di3��

i 1

Ntot

di2. (4)

The influence of Ntot, �, and � on the global rainbowpattern is investigated.

To compute the total scattering diagram of therainbow pattern we sum the scattered intensities ofthe individual droplets. Here, optical interferencebetween droplets is neglected because of the arbitrarypositions of the scatterers in the spray. The patternfor a single droplet can be simulated by variousmeans with different degrees of complexity. Herewe use and compare the efficiencies of the Airy andthe Lorenz–Mie theories.

In the Airy approach we use the normalized Airyfunction Ai�x, d�, with x � � �rg �see Ref. 13�. Withthis function, only the Airy fringes and no ripplestructure are taken into account. Then the averagescattered-light intensity distribution RnbwAi�x� be-comes

RnbwAi� x� � �i 1

Ntot

�Ai� x, di�2di

7�3�, (5)

where the Airy function accounts for the normalizedangular variation of the scattering diagram, whereasthe factor di

7�3 describes the dependency of the in-tensity on the droplet diameter within the primaryrainbow region.14 Laser beam characteristics andthe receiving optics are not taken into account.

The Lorenz–Mie approach consists in computingnumerically the full electromagnetic field contribu-tions. The program imagshtj.f90 is employed,15,16

which can include a single receiving lens, laser beamparameters, and individual droplet positions in the

spray. For this approach the average scattered-light intensity distribution RnbwLM�x� becomes

RnbwLM� x� � �i 1

Ntot

��E� x, di, zi��2�, (6)

where E is the scattered electromagnetic field, whichis a function of droplet size di and of arbitrary posi-tion zi in the spray. We remark again that the op-tical interference term in both models is neglected.

B. Numerical Results

In the Airy and Lorenz–Mie approaches, we investi-gate the influence of Ntot, �, and � on the globalrainbow pattern. For all simulations in this paper,wavelength � is set at 514.5 nm. Computations aremade for three numbers of droplets, Ntot. The beamradius is 7.5 mm and the lens radius is 75 mm. Thedistance between lens and probe volume equals thefocal length of the single receiving lens, which is 17.3mm. No spatial filter is included so far. The indi-vidual droplets are arbitrarily positioned in the beamwaist within a, more or less cubic, probe volume of 10mm3. Beam and lens characteristics and droplet po-sitions are chosen such that they do not influence therainbow pattern within an angular range of 20°, be-cause these parameters are also not included in themodel that uses the Airy function.

Figure 4 depicts �for Lorenz–Mie theory� the evo-lution of the normalized intensity diagram for param-eters � and � fixed at 50 �m and 0.2, respectively.Note that for these parameters the droplet-size dis-tribution ranges from 35 to 70 �m. The simulatedprofile shows clearly a strong decrease in the impor-tance of the ripple structure when the number ofdroplets Ntot increases. As mentioned above, thisdecrease improves accuracy in the detection of inflec-tion points of the principal rainbow, which is neededto invert the global rainbow pattern to mean dropletsize and temperature.

In Figs. 5 and 6 we show the influence of meandroplet diameter � and dispersion � on the two meth-ods, Airy and Lorenz–Mie. The total number ofdroplets Ntot is fixed at 100, which is sufficient to

Fig. 4. Simulation of global rainbow patterns by the Lorenz–Mieapproach for three numbers of droplets Ntot at constant meandiameter � 50 �m and constant dispersion � 0.2.

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reach a converged solution. The principal rainbowmaximum tends to move toward the geometricalrainbow angle ��rg 138°�, and the width of thismaximum is reduced as � increases. Moreover, thevisibility of the fringes diminishes for higher scatter-ing angles � as dispersion � increases.

The agreement between the Airy and Lorenz–Mietheories is perfect with respect to the principal rain-bow maximum. This implies that data inversionschemes should be related to this peak if one desiresto employ the Airy function. Farther away from thegeometric rainbow angle the agreement degrades, inagreement with previous comparisons between Airyand Mie computations.13,14,17,18

From a numerical point of view there is a clearadvantage in using the Airy model for further studyof GRT data inversion. For droplet diameters largerthan 33 �m the Lorenz–Mie approach requires im-portant CPU resources. To compute the field com-ponents accurately, the method uses an iterativeprocedure whose convergence time is proportional to�D���5�2. Computations have been performed on ahigh performance computer, ES40 Alpha EV67�667MHz. For Ntot 100 and dispersion � 0.2, the

CPU times are 30, 85, and 480 h for mean diameters� 33 �m, � 50 �m, and � 100 �m, respectively.In spite of this fact, only the Lorenz–Mie approachprovides an exact solution for the current problem.

4. Data Inversion Algorithm

From the simulations of the global rainbow patternby use of the Airy theory, one can evaluate the meandiameter and temperature obtained from the globalrainbow. The aim is to look for a data inversionalgorithm that is independent of spray-dispersion pa-rameter � �Eq. �1��. Several schemes were studiedin the past.19 They varied in the characteristic in-formation deduced from the signal.

The most interesting results were obtained for thedata inversion algorithm based on the inflectionpoints about the main rainbow maximum �see, e.g.,Fig. 5�, i.e., �inf1 and �inf2. The relationship betweenthese points and Drainbow and �rg can be constructedanalytically from the Airy theory applied to a singledroplet.13 For a water droplet in air it follows that

Drainbow � 531.6���inf2 � �inf1��3�2, (7)

�rg � �inf1 � 13.91���Drainbow�2�3, (8)

where �rg is the so-called geometrical rainbow angle,which depends on the refractive index, and thus onthe droplet temperature, and is therefore an impor-tant parameter for the rainbow technique. For atemperature variation of 40 °C in a water spray, therainbow pattern, i.e., �rg, shifts �1°. Roth et al.20

have noted that �inf1 lies close to �rg. Hence the firstinflection point is a good indicator for the droplettemperature, especially when the diameter’s influ-ence is corrected for by the second term on the right-hand-side of Eq. �8�. For that equation one needs toknow droplet diameter Drainbow, which can be deter-mined from Eq. �7� without prior knowledge of thetemperature, as explained by van Beeck and Rieth-muller8 and van Beeck.13

One hopes that the formulas above apply not onlyto a single droplet but also to an ensemble of droplets.This means that Drainbow and �rg should yield physi-cally meaningful average quantities, an issue that weaddress below.

Figure 7 shows Drainbow as a function of � for � 100 �m. Drainbow exceeds Sauter mean diameterD32 by a considerable amount. This excess is relatedto the decrease in the width of the principal rainbowmaximum for increasing �, as illustrated in Fig. 6.This result leads to a smaller distance between theinflection points and thus to a larger droplet diameter�Eq. �7��. Figure 8 depicts, for three droplet diame-ters, the deviation of �rg from its value for a mono-dispersed droplet ensemble �i.e., � 0�. Thisdeviation is never larger than 0.025°, which meansthat the algorithm ensures an accuracy of �1 °C inthe temperature measurement for water droplets.This accuracy is even higher for, e.g., fuel spray drop-lets, because of a stronger dependency of the fuel’srefractive index on temperature.

Fig. 5. Simulation of global rainbow patterns by Lorenz–Mie �lm�and Airy �ai� approaches for several mean diameters � at constantdispersion � 0.2 and for a total number of droplets Ntot fixed at100.

Fig. 6. Simulation of global rainbow patterns by Lorenz–Mie �lm�and Airy �ai� approaches for several dispersions � at constant meandiameter � 50 �m and for a total number of droplets Ntot fixed at100.

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5. Experimental Results in a Water Spray

A. Experimental versus Numerical Rainbow Patterns

Figure 9 shows an experimental rainbow interferenceprofile. It was measured at 40 cm from the nozzle onthe center line of an isothermal flat fan water sprayoperating at 3 bars �3 � 105 Pa�. The figure depictstwo attempts with the Airy approach to simulate theexperimental pattern, one with a mean diameter of� 45 �m and the other with � 50 �m. The

standard deviations about these diameters are � 0.4 and � 0.3, respectively. In any case, only themain rainbow maximum is well captured. It is suf-ficient for data inversion toward mean diameter andtemperature, because the inversion algorithm isbased only on the inflection points of this peak.However, from a fundamental point of view it is in-teresting to try to understand the differences be-tween the experimental and the numerical patterns.An explanation cannot be related to temperaturevariations because the spray operates at isothermalconditions. The present authors believe that thereason for the differences in the higher-order maximaof the rainbow patterns should be sought in the drop-let shape. Whereas nonspherical droplets are sup-posed to contribute to a uniform background, nearlyspherical droplets could still interfere constructivelyto the interference pattern, leading to peak broaden-ing. We intend to investigate the quantification ofthe influence of spheroids on the pattern.

B. Comparison of Global Rainbow Thermometry withPhase Doppler Anemometry

Figure 10 shows the rainbow diameter, Drainbow,along a radial profile from the center to the edge ofthe water spray, operating at the conditions de-scribed above. Drainbow is close to Sauter mean di-ameter D32 but is much larger than arithmeticdiameter D10. However, toward the edge of thespray all diameters are identical within �3%, indi-cating that the droplet-size distribution is nearlymonodisperse. The perfect agreement between therainbow diameter and that measured by PDA at theedge of the spray establishes confidence in the datainversion schemes of both techniques. However, forpolydispersed droplet ensembles, i.e. in the spraycore, it is questionable why the rainbow diameter isclose to D32 and not much bigger, as was simulated inFig. 7. This question is answered by Fig. 11, whichshows center line measurements of the droplet diam-eter made with PDA and GRT. It can be seen thatcloser to the nozzle the rainbow diameter Drainbowapproaches D10. This seems to be a general ten-

Fig. 7. Rainbow diameter, computed from �inf2 � �inf1, as a func-tion of spray dispersion � for various numbers of droplets Ntot �Eq.�7��.

Fig. 8. Deviation of �rg, based on �inf2 and �inf1, from its value at� 0, as a function of � for three mean diameters � �Eq. �8��.

Fig. 9. Experimental versus numerical rainbow patterns in thecore of the spray at z 40 cm from the spray nozzle.

Fig. 10. Comparison of GRT and PDA. The rainbow, Sautermean, and arithmetic mean diameters are presented along a radialprofile in a flat fan water spray at z 40 cm from the waterspray nozzle; p is nozzle pressure.

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dency. Closer to the spray nozzle, the fraction ofspherical droplets that generates the global rainbowpattern decreases. The spherical droplets tend to bethe smaller ones in the droplet-size distribution.The rainbow technique measures a smaller mean di-ameter than PDA because the latter is less sensitiveto droplet nonsphericity than GRT.21 Because theglobal rainbow technique is sensitive to the fractionof nonspherical droplets in a spray, one could try todeduce this fraction from the technique. As men-tioned above, we intend to study this issue, concen-trating on the intensity of background light in theAlexander dark band. Because there are no impor-tant rainbow rays in that region, it could serve asindicator of the effect of asphericity on GRT.

C. Accuracy in Rainbow Temperature Measurement

Figure 12 shows the variation of the mean droplettemperature along two radial profiles, at 20 and 40cm from the nozzle. Because the humidity is almost100%, the droplet temperature should equal the am-bient temperature and should therefore be uniformover the profiles. Nevertheless, variations of �4 °Care observed, which thus represent the precision inthe droplet temperature measured by GRT. Thisprecision is lower than expected from the inversionalgorithm because of the measurement uncertainty

in the determination of the inflection points, forwhich the derivative of a smoothed rainbow patternhas to be evaluated. The value �4 °C does not nec-essarily represent the accuracy of experiments per-formed at nonisothermal conditions: For instance,for rapidly evaporating droplets a strong tempera-ture gradient inside the droplet could affect the meantemperature measurement. This effect has longbeen known5–7 and unfortunately has not yet beenovercome by GRT.

6. Discussion and Conclusion

Global rainbow thermometry has been evaluated nu-merically and experimentally. Through the use of aspatial filter the technique measures the local meandiameter and the temperature of an ensemble oftransparent spheroidal particles, which in thepresent case are water droplets generated by a flatfan water spray. The probe volume is much smallerthan the typical spray dimension. The global rain-bow pattern consists of constructive interference ofthe rainbows coming from spherical droplets. Thenonspherical droplets are assumed to create a uni-form background. This solves the problem relatedto the dependence of the pattern on the droplet shape,which degrades the temperature measurement whenthe rainbow pattern that is generated by a singledroplet is employed.

We used Lorenz–Mie theory and Airy theory tosimulate a global interference pattern to assess theapplicability of the latter theory in establishing datainversion schemes. It appears that the main rain-bow maximum can always be well simulated with thehelp of Airy functions. The Airy approach degradesfor higher-order fringes; this is why only the mainrainbow maximum is used in the data inversionscheme. One determines the two inflection pointsabout this maximum, from which one derives a meandiameter and subsequently a mean temperature.This procedure ensures proper temperature mea-surement within �1 °C. However, the experimentswith the flat fan water spray have shown that theaccuracy is rather of the order of �4 °C because ofuncertainty in the experimental determination of theinflection points.

Attempts were made to fit the Airy model to anexperimental rainbow pattern. Again, the mainrainbow maximum was correctly simulated, whereasthe higher-order fringes were only reasonably wellcaptured. The differences between numerical andexperimental patterns are larger than the differencesbetween the Airy and Lorenz–Mie simulations. Thenumerical model assumes perfectly spherical drop-lets. In reality, spheroids might still influence theinterference pattern, resulting in extra peak broad-ening and subsequent reduction in fringe visibility.

The mean diameter measurement by GRT, Drainbow,lies between the arithmetic mean diameter D10 andthe Sauter mean diameter D32, measured by phaseDoppler anemometry. The rainbow diameter differsless than �3% from the PDA measurements at loca-tions in the spray where the droplet-size distribution

Fig. 11. Comparison of rainbow, Sauter mean, and arithmeticmean diameters for an axial profile; p is the nozzle pressure.

Fig. 12. Temperature variations measured by GRT in an isother-mal spray at two different axial positions; p is the operating pres-sure.

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is close to monodisperse. At locations where thedroplet ensemble is polydisperse, the rainbow diam-eter is smaller than what was expected from simula-tions. This is so because the GRT signal isconstructed by only the most nearly spherical, thussmaller, droplets. We confirmed this tendency ex-perimentally by using a spray nozzle, for which therainbow diameter diminishes because the fraction ofspherical droplets decreases.

Th. Grosges was supported by grant 991�4264 fromthe Region Wallonne of Belgium. The authors aregrateful for fruitful discussions with G. Grehan of theLESP of the University and INSA of Rouen. Use ofthe computer facilities of the Centre de Calcul Inten-sif, University of Louvain, and of Fonds de la Recher-che Fondamentale Collective project 2.4556.99,“Simulations numeriques et traitement des donnees”of the Belgian Fonds National de la Recherche Sci-entifique, is acknowledged.

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