Heat-Flux Measurements with Temperature-SensitivePaint in a Mach-6 Quiet Tunnel

Tianshu Liu∗

Western Michigan University, Kalamazoo, Michigan 49008

and

Christopher A. C. Ward,† Justin Rubal,‡ John P. Sullivan,§ and Steven P. Schneider¶

Purdue University, West Lafayette, Indiana 47907

DOI: 10.2514/1.A32311

A procedure based on an analytical inverse method is developed to calculate heat-flux distributions on a model

using a time sequence of temperature-sensitive-paint images acquired in the Purdue quiet Mach-6 Ludwieg tube.

Temperature-sensitive-paint measurements are conducted on a 7-deg half-angle sharp circularmetal cone atMach 6

to evaluate the accuracy of this method. It is found that the historical effect of the warming-up process of a model in

sequential runs becomes significant. Therefore, it must be corrected by in situ calibration to obtain accurate heat-flux

distributions. After this historical effect and the effect of the nonuniform coating thickness and other potential factors

are corrected, the temperature-sensitive-paint-derived heat-flux distributions on the cone are in good agreementwith

those given by the similarity solution and the reference temperature method. The error sources in temperature-

sensitive-paint heat-flux measurements in the Ludwieg tube are discussed.

Nomenclature

a = thermal diffusivity, m2 · s−1

c = specific heat, J · kg−1 · K−1

D = cone base diameter, mh = static enthalpy, m2 · s−2

I = luminescent intensity,W · m−2 · sr−1

k = thermal conductivity,W · m−1 · K−1

L = polymer thickness, mLc = cone length, mp0 = total pressure, N · m−2

qs = surface heat flux, W · m−2

ReD = Reynolds number, ReD � ρ∞U∞D∕μ∞St = Stanton number, St � qs

ρeue�haw−hw�T = temperature, Kt = time, sTin = initial temperature or ambient temperature, KTw = wall temperature, KT0 = total temperature, Ku, v = velocity components in x and y coordinates, m · s−1

Ue = velocity at edge of boundary layer, m · s−1

x = cone surface coordinate or axisymmetrical coordinate, my = coordinate normal to surface, mδ = cone half angle, radε =

��������������������������������

kpρpcp∕kbρbcbp

�ε = �ε � �1 − ε�∕�1� ε�θ = temperature change from the ambient temperature, K

μ = dynamical viscosity, N · m−2 · sρ = density, kg · m−3

Subscripts

b = basee = edge of boundary layero = total conditionp = polymerps = polymer surfaceref = referencerun = time at startup of tunnels = surfacew = wall∞ = freestream

I. Introduction

H EAT transfer measurements in hypersonic flows have beenconducted to understand laminar-turbulent transition in bound-

ary layers and assess turbulence models. Thermocouples distributedon a surface of a model have been traditionally used to measureheat flux [1–3]. To obtain global experimental aeroheating data, thetwo-color thermographic phosphor measurement technique [4–7],thermochromic liquid crystals [8–10], and temperature-sensitivepaint (TSP) [11–14] have been used in hypersonic wind tunnels.Most quantitative heat measurements have been conducted inhypersonic tunnels where the total pressure and temperature arerelatively high.The Boeing/Air Force Office of Scientific ResearchMach-6 Quiet

Tunnel at Purdue University (simply referred to as the Ludwieg tubein this paper) is unusual compared with other hypersonic tunnelsbecause the total pressure and temperature are much lower. Thisfacility is the largest operational hypersonic tunnel with low noisecomparable to flight [15], and the only one with good optical accessfor global flow diagnostic techniques like TSP. TSP has been used instudies of flow transition in the Ludwieg tube, providing global high-resolution flow diagnostics with simple instrumentation and asmooth surface coating [16–20]. Another advantage of TSP is that itworks readily with thick conformal Plexiglas windows because Ru(bpy)-based TSP can be excited by blue light at 465 nm from a light-emitting diode (LED) array that can penetrate through ordinaryPlexiglas. These characteristics are required to preserve low-noiseflow andwithstand the high stagnation pressures that are present priorto the bursting of the downstreamvalve.Qualitative information from

Received 6 January 2012; revision received 2 April 2012; accepted forpublication 4 April 2012; published online 23 November 2012. Copyright ©2012 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-6794/12 and $10.00 in correspondence with the CCC.

*Professor, Department of Mechanical and Aeronautical Engineering,Parkview Campus; tianshu.liu@wmich.edu (Corresponding author). SeniorMember AIAA.

†Graduate Student, School of Aeronautics and Astronautics, 701W. Stadium Ave. Student Member AIAA.

‡Graduate Student, School of Aeronautics and Astronautics, 701W. Stadium Ave. Student Member AIAA.

§Professor, School of Aeronautics and Astronautics, 701 W. Stadium Ave.Fellow AIAA.

¶Professor, School of Aeronautics and Astronautics, 701 W. Stadium Ave.Associate Fellow AIAA.

282

JOURNAL OF SPACECRAFT AND ROCKETS

Vol. 50, No. 2, March–April 2013

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TSP measurements has been very valuable in the past and continuesto be very useful [20]. However, quantitative global heat-fluxmeasurements using TSP in the Purdue Ludwieg tube have not beensystematically achieved due to some unique problems associatedwith the unusual design that was necessary to achieve quiet flow.Because the total pressure and total temperature in the Ludwieg

tube are much smaller than those in typical hypersonic tunnels,surface heat-flux is on the order of 1000 W∕m2 in laminar boundarylayers on slender bodies (two to four times higher for blunt bodies),and the corresponding surface temperature change is a few degrees,even on a low-conductive material like a nylon model. This makesquantitative TSP heat-flux measurements challenging. Some errorsources that could be ignored in typical hypersonic tunnels becomesignificant in measurements in the Ludwieg tube. In particular, thebodywarms up in sequential runs, and this historical effect will causea systematic error if it is not corrected. The nonuniformity of thecoating thickness also introduces a spatial variation of the measuredheat-flux distributions. Other potential error sources that maycontribute to the heat-flux measurement uncertainty include thestartup of the Ludwieg tube, unsteady model boundary and initialconditions, pressure sensitivity of TSP, interscattering betweenpolished stainless steel surfaces of the test section, filter leakage,variation in material properties, and paint preconditioning. Theobjective of this work is to develop a procedure to improve theaccuracy of quantitative heat-flux measurements using TSP inthe Ludwieg tube.

II. The Purdue Ludwieg Tube

The Purdue Ludwieg tube is shown in Fig. 1 [17–19]. To run thetunnel, high-pressure air at an elevated temperature (approximately433 K) is added from the left side of Fig. 1. A double-burst-diaphragm apparatus is located downstream of the contraction andnozzle, so that the flow can be initiated very rapidly in a well-controlledway that does not preclude quiet flow in the nozzle. A slowgate valve is located downstreamof the burst-diaphragm apparatus sothevacuum pumps can continue to exhaust the vacuum-tank air whilethe burst diaphragms are replaced. The model is placed in thedownstream end of the nozzle. The 113 −m3 vacuum tank is broughtdown to roughly 1–3 kPa before each run, depending on the stag-nation pressure. For quiet runs, the boundary layer on the contractionwall is removed via the bleed slots [15]. Typical characteristicproperties of the tunnel can be found in Table 1. The low value ofsurface heat flux and the subsequent small changes in surface temper-ature make quantitative TSP heat-flux measurements challenging.The detailed description of the working principles of the Ludwieg

tube is given by Schneider [15]. Initiation of supersonic flow involvescomplex three-dimensional, unsteady shocks and expansion waves.When the diaphragmbreaks, an expansionwave travels upstream andthen reflects between the end of the driver tube and the contraction.The pressure drops with each reflection, until it decreases to the pointthat the tunnel unstarts. Run times of 3 to 5 s are typical. The PurdueLudwieg tube has an additional effect because of the differentialheating along the tube. The contraction and driver tube, upstream ofthe nozzle, are heated, but the test section and model are not.Therefore, the initial expansion wave during the startup cools the test

section andmodel, and this is followed by heating as the hot gas fromupstream enters the test section. Such a large oscillating change inheat flux during the startup is unusual in hypersonic tunnels. Itappears to be the result of placing the burst-diaphragm apparatusdownstream of the nozzle, to avoid disturbing quiet flow. Further-more, the downstream end of the nozzle is unheated to simplifyoperations and permit the use of conformal Plexiglas windows.Before the run begins, the nozzle and contraction are at stagnationpressure, and the nozzle air is mostly at room temperature. This high-pressure air must be exhausted during the startup process, generatingthe large oscillating flux in heat transfer. Most hypersonic tunnelsbeginwith vacuum in the nozzle and test section, and use an upstreamvalve. Although this simplifies the design of the test section, nozzle,and models, and reduces the startup duration, such a design is notfeasible for use in a quiet Ludwieg tube. In addition, determiningaccurate heat flux usually requires that a model be in thermalequilibrium just prior to tunnel start and that the model temperaturebe accurately known. For the first run of the day, these conditions aresatisfied.However, at the end of each run, transonic flow in the nozzleleads to high heat fluxes and the model warms up considerably.The model cools during preparation for the next run and during thepump-up for the run. A good method for obtaining heat flux fromTSP measurements in the Ludwieg tube must cope with theseissues.

III. Theoretical and Numerical Solutions

The purpose of this section is to assess the accuracy of thesimilarity solution and the reference temperature method for ahypersonic laminar boundary layer on a sharp circular cone forcomparison with TSP data. A similarity solution exists for ahypersonic laminar boundary layer on a circular cone at zero angle ofattack [21,22]. This similarity state seems to be quickly approachedand sustained in the Ludwieg tube even though the total enthalpy andwall temperature fall moderately with time. In general, a similaritysolution is sought under the constant wall temperature condition. Thereference temperature method for a circular cone is also used [21].For both the similarity solution and reference temperature method,the Taylor–Maccoll solution is used for the conical outer flow

Fig. 1 Schematic of the Purdue Ludwieg tube.

Table 1 Typical running characteristics

for the Purdue Ludwieg tube

Tunnel ResultsTotal pressure 1034 kPaTotal temperature 433 KReynolds number based on one foot 3.0 × 106

Run time ≈5 sOn a 7-deg sharp coneΔT on nylon cone ∼10 KΔT on aluminum cone ∼2 KHeat flux in laminar boundary layer∼1000 W∕m2

Stanton number ∼3 × 10−4

Recovery temperature 377 KSkin friction coefficient 4.7 × 10−4

Skin friction 11 Pa

Note: The Ludwieg tube can have one run per hour.

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conditions [23,24]. Furthermore, the similarity solution and thereference temperature method for hypersonic flows over circularcones are examined by them comparing with numerical Navier–Stokes (NS) solutions for sharp cones, as shown in Fig. 2 [25].**

Computations of hypersonic flows and heat transfer are sensitive tothe temperature dependencies of the viscosity and thermal conduc-tivity of air. The relatively newmodels of Lemmon and Jacobson [26]for the thermodynamic properties are used in the similarity solutionsand the reference temperature method. The viscosity of air as afunction of temperature is given by a two-piece power-law function tofit Lemmon and Jacobson’s data [26]. The temperature dependencyof the thermal conductivity is given by a power-law function and thespecific heat is given by a quadratic function of temperature to fittheir data.Kimmel’s measurements at Mach 8 are used to evaluate the heat

transfer computationmethods [2]. The total pressure and temperatureare p0 � 443 psi and T0 � 722 K, respectively, and the walltemperature is Tw � 303 K. Figure 2a shows comparisons of theheat-flux distributions given by the similarity solution, referencetemperature method, numerical NS solutions [27], and Kimmel’sdata [2] in the laminar boundary layer. The NS solutions, similaritysolution, and reference temperature method are consistent. Therelative differences in the similarity solution and the referencetemperature method are less than 5% compared with the NSsolutions.Chien’s laminar results at Mach 7.9 on a 5-deg cone provide

another case for the evaluation of the heat transfer computationmethods since data from thermocouples are available [1]. Figure 2bshows Stanton number distributions given by the similarity solution,the reference temperature method, and a numerical NS solutioncompared with Chien’s data [1]. Here, the Stanton number is definedbased on the outer flow conditions and the Reynolds number is basedon the freestream conditions. The experimental data in Fig. 2b arecollected from runs 21, 23, 24, 26, and 29 [1], where Tw∕T0 � 0.35.In runs 21, 23, and 24, p0 � 2190 psi and T0 � 1470°R. In runs 26and 29,p0 � 1481 psi and T0 � 1450°R. These results indicate thatthe similarity solution and the reference temperature method aresufficiently accurate in predicting laminar heat flux in hypersonicflows over a sharp circular cone at zero angle of attack. Therefore,they will be used as a reference standard for the development of thepresent method.

IV. Determination of Heat Flux from TSPMeasurements

TSP is a thin polymer layer doped with certain luminescentmolecules whose emission is sensitive to temperature. TSP is usuallyair brushed directly on a model surface or the surface of an insulatinglayer used to coat a metal model. Illumination lights with a suitablewavelength are used to excite TSP. Once the TSP coated on a surfaceis calibrated, the surface temperature fields can be measured bydetecting the luminescent emission. Digital cameras with opticalfilters are used to image the TSP. From a time sequence of the surfacetemperature fields in a hypersonic tunnel, an analytical inversemethod can be used to determine the heat flux on a surface.It is assumed that a model is in thermal equilibrium with the

ambient conditions at t � 0. In other words, the initial temperature ofa model including both the insulating layer and base is assumed to bethe ambient temperature Tin (the initial temperature) before adiaphragmbreaks in a specific run. Under this condition, the transientsolution of the one-dimensional time-dependent heat conductionequation has been obtained for a thin polymer (TSP or TSP/insulator)on a semi-infinite base [28–30]. The heat flux at the polymer surfaceis given by

qs�t� �kp�1 − �ε2�

���������

πapp

Z

t

0

�W�t − τ; �ε�����������

t − τp dθps�τ�

dτdτ (1)

where θps�t� � T�t; L� − Tin is the temperature change at thepolymer surface from the ambient temperature,L is the polymer layerthickness, and

�W�t; �ε� � 2���

πp

Z

∞

0

exp�−ξ2�d ξ1� �ε2 − 2�ε cos�2Lξ∕ �������

aptp � (2)

The parameter �ε is defined as �ε � �1 − ε�∕�1� ε�. A discrete formofEq. (1) is given by

qs�tn� ≅kp�1 − �ε2�

���������

πapp

X

n

i�1

θps�ti� − θps�ti−1��������������

tn − tip � �����������������

tn − ti−1p � �W�tn − ti�

� �W�tn − ti−1�� (3)

For �ε � 0, Eq. (3) recovers the Cook–Felderman method for a semi-infinite base because �W�t; 0� � 1 [31]. This means that the function�W�t; �ε� represents the effect of the polymer layer (or TSP itself) on thedetermination of heat flux, which depends on �ε, ap, and L. In ourmeasurements, the thermal properties of the TSP/insulator areassumed to be the same as those of Mylar. The nose tip and the main

Fig. 2 Comparisons of heat-flux computations with a) Kimmel’s dataon the 7-deg cone at Mach 8, and b) Chien’s data on the 5-deg cone atMach 7.9.

**Personal communication with H. Johnson of University of Minnesota2009.

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cone body are made of stainless steel and aluminum, respectively.Table 2 lists the thermal properties of the polymer Mylar and basematerials.

V. TSP Measurements on 7-Deg Half-AngleAluminum Cone

A series of runs at Mach 6 was performed with a 7-deg half-angle aluminum cone in the Ludwieg tube. The first cone had a totallength of 0.4 m and the radius of the stainless steel nose tip wasapproximately 1 mm. The interchangeable nose was 0.15 m long.Thewhole conewas coated with LustreKote paint [made by Top Flite(TF)] that was composed of white primer and glass MonoKote. Thepurpose of this coating is twofold. Thewhite coatingwill enhance theluminescence emission of TSP detected by a camera for achievinga high signal-to-noise ratio. Furthermore, for a metal model, aninsulating coating on the surface will increase the change of surfacetemperature for a given heat flux to improve the accuracy in the heat-flux calculation. It is speculated that the thermal properties of theinsulating layer are similar to those of a typical polymer. Then, theTSP, Ru(bpy) in Chromaclear auto paint, was coated on the top ofthe LustreKote paint. Themean paint thickness from x∕Lc � 0.2 − 1is approximately 250 μm based on paint thickness measurementsusing an eddy current gauge. The temperature calibration data for Ru(bpy) in Chromaclear auto paint are fitted by a polynomial [19]:

T∕Tref �X

4

n�0

Cn�I∕Iref�n (4)

where C0 � 1.2342, C1 � −0.3804, C2 � 0.3443, C3 � −0.2543,and C4 � 0.0542. Equation (4) is applicable in a rangeof 0.9 < T∕Tref < 1.1.Table 3 lists the test conditions for four sequential runs: runs 1, 2, 3,

and 4, including the initial total pressure and temperature, the prerunwall (surface) temperature measured using a thermocouple atx∕Lc � 0.75, the Reynolds number based on the base diameter, andthe times at which the tunnel started. The prerun surface temperaturewas 296 K in run 1, and increased to 307 K in run 4. Figure 3 showsthe total pressure and temperature that are a decreasing functionof time in run 1, where the pressure was measured using a Kulitepressure transducer (XTEL-190-500A) on thewall of the contractionsection, and then the temperature was calculated using the isentropicrelations. For the other runs, the total pressure and temperature havesimilar behaviors. The decay rates of the total pressure for runs 1, 2, 3,and 4 are approximately 25, 34, 21, and 57 kPa∕s, respectively. Thedecay rates of the total temperature for runs 1, 2, 3, and 4 areapproximately 5, 5, 5, and 7 K∕s, respectively. Figure 4 shows timehistories of surface temperaturemeasured using the thermocouple forruns 1, 2, 3, and 4. In run 1, the initial model surface temperature wasequal to the ambient temperature Tin � 298 K. Then, the modelsurface temperature T�trun� before a specific run increased in thesequential runs (runs 2, 3, and 4), where t � trun is the time just before

a diaphragm breaks in the run. The large temperature increaseindicated in Fig. 4 occurs during the shutdown of the tunnel after aMach 6 run, when a transonic flow with a high mass flow rate passesthe model.Figure 5 shows a generic setup of a charge-coupled device (CCD)

camera with a bandpass optical filter for detecting the luminescentemission from theTSP, and a blueLEDarray for illuminating theTSPon a model. The 16-bit camera (POC 1600) and blue LED array(LM2X-460), whose emission peak is at 460 nm,were used. They areproduced by Innovative Scientific Solutions Inc. A time sequence of100 images was acquired by the camera for each run at 32 f∕s. Thebasic testing procedures for TSP in awind tunnel are described byLiuand Sullivan [32]. A surface temperature image is obtained by usingthe calibration relation [Eq. (4)] based on a ratio between a wind-onTSP image and a wind-off TSP image (a TSP image at a constantreference temperature). Then, from a time sequence of the surfacetemperature images, the corresponding heat-flux images arecalculated at every pixel by applying Eq. (3) to the selected imagedomain.For the purpose of comparison and in situ correction, a thin-film

heat-flux sensor (Omega HFS-4) was attached on the back surface ofthe model (relative to the camera) at x∕Lc � 0.67. To monitor thechanges in the local surface temperature of the model during andbetween runs, a surface thermocouplewas located at x∕Lc � 0.67 ontop of the TSP covered with a 0.3-mm acrylic adhesive layer. A base

Table 3 Test conditions for the aluminum cone

Properties Run 1 Run 2 Run 3 Run 4

p0 (initial), kPa 625 896 521 896T0 (initial), K 432 432 435 430Tw (prerun), K 298 301 305 307ReD 5.9 × 105 8.5 × 105 4.8 × 105 8.3 × 105

Time, h 12:59 13:48 14:30 15:10

Fig. 3 Total pressure and temperature as a function of time in run 1.

Table 2 Thermal properties of polymer andbase materials

Properties Mylar Aluminum Stainless steel

k�W∕m K� 0.15 204 16ρ�kg∕m3� 1420 2700 7900cp�J∕kg K� 1090 904 500

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thermocouple was in contact with the base of the cone and insulatedfrom the air by five layers of 25-μ-thick adhesive Mylar. Thethermocouple signals were amplified 25 times and recorded on adigital oscilloscope.For a specific run (such as run 4), the camera records a time

sequence of TSP images from t � trun just before the diaphragmbreaks in this run. Based on these images, we actually calculate thefollowing integral:

qs;run�t� �kp�1 − �ε2�

���������

πapp

Z

t

trun

�W�t − τ; �ε�����������

t − τp dθps;run�τ�

dτdτ (5)

where θps;run�t� � H�t∕trun − 1��Tps�t� − Tin� is the change in thesurface temperature during this run, and H�ξ� is the Heavisidefunction [H�ξ� � 0; if ξ < 0 andH�ξ� � 1; if ξ ≥ 0]. Note that thevalue of Tin does not change the value of the integration. In general,Eq. (5) is not equivalent to Eq. (1) unless the paint layer and thealuminum base are in thermal equilibrium at t � trun. This problemwill be further discussed in Sec. VI.Figure 6 shows typical images of the changes in surface

temperature and heat flux at t � 1.6 s, where the heat flux images arecalculated by using Eq. (5) from a time sequence of 100 TSP images.Note that thewhole cone from the tip to the end is shown in Fig. 6, andan artifact on the images appears at the junction between the stainlesssteel nose tip and the aluminum cone body. Figure 7 shows timehistories of the surface temperature change and estimated heat flux at

Fig. 5 A generic setup of a CCD camera and a UV LED array forilluminating the TSP on a model at the Ludwieg tube.

Fig. 6 Run 1 at t � 1.6 s a) image of surface temperature change, and

b) heat-flux image.

Fig. 7 Time histories of a) surface temperature change and b) estimatedsurface heat flux at x∕Lc � 0.59 in run 1.

Fig. 4 Time histories of surface temperature measured using a

thermocouple for runs 1, 2, 3, and 4.

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x∕L � 0.59 on the centerline of the cone,whereLc is the cone length.The relatively large negative and positive heat transfer during thestartup processwas a surprise. After the transient startup, the heat fluxonly slightly decreases with increasing time. This gradual decrease inthe heat flux corresponds to the decrease in the total pressure andtemperature during a run (see Fig. 3).To compare to the steady-state theoretical solutions, the time-

averaged heat flux from t � 0.6 − 2.75 s is calculated at eachlocation. Figure 8 shows the time-averaged heat-flux distributionsalong the centerline of the cone for runs 1, 2, 3, and 4 in comparisonwith those given by the similarity solution and the referencetemperature method. Note that the blunt tip of a cone has some effecton heat flux near the tip. Numerical NS solution†† for a blunt conewith a nose radius of 1 mm predicts a lower heat flux than that for asharp cone for the front 20% portion.In run 1, the heat-flux distribution calculated using Eq. (5) is close

to the theoretical solutions. In run 1, as indicated in Fig. 4, thetemperatures of the paint layer and the aluminum base are equal tothe ambient temperature, i.e., T�trun� � Tin just before a diaphragmbreaks at t � trun. Physically, this means that the model is in thermalequilibrium with the ambient conditions in run 1. However, Fig. 8shows that the TSP-derived heat-flux distributions obtained usingEq. (5) deviate more and more from the theoretical solutions as timegoes on. For run 4, the TSP-derived heat flux even becomes negative,

which is physically unrealistic in this case. This nonphysical shift inthe estimated heat flux after the first run has not been noticed inprevious measurements in hypersonic tunnels. In the followingsection, we argue that it is related to the historical effect of the modelwarming-up process in the Ludwieg tube (see Fig. 4). This effect onthe heat-flux calculation may be significant when the heat flux to bemeasured is small. In addition, a repeatable pattern in the heat-fluxdistributions is observed in all cases in Fig. 8. This streamwisevariation is generated by a nonuniform coating of insulator.

VI. Historical Effect of Model Warming-Up Process

As indicated previously, the condition for Eq. (1) or (3) is that, att � 0, both the polymer (paint) and base temperatures are equal to theambient temperature. However, as indicated in Fig. 4, measurementsby a thermocouple on the surface show that the prerun modeltemperature rises from the ambient temperature during the day fromrun 1 to run 4. The calculation using Eq. (5) underestimates heat fluxfor runs 2, 3, and 4. Thus, the effect of the increased prerun modeltemperature could produce a significant systematic error if it is notcorrected in the heat-flux calculation.To give an estimate of the historical effect of the model warming-

up process before a specific run, the evolution of the polymer surfacetemperature can be decomposed into a prerun process and a runprocess for a specific run. The virtual initial time (t � 0) is generallyset when the temperatures of the polymer and base are equal to the

Fig. 8 Heat-flux distributions along a ray on the aluminum cone in a) run 1, b) run 2, c) run 3, and d) run 4.

††Given by H. Johnson in an unpublished report.

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ambient temperature Tin, and trun denotes the time at which adiaphragmbreaks in a specific run. An expression for the time historyof the surface temperature on the polymer from t � 0 to the currenttime in a specific run is

θps�t� � θps;run�t� � θps;pre−run�t�� H�t∕trun − 1��Tps�t� − Tin�� �1 −H�t∕trun − 1���Tps�trun� − Tin�f�t∕trun� (6)

where f�t∕trun� � �Tps�t� − Tin�∕�Tps�trun� − Tin� is a nondimen-sional function describing the model warming-up process before thisrun. The first and second terms in Eq. (6) are the time histories in thisrun and before this run, respectively. This decomposition is exactmathematically, and serves as a reasonable model for a rapid startupprocess of a hypersonic tunnel like the Ludwieg tube. The timehistory f�t∕trun� of the model warming-up process before a specificrun is not known, which includes the histories in preceding runs andintervals between runs.For t > trun, substitution of Eq. (6) into Eq. (1) yields

qs�t� �kp�1 − �ε2�

���������

πapp

�Z

t

trun

�W�t − τ; �ε�����������

t − τp dθps;run�τ�

dτdτ

�Z

trun

0

�W�t − τ; �ε�����������

t − τp dθps;pre−run�τ�

dτdτ

�

(7)

The first term in Eq. (7) is just Eq. (5). The second integral in Eq. (7)presents the historical effect of the prerunmodel warming-up processon the heat-flux calculation. Further, the second integral is written asa function of a nondimensional time �τ � τ∕trun, i.e.,

Z

trun

0

�W�t − τ; �ε�����������

t − τp dθps;pre−run�τ�

dτdτ �

Tps�trun� − Tin�������

trunp

×

Z

1

0

�W�t∕trun − �τ; �ε��������������������

t∕trun − �τp

df��τ�d�τ

d�τ

�Tps�trun� − Tin

�������

trunp

�

A0 � A1

�

t

trun− 1

�

� A2

�

t

trun− 1

�

2

· · ·

�

(8)

In Eq. (8), a Taylor expansion near the startup time t � trun is used.Because τ∕trun is close to one when trun is much larger than the runduration (approximately 2–5 s), an approximation is that only the firstterm in the Taylor expansion is retained in Eq. (8). Note that thecoefficients in the Taylor expansion depend on the coating thickness.Therefore, Eq. (7) becomes

qs�t� �kp�1 − �ε2�

���������

πapp

�Z

t

trun

�W�t − τ; �ε�����������

t − τp dθps;run�τ�

dτdτ

� B0� �L∕L��Tps�trun� − Tin��

(9)

where B0 is a proportional coefficient to be determined (its unit iss−1∕2), L is the local coating thickness, and �L is the averaged coatingthickness over a region. The ratio L∕ �L is the normalized thicknessdistribution, which represents the effect of the coating nonuniformity.The second term in Eq. (9) is considered as a correction for the effectof the increased prerun polymer surface temperature Tps�trun� fromthe ambient temperature Tin and the effect of the nonuniform coatingthickness.According to the previously described argument, the heat-flux

distribution given by TSP for run 1 is consistent with the theoreticalsolutions because the prerun surface temperature is equal to theambient temperature (297 K). In heat-flux calculations, the ratio(kp∕ �L) between the thermal conductivity and thickness of thepolymer (Mylar) is 600, where the averaged coating thickness is�L � 250 μm and kp � 0.15 W∕m · K. For Tps�trun� � Tin � Tb in

run 1, this value is confirmed by using the discrete Fourier lawkp∕ �L � qs∕�Tps − Tb�, in which qs is given by a thin-film heat-fluxsensor (Omega HFS-4) located at x∕Lc � 0.65 and Tps is thetemperature measured by TSP at the same location.As indicated before, because the time history f�t∕trun� of the

model warming-up process is not known, the coefficient B0 in thecorrection term has to be determined in situ using the data given bythe thin-film heat-flux sensor mounted at x∕Lc � 0.65 on thesurface. Figure 9 shows the time histories of surface heat flux givenby the heat-flux sensor. The difference between the time-averagedheat-flux values given by TSP using the first term in Eq. (9) and theheat-flux sensor is shown in Fig. 10 as a function of Tps�trun� − Tin.The data for these runs are approximately on a straight line asindicated by Eq. (9). Overall, the correlation between the runs inFig. 10 is reasonably good, and the estimated value of B0 is 1.3.Without suitable correction, the relative systematic errors to themeanheat flux of the similarity solution in x∕Lc � 0.1 − 0.4 in runs 1, 2, 3,and 4 are 8, 30, 34, and 70%, respectively.As shown in Fig. 8, after the single value of B0 � 1.3 is applied to

all the runs and the measured thickness distribution L in Sec. VII isused in Eq. (9), good correction is achieved, indicating that thesystematic error due to themodel warming-up process is reduced andthe spatial variation due to the nonuniform thickness is reduced. Theabsolute root-mean-squared (rms) error after the systematic error iscorrected is 220-292 W∕m2 in x∕Lc � 0.1 − 0.4, and the corre-sponding relative rms error to the mean heat flux of the similaritysolution in x∕Lc � 0.1 − 0.4 is approximately 20–27%.As indicatedin Fig. 8b, boundary-layer transition may be observed since the heatflux increase after x � 0.32 m in run 2. Figure 11 shows thedistributions of the heat transfer parameter St × Re

1∕2D given by TSP

compared with the theoretical solutions for runs 1, 2, 3, and 4. Here,the Stanton number is defined as

St � qsρeue�haw − hw�

(10)

where ue and ρe are the velocity and air density at the edge of theboundary layer, respectively, and haw and hw are the enthalpies at theadiabatic wall and wall, respectively. The Reynolds number isdefined as ReD � ρ∞U∞D∕μ∞, whereD is the base diameter of thecone, and U∞, ρ∞, and μ∞ are the freestream velocity, density, anddynamic viscosity, respectively. The heat transfer parameter St ×Re

1∕2D takes the effect of the Reynolds number into account in

aeroheating scaling. The relative rms error in the heat transfercoefficient in all the runs is 22% in x∕Lc � 0.1 − 0.4. Note that,when x∕Lc < 0.1 near the cone tip, the assumptions of the semi-infinite base and one-dimensionality are not good.

Fig. 9 Histories of heat flux given by the heat-flux sensor at x∕Lc �

0.65 for the four runs.

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The previously described correction procedure is not purelyempirical, and it has a rational foundation based on an estimateB0� �L∕L��Tps�trun� − Tin� in Eq. (9) for the historical effect of themodel warming-up process and the effect of the nonuniformthickness. The historical effect is evident in the Ludwieg tube becausethe aerodynamic heating flux on a model during a specific run isrelatively small (1000-5000 W∕m2), and the heating between thehypersonic portions of the runs is not negligible partially due to thedownstream valve that is necessary to achieve quiet flow. In previousTSP measurements in hypersonic tunnels [11–14], the effect has notbeen noticed because the hypersonic aerodynamic heating is sodominant. In addition, this effect disappears when a model coolsdown to the ambient temperature if the interval between twosequential runs is sufficiently long. However, this is not practical fortests in the Ludwieg tube because the productivity of wind tunneltesting is reduced.

VII. Nonuniformity of Coating Thickness

A fixed pattern of spatial variation is observed in the heat-fluxdistributions in all the cases in Fig. 8. This error is caused by thenonuniformity of the coating thickness. For a TSP on a metal model,TSP heat-flux measurement is more sensitive to the coating

thickness. This variation is enlargedwhen the total pressure is higher,as indicated in Figs. 8b and 8d. To correct the effect of the nonuniformthickness of a coating, the thickness distribution of the coating shouldbe known. It is noticed that the startup process experiences a suddenheating on a model when the hot gas enters the test section fromupstream. Then, the model surface temperature decays due to heatconduction to a higher-conductive aluminum model, which dependson the coating thickness. Figure 9 shows the time histories of heatflux given by theOmega thin-filmheat flux sensor at x∕Lc � 0.65 onthe cone in runs 1, 2, 3 and 4. A question is whether the coatingthickness distribution can be determined based on TSP measure-ments in the startup process. This possibility is explored in this paper.For a thin polymer on a high-conductivemodel, Liu et al. [33] give

an approximate solution for the polymer surface temperature changeθps�t� � T�t; L� − Tin from the initial temperature:

θps�t� ≅1

������������������

πkpρpcpp

Z

t

0

qs�τ�S�t − τ�����������

t − τp dτ (11)

where

S�t − τ� � 1.163� 3.676 exp

�

−�0.5434L�24ap�t − τ�

�

− 1.3 exp

�

−�0.2592L�24ap�t − τ�

�

− 3.576 exp

�

−�0.8421L�24ap�t − τ�

�

� ε exp

�

−L2

4ap�t − τ�

�

(12)

Note that S�t − τ� represents the effect of the thin insulating layer.Here,ε �

��������������������������������

kpρpcp∕kbρbcbp

, and kp, cp, and ρp are the thermalconductivity, specific heat, and density of the polymer, respectively;kb, cb, and ρb are the thermal conductivity, specific heat, and densityof the base material, respectively; and ap � kp∕cpρp is the thermaldiffusivity of the polymer. To model the heating peak in Fig. 9,impulse heating qs � qsoδ�t� is used, where δ�t� is the Dirac-deltafunction. Thus, Eq. (11) can be written as

θps�t� �qs0L

2���

πp

kp�t∕τp�−1∕2S�t∕τp� (13)

where τp � L2∕4ap is the characteristic timescale of the polymerlayer. Even though the delta function is a highly ideal model, it isreasonable because we focus on the temperature decay after theimpulse heating.To determine the coating thickness conveniently in practice,

Eq. (13) should be suitably approximated. In the first method,S�t∕τp� is approximated by a power-law function S�t∕τp�≈0.96�t∕τp�−0.8, where the effect of ε is neglected because it issmall for a polymer layer on a high-conductivemodel. Thus, we have

θps�t� �0.48qs0L

3.6

���

πp �4ap�1.3kp

t−1.3 (14)

Equation 14) indicates that the surface temperature of a polymer layerdecays in a−1.3 power law after the impulse heating. For linear least-squares regression of data, Eq. (14) is conveniently written as

ln�θps�t�� � −1.3 ln�t� � ln

�

0.48qs0L3.6

���

πp �4ap�1.3kp

�

(15)

By fitting the TSP data in the decay using Eq. (15), the coatingthickness L can be estimated from the intercept when qso, ap, and kpare given. As shown in Fig. 12, the decay of the surface temperaturecan be fitted using the −1.3 power law. Thus, the intercept can bedetermined and the coating thickness can be estimated. Based on thedata given by the heat-flux sensor in Fig. 9, an estimated peak heat-flux value is qs0 � 10 kW∕m2. It is also assumed that the suddenheating due to the passing of the hot gas is uniform over the surface,

Fig. 11 The distributions of the heat transfer parameter given by TSPprocessed with the analytical method with correction compared with thetheoretical solutions for uns 1, 2, 3, and 4.

Fig. 10 Difference between the time-averaged heat flux values given byTSP processed with the analytical method without correction and theheat flux sensor.

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and the decay of the polymer surface temperature ismainly caused byheat conduction to the aluminum model. Aerodynamic skin frictionheating is neglected in this initial stage of the decay compared withthe heat conduction to the model. This method is called the intercept-based method.In the second method, we use the following exponential

approximation:

�t∕τp�−1∕2S�t∕τp� ≈ 11.7 exp�−γ�t∕τp�β�

where γ � 2.54 and β � 0.384 are determined by least-squaresestimation. Similar to Eq. (15), we have

ln�θps�t�� � −γ�t∕τp�β � ln

�

11.7qs0L

2���

πp

kp

�

(16)

By fitting theTSPdata in the decay usingEq. (16), the slope−γ�τp�−βcan be determined in a plot of ln�θps�t�� as a linear function of tβ,which is related to the coating thickness L. This method is calledthe slope-based method, which has an advantage in that thedetermination of the thickness does not depend on the value of qso,unlike the intercept-based method.Figure 13 shows the coating thickness distributions along the

centerline of the cone in images obtained using the intercept-basedmethod (method 1) and the slope-based method (method 2) based onthe TSP data in run 4. The averaged thickness distribution from thetwo distributions extracted using the two methods and the data givenby an eddy-current gauge (Elcometer 456 coating thickness gauge)on the aluminum section of the cone are also plotted for comparison.The slope-based method is more sensitive to the thickness variationand other factors. In Fig. 13, the random noise is filtered in theaveraged distribution. The pattern in the estimated thicknessdistribution corresponds to that in the heat-flux distribution. Themean thickness on the aluminum section (x � 0.15 − 0.4 m) is 250–300 μ, which is consistent with that given by the Elcometer thicknessgauge. Because the Elcometer thickness gauge gives a mean value inan area of approximately an 8-mmdiameter, the gaugemeasurementsdid not sufficiently resolve the spatial thickness variation. Theaccuracy of the gauge claimed by the manufacturer is 1–3%. Thecoating thickness distribution averaged from the results given bymethods 1 and 2 is used for the thickness correction in heat-fluxcalculation using the inverse method. Figure 8 shows the heat-fluxdistributions with the corrections for both the historical effect and thecoating thickness variation for runs 1, 2, 3, and 4. The spatial patternsin the heat-flux distributions for these runs are considerably reduced.

VIII. Simple Discrete Fourier Law MethodCombined with In Situ Calibration

The aforementioned analysis identifies the main issues in TSPheat-flux measurements in the Ludwieg tube, and therefore it sets astage for further simplification with in situ calibration. This sectionwill show that a discrete Fourier law can be used as a simplifiedversion of Eq. (9) for a metal model. From a viewpoint ofimplementation, the discrete Fourier law is preferred due to itssimplicity and reasonable accuracy in ametal model. A typical case isconsidered, in which a thin TSP/insulator layer is coated on a highlyconductive model (e.g., metal model). When the conditions ε ���������������������������������

kpρpcp∕kbρbcbp

<< 1 and L�����������

s∕app

<< 1 are satisfied, where s isthe variable of the Laplace transform in the time domain, anapproximate expression of Eq. (1) is [33]

qs�t� ≅kp

Lθps�t� −

����������������

kpρpcp

π

r

�

θps�t���

tp � 1

2

Z

t

0

θps�t� − θps�τ��t − τ�3∕2

dτ

�

(17)

The first term in the right-hand side of Eq. (17) is a discrete Fourierlaw when the base remains at the initial temperature Tin. In reality,when the base temperature is slightly changed, the effect of thebase on heat-flux computation is contained in the second term.Interestingly, the magnitude of the second term equals the heat fluxinto a presumed semi-infinite polymer slab that is relatively smallfor a good insulator and can be neglected in the first-orderapproximation. For an insulating polymer on a metal base, the firstcondition ε << 1 is usually met. Because s ∼ t−1, the secondcondition L

�����������

s∕app

<< 1 can be written as τp∕t << 1, where τp �L∕a2p is a diffusion timescale of the polymer layer.When the polymerlayer is sufficiently thin, τp=t << 1 should be met after a transientstage of a run. Therefore, Eq. (17) establishes the legitimacy of usingthe discrete Fourier law in TSP heat-flux measurements on ametal model.For a thin polymer layer on a high-conductive metal base, such as

aluminum and stainless steel models, the first term in Eq. (9) can beapproximated by the discrete Fourier law. Therefore, Eq. (9) can bewritten as the following approximate form:

qs�t� � �kp∕ �L��Tps�t� − Tb�trun�� � �B0� �L∕L��Tps�trun� − Tin�(18)

where �B0 � kp�1 − �ε2�B0∕���������

πapp

(the unit of B0 is s−1∕2). Thediscrete Fourier law has been used to estimate the heat

Fig. 13 The coating thickness distributions along the centerline of the

cone in images obtained by using the intercept-based method (Method 1)and the slope-based method (Method 2).

Fig. 12 The surface temperature decays fitted using the−1.3 power lawat two locations on a circular aluminum cone for run 4.

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flux distributions on metal models in TSP measurements inconventional hypersonic tunnels [11,12]. However, in the Ludwiegtube, more factors (such as the historical effect of the modelwarming-up process) affect heat-flux calculation. For kp∕ �L �600�W · m−2 · K−1�, Eq. (18) gives results that are consistent withthose given by Eq. (9) for runs 1–4, where the coefficient B0 isdetermined by fitting the data, and a linear fit givesB0 � 2 for the useof the discrete Fourier law. It is noted that it differs fromB0 � 1.3 forthe complete inverse solution Eq. (9). Because the previous errorsources cannot be completely corrected a priori, the TSP is calibratedin situ using heat-flux data obtained by Schmidt–Boelter (SB) gaugesat several locations [34]. The aluminum base has a high thermaldiffusivity so that the effect of the startup heat pulse dissipatesquickly. The factors kp∕ �L and �B0� �L∕L��Tps�trun� − Tin� in Eq. (18)can be considered collectively as two unknown coefficients to bedetermined by in situ calibration to match the data from the SBgauges. The accuracy of such in situ calibration depends on that of theSB gauges.A series of additional runs was performed with a sharp 7-deg half-

angle cone at zero angle of attack [34]. Figure 14 shows typical time-dependent data from an SB gauge [SB gauge D (SB-D) located0.28 m from the cone tip] fitted by TSP data at a location in a runwhere the total pressure is 903 kPa and the total temperature is 430K.The two unknown coefficients in Eq. (18) are determined by least-squares fit, and Eq. (18) is then applied to other locations to obtain aheat-flux field. Figure 15 shows the heat-flux distributions along a rayon the cone obtained using the discrete Fourier law with in situcorrection based on the heat-flux data from SB-D. The error is lessthan 20% if the SB gauge is accurate. This method is particularlysimple, and therefore it is now used in TSP measurements in theLudwieg tube [34].

IX. Uncertainty

The uncertainty analysis has two related parts: uncertainty in TSPtemperature measurement and uncertainty in heat-flux computationfrom surface temperature. The error propagation equation of TSPtemperature measurement and the sensitivity coefficients have beengiven by Liu and Sullivan [32]. The elemental error sources areassociated with model deformation, unstable illumination, paintphotodegradation, filter leakage, and luminescence measurement.At nominal zero angle of attack (AOA) in the present tests, the coneand support are considered to be rigid such that the major errorsources related to model deformation are small. For large modeldeformations, an image registration procedure should be applied toalign the wind-on and wind-off images. The temporal intensityvariation of the blue LED array for illuminating the TSP was notmonitored during the tests. However, because the time-averaged heatflux is obtained, the random variation is removed by averaging. Theminimum temperature resolution limited by the photon shot noise ofa typical CCD camera is approximately 0.2 K [32].

The sensitivity analysis has been given by Liu et al. [28] to assessthe uncertainty of heat flux extracted using Eq. (1), which depends onthe elemental error sources in the parameters L, kp, �ε, and ap.Simulations are conducted to quantitatively determine theheat-flux uncertainty Δqs∕qs as a function of the elemental errorsΔL∕L, Δkp∕kp, Δ�ε∕�ε, and Δap∕ap for a thin polymer layer on asemi-infinite aluminumand nylon bases. The sensitivity ofΔqs∕qs tothe elemental errorsΔL∕L,Δkp∕kp, andΔ�ε∕�ε is weak for a polymerlayer on the nylon base. In contrast, for a polymer layer on analuminum base, Δqs∕qs is proportionally dependent on ΔL∕L andΔkp∕kp according to the discrete Fourier law Eq. (17). Of course, thesystematic error due to the historical effect of the model warming-upprocess should be considered as well.Other possible error sources in our measurements are discussed in

the following sections. These errors are not corrected in this work,and estimates of these errors will require further investigation.

A. Interscattering Between Polished Stainless Steel Surface

in Test Section and Filter Leakage

The inside of the Ludwieg tube test section is polished stainlesssteel, so that interreflections of the emitted luminescence from theTSP are detected by the CCD camera. In addition, reflections ofthe incident LED light leak through the CCD camera filter. It isclear from many tests that both of these effects contribute to errors.But the extent is not exactly known yet, and it is complicatedto correct the effect of interscattering. It is also not trivial to blackenthe inside of the nozzle without risk of damaging the polishedsurfaces.

B. Paint Preconditioning

Past calibration results have shown that a polymer in TSPundergoes a physical change when the temperature exceeds itsglass transition temperature during the first time the TSP is heated,which changes the calibration [32]. Then, in subsequent heating andcooling cycles, the calibration curve becomes stable. It may benecessary to heat newly coated TSP on a model beyond its glasstransition temperature in an oven prior to the first Ludwiegtube run. In our experiments, such paint preconditioning is notperformed.

C. Pressure Sensitivity of TSP

The pressure sensitivity of TSP is relatively unknown. Ruthenium-based luminophores are susceptible to the effects of pressure throughFig. 14 Data from an SB-D gauge fitted by TSP data at a location.

Fig. 15 Heat-flux distributions along a ray on the aluminum coneobtained using the discrete Fourier law with in situ correction based on

heat-flux data from SB-D.

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the oxygen quenching process. This is highly undesirable for TSPbecause the pressure effects will introduce errors in temperaturemeasurements if they are not corrected. This is particularly importantin the Purdue Ludwieg tube, where the model sits at stagnationpressure for an extended time before the run begins. Therefore, abinder that is oxygen impermeable must be chosen for TSP.Matsumura et al. [16] noticed that the TSP [specifically, Ru(bpy) in apolyurethane binder with titanium-dioxide powder] did show aneffect due to an increase in pressure. The pressure effect is consideredas a major source contributing to the large systematic error in theirTSP heat flux measurements. It is speculated that a mixture of thebinder with titanium-dioxide powder could enhance the pressuresensitivity of the TSP. However, experiments have been performed todetermine the oxygen sensitivity of the TSP [Ru(bpy) in automotiveChromaclear coat]. It has been found that there is virtually nosensitivity of the TSP to oxygen.

X. Conclusions

TSP heat-fluxmeasurements have been conducted on a 7-deg half-angle aluminum circular cone at Mach 6 in the Purdue Ludwieg tubeto evaluate the accuracy of the analytical inverse method incalculating heat flux from a time sequence of TSP images. The lowheat flux and small surface temperature change, along with theunsteady oscillating thermal startup process in the Ludwieg tube,pose challenges to quantitative global heat-flux measurements.The similarity solution and the reference temperature method forhypersonic laminar boundary layers on sharp circular cones areexamined by comparing them with numerical NS solutions. It isshown that the theoretical solutions are accurate enough to serve as areference for comparison with the TSP-derived heat-flux distri-butions on a circular cone.Four tests are conducted sequentially under different total

pressures and temperatures in the Ludwieg tube. It is found that theTSP-derived heat-flux distribution increasingly deviates from thetheoretical solutions as the prerun surface temperature of the coneincreases during the sequential runs. The relative systematic errors tothemean heat flux of the similarity solution in runs 1, 2, 3, and 4 are 8,30, 34, and 70%, respectively. This nonphysical shift is caused by thehistorical effect of the warming-up process of the cone model. Thishistorical effect on the time integration in the analytical inversemethod is estimated as a linear term of the difference between theprerun surface temperature and the ambient temperature, wherethe effect of the nonuniformity of the coating thickness is included.The unknown coefficient in this term is determined in situ using thedata given by a heat-flux sensor mounted on the cone surface. Theanalytical inverse method, coupled with an in situ correction schemefor this historical effect and the effect of the coating nonuniformity,can improve the accuracy of TSP heat-flux measurements in theLudwieg tube. In addition, the coating thickness distribution isdetermined using the decay of the coating surface temperatureimmediately after the rapid heating in the startup process of theLudwieg tube. After the corrections aremade, the relative root-mean-square error in heat flux is approximately 22% compared to thesimilarity solution on the cone.The analysis given in this paper provides a good understanding

into the main issues that affect the accuracy of TSP heat-fluxmeasurements in the Ludwieg tube. It is indicated that in situ TSPcalibration using several heat-flux gauges is necessary. Therefore, thesimple discrete Fourier lawmethod coupled with in situ calibration isjustified for highly conductive models. A small number of Schmidt–Boelter (SB) heat-transfer gauges are now installed in everymodel, toget accurate single-point results that can be used for in situ TSPcalibrations on every tunnel run. The models are now made ofaluminum covered with insulating paint, because it was found thatthe startup heat-transfer pulse diffuses rapidly into the aluminum, sothat it can be neglected during the run. New kinds of paint are usedto insulate the model and bind the TSP, improving uniformity,repeatability, ease of use, and adherence. These factors in combi-nation have resulted in a simple and practical method that has yieldedpromising results for simple geometries.

Acknowledgments

This work was supported by a NASA Research Announcementgrant (NNX08AC97A). We would like to thank H. Johnson of theUniversity of Minnesota and C. Roy of Virginia Tech for kindlyproviding their numerical results for comparison.

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M. MacLeanAssociate Editor

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