Uncertainty in velocity measurement based ondiode-laser absorption in nonuniform flows

Fei Li,1 Xilong Yu,1 Weiwei Cai,2 and Lin Ma2,*1State Key Lab of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences,

Beijing 100190, China2Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA

*Corresponding author: linma@vt.edu

Received 16 February 2012; revised 14 May 2012; accepted 14 May 2012;posted 14 May 2012 (Doc. ID 163180); published 9 July 2012

This work investigates the error caused by nonuniformities along the line-of-sight in velocity measure-ment using tunable diode-laser absorption spectroscopy (TDLAS). Past work has demonstrated TDLASas an attractive diagnostic technique for measuring velocity, which is inferred from the Doppler shift oftwo absorption features using two crossing laser beams. However, because TDLAS is line-of-sight innature, the obtained velocity is a spatially averaged value along the probing laser beams. As a result,nonuniformities in the flow can cause uncertainty in the velocitymeasurement. Therefore, it is the goal ofthis work to quantify the uncertainty caused by various nonuniformities typically encountered inpractice, including boundary layer effects, the divergence/convergence of the flow, and the methods usedto fit the Doppler shift. Systematic analyses are performed to quantify the uncertainty under variousconditions, and case studies are reported to illustrate the usefulness of such analysis in interpretingexperimental data obtained from a scramjet facility. We expect this work to be valuable for the designand optimization of TDLAS-based velocimetry, and also for the quantitative interpretation of themeasurements. © 2012 Optical Society of AmericaOCIS codes: 120.7250, 280.3420, 300.1030, 300.6260.

1. Introduction

Among all the laser diagnostics developed for com-bustion flows, tunable diode-laser absorption spec-troscopy (TDLAS) has been demonstrated as anattractive technique offering unique advantagessuch as fast temporal resolution, quantitative mea-surements, and low cost. As a result, variations ofTDLAS have been developed to monitor multipleflow parameters, including temperature, pressure,velocity, density, and flow rate, and applications ofTDLAS have been demonstrated in a wide spectrumof combustion systems ranging from aircraft engine,IC engine, high enthalpy wind tunnel, and superso-nic combustion flow. Readers interested in a compre-hensive and in-depth discussion of the capabilities

and applications of TDLAS are referred to a reviewpaper [1].

Despite these unique advantages, the limitation ofTDLAS is well recognized: it is a line-of-sight techni-que in nature and hence its application is limitedto flows with negligible nonuniformity. A consider-able amount of research efforts have been investedin overcoming this limitation. Past efforts can bebroadly divided into two categories. Efforts in thefirst category analyze the uncertainty caused by non-uniformities, so that measurement uncertainty canbe quantified and minimized. For example, after an-alyzing effects of the thermal and concentrationboundary layer in concentration and temperaturemeasurements, the uncertainties caused by such aboundary layer can be quantified, and furthermoreoptimal wavelengths can be chosen to minimizethe uncertainties [2]. The work presented here fallsin this category: we analyzed the effects caused by

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nonuniformities to quantify uncertainties in velocitymeasurements, and these analyses also suggestedways to minimize the uncertainties. Efforts inthe second category attempted to obtain spatialresolution by combining TDLAS with tomographyinversion [3–5]. This approach typically requiresmeasurements using multiple probing beams andis not the focus of this study.

In this work, we focused on analyzing the uncer-tainties of velocity measurements using TDLAS,especially when applied in high speed flows suchas those encountered in scramjet facilities [6,7].Velocity is a critical parameter for propulsion study,and various mechanisms can cause nonuniformity insupersonic flows. Thus, the effects of nonuniformitieson velocity measurement merit careful examination.Recent work includes that performed by Changet al. [8,9], where the performance of TDLAS-basedvelocimetry was evaluated in a flow tunnel, and thatperformed by Brown and Barhorst [10], where theaverage flow rate in a flight test was estimatedassuming nonuniform pressure distribution obtainedfrom computational fluid dynamics (CFD) simula-tion. These works analyzed the velocity-error causedby boundary layers in nonreacting flow fields, andthe influence of nonuniform pressure because ofthe pressure-induced frequency-shift. These past re-sults were usually application-specific, and thereforethis work aims at providing a more general analysisbased these past efforts. The approach adopted inthis work is to analyze the effects of nonuniformitiesin general quasi—two-dimensional flows, and themethodology is to decouple the nonuniformities intothose normal to the overall flow direction (e.g., thosecaused by boundary layer effects), and those alongthe overall flow direction (e.g., those caused by flowdivergence/convergence or heat transfer). The re-sults obtained from this approach are expected tobe applicable to an expanded range of applications.

For the remainder of this paper, Sec. 2 first sum-marizes the fundamentals of velocity measurementsbased on TDLAS, with an emphasis on introducingtwo approaches of obtaining the Doppler shift: thescanned direct-absorption (DA) approach and thewavelength-modulation (2f ) approach. Section 3analyzes the effects caused by nonuniformities typi-cally encountered in scramjet flows. Results are re-ported to illustrate the measurement uncertaintiescaused by boundary effects, divergence/convergenceof the measurement region, and the variation flowproperties. Section 4 applies the results obtainedin Sec. 3 to experimental data to illustrate the use-fulness of the results in interpreting practical mea-surement data.

2. Principle of TDLAS Velocimetry

The use of TDLAS was first demonstrated by [11]and thorough description of the technique can befound in [12]. A brief summary here is providedto facilitate the discussion in subsequent sections.Figure 1 illustrates a typical setup of TDLAS-based

velocimetry. A laser beam was split to two probingbeams, labeled as beam 1 and beam 2, respectively,in Fig. 1. These probing beams cross in the region ofinterest and define a plane of measurement. Two an-gles, θ1 and θ2, are then defined in the plane of mea-surement to specify the direction of the probingbeams relative to the flow (assumed to have uniformvelocity and pressure distributions). This work de-fines θ1 and θ2 as the angles between the laser beamsand the direction normal to the flow. To facilitate thediscussion, the direction of the flow is taken to be thex direction, and direction normal to the flow the ydirection.

The wavelength of the laser is modulated to scanan absorption feature of a target species, typicallywater vapor in combustion flows. The absorption fea-ture measured by the two probing beams, due to theirdifferent orientation relative to the direction of theoverall flow, will show a frequency shift because ofthe Doppler effects. The frequency shift is propor-tional to the velocity of the flow (V) as shown bythe following equation:

ΔνDoppler �Vc~ν0�sin θ1 � sin θ2�; (1)

where ΔνDoppler is the Doppler shift between theabsorption features measured by the two probingbeams, c the speed of light, and ~ν0 is the line-centerfrequency of the absorption line. Practical implemen-tation often sets θ1 � θ2, and then Eq. (1) becomes

ΔνDoppler �2Vc

~ν0 sinθ2; (2)

where θ � θ1 � θ2.In the presence of nonuniformity along the path of

the probing beams, Eqs. (1) and (2) need to be mod-ified and the modification is detailed in Appendix A.The nonuniformity could be in terms of temperature,pressure, and/or velocity. Furthermore, such nonuni-formity also causes distortion in the shape of absorp-tion feature, leading to uncertainties in determiningthe absorption peak and eventually in determiningthe Doppler shift. Analyzing such uncertainties inTDLAS-based velocimetry is the focus of this work.

In our analyses, we decoupled the nonuniformitiesto provide insights of the uncertainties and to facil-itate the application of the results in practice. Thenonuniformities are decoupled into those along thedirection of the flow (i.e., the x direction shown inFig. 1), and those normal to the flow (i.e., the y direc-tion). In practice, nonuniformities in the y direction

Fig. 1. (Color online) Schematic of TDLAS velocimetry.

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are typically caused by the existence of boundarylayers, and those in the x direction are caused bythe divergence/convergence of the flow, variationsof the boundary layer thickness along the x direction,and/or variations in flow properties due to heattransfer. In this work, uncertainty due to each typeof nonuniformity is analyzed in isolation, based onwhich it is straightforward to analyze multiple typesof nonuniformity simultaneously, as illustratedin Sec. 4.

Before proceeding further, the assumptions in-voked in this work are summarized. This work as-sumes a rectangular flow cross-section and the planeof the optical beams is immune from corner flow.Furthermore, the flow is assumed to be unseparated.

As shown in Eqs. (1) and (2), TDLAS-based veloci-metry essentially measures the Doppler shift(ΔνDoppler) from the absorption signals registered atthe two probing beams. And several approaches existin practice to obtain the Doppler shift. This workanalyzes two of them: DA and 2f . In the DA ap-proach, the absorption spectra at both probing beamsare first calculated, and then ΔνDoppler is determinedby the frequency difference of an absorption peakon those two spectra. The DA approach is simplewhen the absorption spectra can be obtained, whichrequires that a baseline can be obtained [13]. In con-trast, the 2f approach utilizes the peaks in thesecond harmonic of the signals at both probingbeams to determine ΔνDoppler [14]. Compared withthe DA approach, the 2f approach can significantlyimprove the signal-to-noise ratio (SNR) and is there-fore suitable for applications with weak absorption,and does not require a baseline under optimizedmodulation parameters, and is therefore applicableunder high pressure [15,16]. A key parameter inthe 2f approach is the so-called modulation depth,and this work uses a modulation-depth of 2.2 follow-ing [17]. Detailed descriptions of simulating theabsorption features in nonuniform flow field and ob-taining the peaks are provided in Appendix A forboth the DA and 2f methods.

3. Uncertainty in Velocity Measurements Caused byNonuniform Flow Parameters

As mentioned above, this work decouples the nonu-niformities in the x direction from those in the ydirection, which are treated in Sec. 3.A and 3.B,respectively. The analyses were conducted underthe context of applications in our scramjet facilityas reported in [18], where the typical flow conditionsat the measurement location are Mach number of1.83, static temperature of 600 K, and static pressureof 1.0 atm. Water vapor absorption feature centeredat 7185.597 cm−1 (i.e.,eν0) was used in this analysis.This transition has been applied extensively in prac-tice due to its relatively strong absorption strength athigh temperatures and its isolation from interferingtransitions in its vicinity [14,18].

Relevant spectroscopic parameters were extractedfrom HITRAN 2008 [19]. The lower state energy of

this absorption line is E00 � 1045 cm−1. To studythe impact of E00 on the TDLAS measurement, weassumed two other values of E00 (2000 and 500 cm−1)for this same transition while keeeping its line-strength and eν0 fixed. This is a hypothetical study(in practice, each transition has its own set of E00,linestrength, andeν0). Nonetheless, the results are va-lid to illustrate the effects of E00 because the TDLASmeasurements [see Eq. (1)] do not depend on theabsorption strength and depend linearly on eν0 (ifthe SNR of the measurements is high enough as dis-cussed in Sec. 3.C). Therefore, it is straightforward toextend the results obtained in such hypotheticalstudy to practical applications.

A. Nonuniformity Along y Direction

Nonuniformity always exists in the y direction due tothe boundary layer effects as illustrated in Fig. 2(a).Figure 2(a) shows a top view of a TDLAS-velocimetryapplied to a flow with boundary layers. Here the topview defined as the view along a direction normal tothe measurement plane. Both velocity and tempera-ture are nonuniform in the boundary layers. Practi-cal applications may have nonunity Prandtl numberand different thickness of velocity and temperatureboundary layers. It is straightforward to extendthe analysis to such cases. To simplify the problemand elucidate the analysis, the velocity and thermalboundary layer are assumed to have the same thick-ness (δ) at both the walls. The velocity was assumedto follow a linear distribution in the boundary layeras shown in Fig. 2(b), and to be constant (at VC �900 m∕s) in the core flow. The temperature distribu-tion in the boundary layers was calculated under theisentropic assumption from the velocity distribution

Fig. 2. (Color online) (a) Illustration of TDLAS velocitymeasurement with boundary layer. (b) Temperature and velocitydistributions assumed in the analysis.

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as shown in Fig. 2(b), and the temperature was takento be constant (TC � 600 K) in the core flow. Such anisotropic approximation will exaggerate the effects ofthe thermal boundary layer, and therefore the uncer-tainty calculated represents a conservative estimate.The static pressure was assumed to be uniform at1 atm (P � 1.0 atm). These conditions correspondto a flow with Mach number (Ma) of 1.83 and a totaltemperature of 960 K, a representative condition atthe inlet of our scramjet combustor [18]. The molefraction of water vapor was assumed to be uniformat 10% (X � 10%) due to the preheating of the inflowby the combustion of hydrogen. These conditions re-present a typical flow condition corresponding to the“starting point” of scramjet engine in fly tests [20].

Figure 3 shows the uncertainty in velocity mea-surements due to the nonuniformity in the boundarylayers. Here the uncertainty is defined as

e � VM − VC

VC; (3)

where VM is the velocity that the TDLAS sensorwould measure. Two approaches, the DA and 2fapproaches, of obtaining the Doppler shift were com-pared for transitions with various lower state ener-gies (E00). Note that the error was negative for allthese cases (i.e., VM is less than VC); the flow isslower in the boundary layer than in the core flow.The results shown in Fig. 3 provide several usefulobservations.

First, obviously, the error caused by boundarylayer increases with the thickness of the boundaryrelative to width of the core flow (quantified byδ∕L1 in Fig. 3). When δ∕L1 � 5% the error can reach∼10% depending on the specifics of the TDLAS sen-sor (e.g., absorption transition used and the methodused to determined the Doppler shift). Figure 3shows typical values of δ∕L1 in the inlet, combustor,and exit sections of our scramjet facility. These re-sults suggest that the analysis of boundary effectsis not trivial, and the simulation results shown herecan be used as guidelines.

Second, the errors are smaller when the 2f methodwas used than when the DA method was used todetermine the Doppler shift. Our explanation forthe superiority of the 2f method here is that the

2f method is less sensitive to the distortion in theshape of the absorption feature. As mentioned inSec. 2, the nonuniformities in the boundary layerscause distortion in the shape of the absorptionfeature. As a result, the absorption shape can nolonger be perfectly fitted with the Voigt profile.The 2f signal (R2f ) is mainly determined by thesecond harmonic of signal (H2), as shown in Eq. (A13)in Appendix A. It corresponds to the derivative ofthe absorption shape and is less sensitive to thedistortion of the absorption shape.

In the past, the 2f approach has been recognizedfor being able to improve the SNR compared to theDA approach for TDLAS-based velocimetry [11].These results suggest that the use of the 2f approachhas an additional advantage in TDLAS-based veloci-metry: the insensitivity to the interference due to theboundary layers. One factor contributing to such in-sensitivity is the fact that the 2f method detects thechange in the absorption spectra, not the absorptionspectra itself [11]. As a result, the 2f method is lesssensitive to the distortion in the lineshape caused byboundary layer effects. The results shown in Fig. 4elucidate such insensitivity. Here, Fig. 4(a) showsthe simulated DA and 2f signals at one probe beamaccording to the method described in Appendix A fora flow with δ∕L1 � 15% and E00 � 2000 cm−1. Theproperties of the main flow are those specified inFig. 2. The DA signal was fit with a Voigt lineshape,which would result in a precise fit should the flow beuniform and have no boundary layer. However, theboundary layer distorts the lineshape, causing a dis-crepancy between the simulated (or distorted) line-shape and the best Voigt fit. The discrepancy (i.e.,the fitting residual) is shown in Fig. 4(b). Similarly,the 2f signal was simulated and fit with a lineshapeassuming no boundary layer effects, and the discre-pancy between the simulated signal and the bestfit are also shown in Fig. 4(b). The results in Fig. 4

Fig. 3. (Color online) Velocity error caused by nonuniformity inthe boundary layers.

Fig. 4. (Color online) Comparison of the lineshape distortioncaused by boundary layer for the DA and 2f methods.

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illustrate that the 2f method is less sensitive to line-shape distortion caused by the boundary layer in twoaspects: (1) as seen from Fig. 4(b), the discrepancy forthe DA method ranges from −0.1% to �0.3% of thepeak value, while that of the 2f method is within�0.1%, and (2) the peak locations determined byfitting the DA and 2f signals were, respectively,3.6 × 10−3 cm−1 and 3.2 × 10−3 cm−1 off the true peaklocation (i.e., the peak location assuming no bound-ary layer effects). Calculation at the other probebeam a yielded similar trend; i.e., the 2f signals suf-fered less lineshape distortion and smaller error indetermining the peak location. After calculatingthe errors in the peak location at both probe beams,the error in velocity shown in Fig. 3 can then be ob-tained (note that under the conditions used here, theDoppler shift at 900 m∕s is 0.011 cm−1). Calculationswere also made at other flow condition (differentboundary layer thickness and lower state energies),and the results all show that the 2f method is lesssensitive to the lineshape distortion caused by flownonuniformities, and yields more accurate determi-nation of the peak location than the DA method.

Also note that in practice, the error caused bylineshape distortion as illustrated in Fig. 4 only re-presents one of the sources of uncertainty. As afore-mentioned, this work attempts to examine thesesources independently first. Section 4 describesexample application of the analysis to experimentalresults where errors caused by multiple sources areconsidered simultaneously.

Third, the error decreases when transition withsmaller E00 is used if the boundary layer is hotterthan the core flow. This is a typical situation forsupersonic test facilities, where the boundary layerhas higher temperature and lower velocity thanthe core flow because the temperature drop at thewall due to heat loss is less significant than the tem-perature rise due to conversion of the kinetic energy.In this case, transitions with smaller E00 are advan-tageous, because it increases the absorption due tothe cooler core flow, and decreases the absorptiondue to the hotter boundary layers, resulting inimproved accuracy of velocity measurements. In ap-plications where the boundary layer is cooler thanthe core flow, the effects of E00 are reversed, and anelaborated discussion can be found in [2].

B. Nonuniform Along x Direction

Nonuniformities along the x direction also occur insupersonic flows due to a variety of reasons. A shock-wave intercepting the probing beams represents anextreme case of nonuniformity along the x direction.Here, we analyze effects caused by other factorsincluding the divergence/convergence of the flow,variations of the boundary layer thickness, and/orvariations in flow properties due to heat release.

First, we consider the effects caused by the conver-gence/divergence of the flow, as schematically shownin Fig. 5, where a TDLAS-based velocimetry is ap-plied to measure the velocity in the section with a

diverging angle of 10 deg (φ � 10 deg). The analysisof a converging section is similar and therefore notpresented here. In this case, the flow properties varyalong the x direction and the variations were mod-eled by assuming an isentropic flow. Figure 6 showsthe distribution of velocity (V), temperature (T), andpressure (P) using the incoming flow conditions asthose shown in Fig. 2(b). The height of the cross-section, labeled as H in Fig. 5, was taken to be 5 cm.Figure 7 shows the error in velocity measurementscaused by such nonuniformities. Here, the error isdefined as,

e � VM − Vavg

Vavg; (4)

where Vavg is the arithmetically averaged velocityalong L2. Note that Eqs. (3) and (4) are independent:they considers the uncertainty caused by nonunifor-mities in the x and y directions, respectively. Thecombined velocity error caused by nonuniformitiesin both directions is considered in Sec. 4.

As shown in Fig. 7, velocity error is generally with-in �1% when the divergence angle is less than20 deg. For practical scramjet applications, the diver-gence angle in the combustor and exit is normallyless than 5 deg as indicated in Fig. 7, which resultsin a velocity error less than 0.2% for the conditionsshown here. Two interesting observations can bemade from these results. First, such an error is

Fig. 5. (Color online) Illustration of TDLAS velocity measure-ment with flow divergence.

Fig. 6. (Color online) Temperature, pressure, and velocitydistribution assumed in the analysis.

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significantly smaller than that caused by nonunifor-mity in the y direction as discussed in Sec. 3.A fortypical scramjet flows, and second, such an error isalso significantly smaller than the variation of Vitself in the x direction (as shown in Fig. 7, V itselfvaries for about 7% along L2). Our explanation forthese two observations is that the probing beamscross the flow at an angle andmeasure a velocity thatis a weighted averaged of absorption coefficient (k)along L2. Even though this average is not an arith-metic average, it is very close to the arithmeticaverage. The following results will show that both ob-servations remain valid for more general cases wherenonuniformities exist in multiple parameters alongthe x direction.

In practice, other factors can cause nonuniformitiesalong the x direction in addition to the divergence/convergence of the flow. For example, in scramjet com-bustors, the static temperature and velocity along thex direction may vary because of heat release or theincreasing thickness of the boundary layers. In thiswork, we analyzed a simple casewhere the static tem-perature (T), pressure (P), and velocity (V) all varyalong the x direction in a linear fashion, as shown inFig. 8. The rate of variation for the static temperature,pressure, and velocity are kT, kP, and kV , respectively.Such a linear variation represents a good approxima-tion for the conditions in our scramjet facility, and the

variations (represented byΔT,ΔP, andΔV in Fig. 8)along L2 typically do not exceed 20% of the incomingconditions at x � 0 (represented by T0, P0, and V0in Fig. 8).

Figure 9 shows the error caused by such simulta-neous variation in T, P, and V along the x direction.The definition of the error is the same as in Eq. (4),defined as the difference between the measured ve-locity relative to the arithmetic average velocityalong L2 as the reference. Figure 9(a) shows the errorwhen the DA approach was used, and Fig. 9(b) whenthe 2f method was used. In Fig. 9, ΔV was taken tobe 0.2 × V0, corresponding to the scenario when themaximum amount of nonuniformity in V wasreached. Again, we see that the error is significantlysmaller than both the error caused by nonuniformi-ties in the y direction and the variation of the velocityalong L2 (20% in this case).

Similar plots can be generated to show the errorversus the variation of V. To aid the visualizationof the results, Fig. 10 shows the maximum absoluteerror (i.e., max jej) when T, P, and V all vary within�20% of their incoming values. As shown, the max-imum error of velocity is less than 1.2%, indicatingthat even the flow parameter varies substantiallyalong L2 (normally smaller than 5 cm); the velocityerror remains less than 1% relative to the arithmeticaverage if a suitable absorption line is used.

C. Consideration of SNR

Analysis thus far has been made under the assump-tion of infinite SNR. In practice, the measurementshave finite SNR due to measurement noises fromvarious sources, such as laser intensity fluctuation,

Fig. 7. (Color online) Velocity error caused by flow divergence.

Fig. 8. (Color online) Illustration of TDLAS velocity measure-ment with nonuniformity simultaneously present in temperature,pressure, and velocity along L2.

Fig. 9. (Color online) (a) Velocity error versus due to variation ofT and P when the DAmethod is used. (b) Velocity error versus dueto variation of T and P when the 2f method is used.

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detector noise, and digitization noise, etc., and themeasurement noises will cause uncertainty in velo-city measurements in addition to those caused byflow nonuniformities. A full-length discussion of theeffect of SNR merits a separate paper. Here webriefly explain our method of considering SNR andreport some sample results.

Our analysis considered the combined effects of allnoises in the transmitted laser intensities (which isthe signal used to infer velocity) according to thefollowing equation:

Imt �ν� � It�ν� � In�ν�; (5)

where Imt is the transmitted laser intensity measuredby the detector. The measurements are composed oftwo parts: the “real” transmitted laser intensity (It)and measurement noises due to various sources (In).Then the SNR of the measurement is defined as

SNR � It∕In: (6)

In the analysis presented in this paper, we as-sumed the measurement noise to be random, whichfollows a normal distribution. The SNR of the mea-surements is then determined by the magnitude ofthe normal distribution. Under this assumption,the analysis can be performed by generating randomnoises (In) artificially and adding them to the signal(It) to obtain the simulated measurements (Imt ). Thesimulated measurements can then be analyzed usingthe procedure detailed in Appendix A to evaluate theeffects of SNR in combination of flow nonuniformi-ties. Elaborated models can also be developed to si-mulate realistic noises that do not follow a normaldistribution. As mentioned before, such full-lengthdiscussion merits a separate treatment. Here, weanalyze an ideal case to illustrate the effects ofmeasurement SNR.

The case analyzed here involves a uniform flow(i.e., without any boundary layer effects), and theflow conditions are the main flow conditions shownin Fig. 2. The transitions used are the same ones

used in previous sections with E00 � 1045 cm−1.The results are shown in Fig. 11. Due to the statis-tical nature of the noise, in Fig. 11, 300 simulationswere performed at each SNR; the relative error (e) iscalculated in each simulation, and the standarddeviation (std) of all 300 simulations is plotted inFig. 11. We conducted analysis under other flow con-ditions, and the results shown in Fig. 11 are quiteinsensitive to the mean flow properties.

Several observations can be made from Fig. 11.First, it shows that the error decreases as SNRincreases as intuitively expected. Second, it showsthat for a SNR larger than 10, the error caused bymeasurement noise is ∼1% for both the DA and 2fmethods. For the experimental data presented inSec. 4, the SNR of ourmeasurements are in the rangefrom 20 to 30. Therefore, we expect the uncertainty tobe dominated by flow nonuniformities under theseexperimental conditions. Third, for velocity measure-ments, Fig. 11 does not suggest a dramatic improve-ment from the 2f method in accuracy relative to theDA method in the presence of measurement noise.

4. Application of Analysis to Experimental Data

This section applies the analysis presented above toexperimental data obtained in our scramjet facility[18,21]. The scramjet facility is a direct-connectedfacility capable of simulating supersonic combustionunder flight condition. Experiments were performedin a two-dimensional combustor fueled with ethy-lene, with a test section of 40 × 85 mm2. The detaileddescription of this facility and the experiments canbe found in [21].

Five representative cases were examined and theresults are summarized in Figs. 12 and 13. These fivecases include velocity measured at the inlet (case 1)and the exit (case 2) of the scramjet combustor atMach 1.8, and velocity measured at the inlet (case 3),the combustor (case 4), and the exit (case 5) of thescramjet at Mach 2.5. Figure 12 shows the relativeerror in velocity for each case; the top panel ofFig. 13 shows the velocity and the bottom panelthe absolute error.

Fig. 10. (Color online) Maximum velocity measurement errorversus kV .

Fig. 11. (Color online) Relationship between velocity error andsignal-to-noise ratio (SNR).

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These error analyses were performed using theresults shown in Figs. 3, 7, and 10. Here we use case4 as an example to elucidate the steps of our ana-lyses. First, the thickness of the boundary layer (δ)is estimated to be 4 mm for the combustor of ourdirect-connected scramjet facility according to CFDsimulations [22]. The measurement pathlength(L1) was 85 mm. Therefore from Fig. 3, the errorcaused by the boundary layer in this case is esti-mated to be 4.8% (the DAmethod was used withE00 �1045 cm−1 in this measurement). Second, the diver-gence angle in the experiments was 2 deg. Then fromFig. 7, the error caused by the divergence angle inthis case was estimated to be 0.1%. Third, alongthe L2 direction, the variation of the parameters inthe combustor was estimated to be ΔT < 0.1 × T0,ΔP < 0.2 × P0, ΔV < 0.05 × V0. Therefore, usingthe results shown in Figs. 7 and 10, velocity errorcaused by nonuniformities along the L2 directionwas determined to be 1%.

With the above relative errors and the velocityshown in the top panel of Fig. 12, it is straightfor-ward to estimate the absolute error shown in thebottom panel of Fig. 12. For example, for case 4, themeasured velocity (nominal velocity) was 560 m∕s,and the absolute errors were estimated to be

∼22 m∕s, in which −27 m∕s was caused by the non-uniformities along the y direction (i.e., boundarylayer effects), and 5 m∕s was caused by the nonuni-formities along the x direction. The error analyses forother cases were estimated similarly.

From these results, it can be concluded that thatthe error in scramjet flows is mainly due to thenonuniformities in the boundary layers. Moreover,the flow fields were assumed to be quasi—two-dimensional in this work; i.e., the flow parametersalong L1 are uniform except in the boundary layers.When significant three-dimensional effects exist(e.g., when a shock wave intercepts the probe beams),both the overall error and the relative contributiondue to x- and y-direction nonuniformities will bedifferent than those reported here, and the interpre-tation of the TDLAS measurements is more compli-cated [10]. Lastly, the assumptions made in ouranalysis should be iterated, including the neglectof corner-flow effects and unseparated flow condition.Also, this work focused on the impact of flow nonu-niformities and neglected other types of uncertain-ties. For example, in practice, beamsteering andthe uncertainty in the angle (θ) between beams (e.g.,caused by mechanical vibration and thermal expan-sion) can lead to additional errors. Our method es-sentially divides errors into two categories: thosecaused by flow nonuniformities and those causedby other effects. The later categories were treatedas measurement noises and discussed in Sec. 3.C.

5. Summary

This work investigated the uncertainty in velocitymeasurements using TDLAS-based techniques un-der the context of scramjet flows. This study focuseson quantifying the uncertainties caused by nonuni-formities in the flow. The nonuniformities typicallyencountered in practice were decoupled into thosein the direction normal to the mean flow (i.e., bound-ary layer effects the y direction) and those in thedirection along the mean flow (the x direction).Two variations of TDLAS, the DA and 2f , wereexamined.

Two main findings can be summarized from theresults obtained in this study. First, under typicalconditions in scramjet flows, the error caused bythe nonuniformities in x direction is significantlysmaller than that caused by the boundary layer ef-fects and also than the variation of the velocity itselfin the x direction. Second, the 2f method is less sen-sitive to the boundary layer effects than the DAmethod in determining the Doppler shift. The bound-ary layers effects distort the absorption line shape todeviate from the Voigt profile. Unlike the DA meth-od, the 2f method measures the second harmonic ofthe absorption feature. It detected the peak by ana-lyzing the derivative of the absorption shape and isless sensitive to the distortion of the absorptionshape. As a result, in our results, the 2f methodyielded more accurate Doppler shift than the DAmethod.

Fig. 12. (Color online) Velocity uncertainty of experimentalresults caused by nonuniformity along L1 and L2.

Fig. 13. (Color online) Velocity uncertainty analysis forexperimental results in scramjet combustor.

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These findings were then applied to analyze theexperimental data obtained in a scramjet facility,illustrating the usefulness of the results reported.We expect this work to be valuable for quantifyingthe experimental data, the data analysis, and alsothe design and optimization of TDLAS-based veloci-metry for high speed flows.

Appendix A: Description of the DA and 2f methods

In anonuniform flow, the absorption of laser radiationis described by the Beer—Lambert as shown below:

τ�ν� � I�ν�I0�ν�

� exp�−

Zb

ak�ν; l�dl

�; (A1)

where τ�ν� is the fractional transmission at frequencyv, I and I0 are the transmitted and incident laserintensities, respectively, k is the spectral absorptioncoefficient, l defines a one-dimensional coordinatealong the path length, and a and b are the beginningand the end of the light path as shown in Fig. 1.Absorption coefficient k is calculated as

k�ν; l� � S�T�l�� · P�l� · X�l� · ϕ�ν − ν0�l��; (A2)

whereS�T�l�� is the line-strength andT�l� emphasizesthe nonuniformity of the flow (i.e.,T depends on l, andthe same notation is used hereafter), P�l� the staticpressure, X�l� the mole fraction of the absorbing spe-cies (i.e., water vapor in this present paper), and ϕ isthe line-shape function. As discussed in Sec. 2, is thefrequency of the line center in the presence of Dopplerand pressure-induced shift:

ν0�l� �~ν0� ~ν0 · V�l�

c · sin θ2� δ ·P�l� for Beam1

~ν0 − ~ν0 · V�l�c · sin θ

2� δ ·P�l� for Beam2; (A3)

where δ the coefficient of pressure-induced shift andother notations have been introduced in Sec. 1already.

Combining Eqs. (A1) and (A2), the fractionaltransmission becomes

τ�ν� � exp�−

Zb

aS�T�l�� · P�l� · X�l� · ϕ�ν − ν0�l��dl

�:

(A4)

Note that in Eq. (A4), the only term in the integralthat depends on ν is the lineshape function. There-fore, Eq. (A4) can be rewritten as

τ�ν� � exp�−A · ϕ�ν��; (A5)

whereA absorbs the terms that have no ν-dependence(S, P, and X) and is essentially the magnitude of thelineshape as shown below:

A �Z �∞

−∞

τ�ν�dν: (A6)

Following Eq. (A5), ϕ�ν� is a normalized lineshapesuch that

R�∞

−∞ϕ�ν�dν � 1.

Based on the above understanding, this work cal-culated Eq. (A4) numerically along the line-of-sight(i.e., l) based on the given distribution of flow param-eters (i.e., temperature, pressure, and velocity).Once Eq. (A4) is evaluated, ϕ�ν� can be obtainedfollowing Eqs. (A4) and (A5).

Once ϕ�ν� is obtained, the DA method seeks thepeak of ϕ�ν� In this analysis, the peak location wasdetermined by first coarsely locating the regionwhere the peak resides, then fitting this region usinga second-order polynomial to obtain an accurate peaklocation. Finally, the peak frequency of ϕ�ν� of beams1 and 2 is then used to calculate the frequency shiftand velocity according to Eq. (2).

In contrast, the 2f method utilizes the peaks in thesecond harmonic of the transmitted signal to deter-mine frequency shift. In practice, the 2f methodmodulates the laser with a sinusoidal wave on topof a triangle wave. The output frequency of the lasercan be expressed by

ν�t� � ν� a cos�2πf mt�; (A7)

where ν is the center laser frequency, a is the mod-ulation amplitude, and f m is the frequency of thesinusoidal modulation. Note Eq. (A7) neglects theeffects due to the triangle modulation since the 2fsignal is detected only at the frequency of the sinu-soidal modulation (f m), which is typically signifi-cantly higher than that of the triangle modulation.

To analyze the 2f signal, Eq. (A7) is used to trans-form Eq. (A4) from the frequency domain into thetime domain so that a Fourier expansion can beperformed as shown below:

τ�ν�t�� � τ�ν� a cos�2πf mt��

� −

Xn��∞

n�0

Hn�ν; a� cos�n2πf mt�; (A8)

where the Hs are the Fourier coefficients, and theyare given by

H0�ν; a� � −12π

Z �π

−πτ�ν� a cos θ� · dθ; (A9)

Hn�ν; a� � −1π

Z �π

−πτ�ν� a cos θ� · cos nθ · dθ�n ≥ 1�;

(A10)

The final 2f signal (R2f ) is then [15]

R2f ��X2

2f � Y22f

�1∕2

; (A11)

while X2f and Y2f are given by

4796 APPLIED OPTICS / Vol. 51, No. 20 / 10 July 2012

X2f �GI02

�H2 �

i02�H1 �H3� cos ω

�; (A12)

Y2f � −GI02

·i02�H1 −H3� sin ω; (A13)

where G is the optical-electrical gain of the detectionsystem and ω the phase shifts between the laserintensity modulation and wavelength modulation(typically equal to π in practice). Note that the abovediscussion is based on the assumption that the 2fmethod completely removes the baseline. This as-sumption is valid when the baseline is constant ora linear function of frequency, which can be realizedin many applications by optimizing the modulationfrequency and modulation depth.

To summarize, this work analyzed the 2f method inthe following steps. First, the fractional transmissionwas calculated according to Eq. (A4) numerically.Second, the results in step 1 were then transformedinto time domain to perform a Fourier expan-sion according to Eqs. (A8–A10). Third, the first threeFourier coefficients (H1, H2, and H3) were thenused to calculate the 2f signal (R2f ) according toEqs. (A11–A13). Lastly, the peak of R2f was deter-mined using the same method as in the DA methodto calculate the frequency shift and the velocity. Weset G � 1 V∕M (volts per watt) in all analysis. Notethat G only scales the magnitude of the 2f signaland does not affect the analysis unless SNR is consid-ered, as briefly discussed in Sec. 3.C. We also set i0 �0.1 andω � π to simulate typical practical conditions.

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