- Email: [email protected]
Temperature measurements in convective heat transfer flows using dual-broadband, pure-rotational coherent anti-Stokes Raman spectroscopy (CARS)
Experimental Thermal and Fluid Science 19 (1999) 13±26
Temperature measurements in convective heat transfer ¯ows using dual-broadband, pure-rotational coherent anti-Stokes Raman spectroscopy (CARS) Sean P. Kearney *, Robert P. Lucht 1, Anthony M. Jacobi Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., MC-244, Urbana, IL 61801, USA Received 20 March 1998; received in revised form 15 November 1998; accepted 29 December 1998
Abstract This paper describes the use of dual-broadband, pure-rotational coherent anti-Strokes Raman spectroscopy (CARS) as a nonintrusive temperature diagnostic for convective-heat-transfer ¯ows. The characteristics of the dual-broadband, pure-rotational CARS technique are discussed, and the technique is compared to other temperature measurement methods. Dual-broadband, purerotational CARS was used to measure mean temperature pro®les in a low-Reynolds-number, turbulent boundary layer. The results are presented in wall units and compared to the thermal law of the wall for zero-pressure-gradient boundary layers. Temperature data were acquired as close as 50 lm (25 lm) to the wall, with a spatial resolution of 50 lm normal to the heat transfer surface, and a 2r precision limit of 4 K. The spatial resolution of this experimental method provides detailed information in complex thermal boundary layers and allows for an estimation of the convective heat ¯ux to within an estimated uncertainty of ÿ5% to +25%. Singlelaser-shot temperature data were acquired in a gas cell, and the potential for measurement of rms temperature ¯uctuations is discussed in terms of the resulting probability density functions. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Laser diagnostics; CARS; Non-intrusive temperature measurements; Boundary layers
1. Introduction Coherent anti-Stokes Raman spectroscopy (CARS) is a well-established technique for probing high-temperature, combustion ¯ows but has been seldom used in ``lower'' (<500 K) temperature convective-heat-transfer and ¯uid-mechanics applications. The ®rst such use of CARS was reported by Kearney et al. [1]. They used a dual-pump (three-laser) CARS technique to map the thermal boundary layer near a vertical cylinder in freeconvection. The present paper presents boundary layer temperature measurements in a forced ¯ow using a dualbroadband, pure-rotational (DBPR) CARS technique, independently suggested by Alden et al. [2] and Eckbreth and Anderson [3,4].
*
Corresponding author. Tel.: +1 217 244 0778; fax: +1 217 244 6534; e-mail: [email protected] 1 Present address: Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA.
CARS oers several advantages over other temperature measurement methods for probing convectiveheat-transfer ¯ows. Thermocouples, thermistors and RTD's, while often highly accurate and simple to use, are intrusive and thereby introduce uncertainties which can be dicult to quantify. CARS is a non-intrusive tool and is free from these types of ambiguities. Conventional optical methods such as Schlieren, shadowgraph photography and interferometry ([5,6]) are non-intrusive and oer the advantage over CARS of ``full-®eld'' views of the boundary layer. However, these more widely used optical methods provide path-averaged data and can be ineective in 3-D ¯ows. CARS oers pointwise measurements with a spatial resolution dictated by a cylindrical probe volume that is typically 50 lm in diameter and 1±2 mm long. Recently, laser-induced ¯uorescence (LIF), another optical technique which, like CARS, has been developed for combustion applications, has been employed in lower temperature ¯ows [7,8]. Temperature information is derived from LIF by exciting a molecule or atom to a higher energy state and detecting the ¯uorescence signal
0894-1777/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 9 ) 0 0 0 0 4 - 7
14
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
as the excited molecules or atoms decay to lower energy states via spontaneous emission. The ¯uorescence signal is proportional to the number density of the initially pumped state, which is a function of temperature. LIF oers the advantage of full-®eld planar imaging without the line-of-sight averaging inherent in the more conventional optical methods discussed above. However, LIF often requires that tracer species be seeded into the ¯ow while CARS does not. This paper begins with an overview of the CARS technique and a discussion of the advantages of DBPRCARS over other CARS implementations for temperature measurement. Following the overview of CARS, the dual-broadband CARS system and wind-tunnel apparatus are described, and the procedures used in conducting the experiments and data reduction are discussed. The results include CARS measurements of mean temperature pro®les, a comparison of the measured mean temperature pro®les to the thermal law of the wall, a determination of the local wall heat ¯ux from the mean temperature pro®les, and a discussion of measurement uncertainty. The potential for single-lasershot CARS measurements for determination of local rms temperature ¯uctuation is also discussed. 2. Description of the CARS technique As shown in Fig. 1(a), a CARS signal is generated by crossing three high-powered, pulsed laser beams, termed pump beam #1, pump beam #2, and the Stokes beam, in a suitable phase-matched geometry. The beam crossing region is at the common focus of the three laser beams and de®nes the CARS probe volume, a 50-lm diameter cylinder that is 1±2 mm in length. The frequency difference x1 ÿ xS is tuned to a Raman-allowed transition of the probed molecule, creating an oscillating polarization which scatters the incident radiation from pump beam #2 to create a CARS signal at frequency xCARS (x1 ÿ xS )+x2 . In practice, broadband laser sources are used to probe several resonances simultaneously, resulting in a broadband CARS signal beam whose spectral signature depends upon temperature, pressure and species concentration. More detailed descriptions of the physics of CARS signal generation and the dependence of CARS spectra on temperature, pressure, and species concentration can be found in the literature. Druet and Taran [9] and Eckbreth [10] present excellent general discussions, while Alden et al. [11], Foglesong et al. [12], and Kearney et al. [1] provide discussions of pure-rotational CARS. 3. Advantages of DBPR-CARS for thermometry The CARS technique presented here diers from usual the implementation in high-temperature environments in two ways. First, two broadband laser sources are used instead of just a single broadband source. Secondly, the Raman resonances probed are pure-rota-
Fig. 1. (a) Shows the arrangement of laser beams for dual-broadband CARS with planar BOXCARS phase matching. The pump #1 and Stokes beams, from the same broadband dye laser source are shown in dark gray, the pump #2 beam, from the spectrally narrow Nd:YAG laser, is shown in light gray, and the CARS signal is in black. A vector diagram for the planar BOXCARS phase matching is given in (b).
tional S-branch (Dv 0, DJ +2) transitions, as opposed to the vibrational Q-branch (Dv 1, DJ 0) resonances typically probed in combustion applications. In many CARS implementations, a single broadband Stokes laser source is typically used so that the frequency dierence x1 ÿ xS is tuned to the range of transitions desired, as shown in Fig. 2(a). When employing a single broadband source, shot-to-shot spectral variations in dye laser output introduce noise in the resulting CARS spectrum because each Raman transition is coupled with only a narrow frequency range of dye laser output. In our study, a dual-broadband technique was used where both the pump #1 and Stokes laser beams originate from the same broadband dye laser. With this dual-broadband method, shown in Fig. 2(b), each Raman resonance is coupled to a large number of frequency combinations in the broadband dye-laser pro®le, resulting in an eective averaging over the spectral noise in the dye-laser output. This spectral averaging minimizes the eect of dye laser spectral noise which is perhaps the largest cause of temperature measurement error. At temperature less than about 500 K, pure-rotational CARS is often preferred over its vibrational
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
15
Fig. 2. Schematic representation of pure-rotational CARS processes. Energy level diagrams and schematic renderings are given for: (a) the widely used broadband CARS technique and (b) dual-broadband CARS. Note the close spectral proximity of the pump #1 and Stokes lasers and the pump #2 and CARS beams characteristic of the small Raman shifts associated with pure-rotational CARS.
CARS counterpart. Both methods have comparable signal strengths at lower temperatures, but pure-rotational CARS spectra are more sensitive to small temperature changes than vibrational CARS spectra; furthermore, pure rotational CARS spectra exhibit wider spectral separation (4B) between neighboring rotational lines. Large spectral separation makes calculation of temperature by comparison to theoretically calculated spectra easier because it is simpler to calculate the theoretical spectra and to use the integrated-lineintensity technique discussed later in this paper. 4. Apparatus 4.1. Lasers and optics A schematic of the DBPR-CARS apparatus is shown in Fig. 3. Eighty percent of the 350 mJ, 10-ns pulse from the second harmonic of a Q-switched, 10 Hz repetition rate, Nd:YAG laser (k2 532 nm, x2 18,797 cmÿ1 ) is used to pump the broadband dye
laser. Using rhodamine 640 dye, the broadband laser output is centered near kS 607 nm (xS 16,474 cmÿ1 ) with a nominal bandwidth of DxS 130 cmÿ1 (FWHM). Ten percent of the Nd:YAG energy is directed to a beam trap and the residual 35 mJ is used to provide the narrowband pump beam #2 in the CARS process. The 20 mJ/pulse output of the broadband dye laser is directed to a 50/50 beamsplitter to provide both the pump #1 and Stokes beams. This beam arrangement results in nominal laser pulse energies of 10, 35, and 10 mJ for the pump #1, pump #2, and Stokes laser beams, respectively. Several mirrors and prisms are used to arrange the three laser beams into the planar BOXCARS con®guration, and a 250-mm spherical focusing lens provides a cylindrical probe-volume that is 50 lm in diameter and 1±2 mm long. The three input laser beams and the CARS signal are recollimated by another 250-mm lens and the Stokes and pump #2 beams are directed into beam traps by prisms. The CARS signal is generated virtually collinear to pump beam #1 and is relayed by a series of eight 532-nm mirrors to a 100-mm spherical
16
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
Fig. 3. Schematic of the dual-broadband CARS system used to measure boundary layer temperature pro®les.
lens, which focuses the signal at the entrance slit of a 1m spectrometer. The eight mirrors are used to attenuate the strength of the remaining pump #1 beam in order to prevent contamination of the signal. This attenuation is possible because the 532-nm mirrors transmit most of the 607-nm light from the pump laser while re¯ecting the majority of the CARS signal, which is centered around 529 nm. The spectrometer diraction grating successfully separates the remainder of the pump beam radiation from the CARS beam and the signal is imaged onto a 14-bit, 512 ´ 512, back-illuminated, unintensi®ed CCD camera through a relay lens pair. 4.2. Wind tunnel A schematic of the low-speed wind-tunnel used for the channel ¯ow experiments is shown in Fig. 4. Air is supplied by a 62-W blower and ¯ows through a ¯exible, 203-mm diameter tube to the 305-mm long settling chamber of 210 ´ 305 mm cross-section. The ¯ow is conditioned by honeycomb ¯ow straightener (75 mm length, 7.5 mm cell dimension) and three ®ne mesh screens. A 4.6 to 1 rectangular nozzle, 450 mm in length, accelerates the ¯ow into the heated test section. The test section is a semi-rectangular channel, consisting of two heated metal plates at the top and bottom, and side windows fabricated from 6.35-mm-thick plate glass. The channel is 146 mm wide and 286 mm long.
The top plate is ¯at and constructed from 10.9-mmthick, 2024-aluminum alloy. The bottom plate, fabricated from 304 stainless steel, is curved so that the CARS laser beams are not clipped as they enter the test section. The channel height is 30 mm at the center-line, where the measurements are made, and 35 mm at the plate glass windows. To ensure that curvature eects are not important, care was taken to guarantee that the local radius of curvature of the steel plate was at least an order of magnitude greater than the half-height of the channel, which represents the maximum attainable boundary layer thickness. A scheme similar to the one used by Swearingen and Blackwelder [13] is used to provide negligible hydrodynamic boundary-layer thickness at the entrance to the test section. Here, the ®rst 25.4 mm of the channel is located inside of the nozzle and a concentric suction slot of 10-mm minimum clearance is provided for the nozzle boundary layer ¯ow to exit into a low-pressure aluminum chamber. Variable-speed axial fans can be used to provide suction from the aluminum chamber to provide a laminar channel ¯ow with a well-de®ned boundary layer origin. In the experiments reported here, the axial fans were not used. With no suction provided, the pressure in the aluminum plenum was not suciently low to remove recirculating ¯uid from the clearance slots. The recirculating ¯uid fed a large-amplitude disturbance into the boundary layer, providing a boundary
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
17
Fig. 4. Wind-tunnel used for CARS boundary layer temperature measurements.
layer ``trip'' at the test section entrance and turbulent boundary layers in the channel ¯ow. The wind-tunnel ¯ow was characterized using hotwire measurements taken in the outer regions (u/ U P 0.6) of an isothermal boundary layer at Reynolds numbers of 4.6 ´ 104 and 8.3 ´ 104 . Mean velocity, streamwise turbulence intensity, and power spectra data are provided in Fig. 5. When plotted in wall units, the mean velocity data exhibit a logarithmic pro®le that is characteristic of turbulent boundary layers. In addition, the turbulence intensity pro®les were found to be similar in shape and magnitude to the results of Klebano [14], as were power spectra obtained at three selected locations in the boundary layer. These hot-wire results and the mean temperature pro®les to be presented indicate that the boundary layer ¯ow was similar in structure to a fully turbulent boundary layer in spite of the low Reynolds number ¯ows, 104 < Rex < 105 , studied. This early transition is a result of the above-mentioned, largeamplitude disturbance present at the leading edge of the channel. A ``bypass'' [15,16] transition of this sort results when a disturbance of sucient magnitude causes the normal instability and transition process to be replaced by a more abrupt transition. 5. Procedure 5.1. Experiment At the beginning of each experiment, the wind-tunnel, plate heaters, and data acquisition system were allowed to warm up for about 1 h so that the temperature of the metal plates forming the top and bottom channel surfaces reached a steady state in a range of 100±150 K above ambient. The criterion suggested by Osborne and Incropera [17] was applied to verify that the free-con-
vection contribution to the heat ¯ux was negligible under the conditions of the experiments to be reported. The CARS probe volume was accurately positioned by keeping the probe volume stationary and moving the wind tunnel, which was ®xed to a three-axis translation stage. The location of the probe volume with respect to the curved plate was determined by viewing the re¯ection of a reduced-power laser beam from the Nd:YAG laser on a white card and moving the wind-tunnel vertically until the image was clipped by the curved plate. This procedure allowed for positioning of the CARS probe volume to within about 50 lm (25 lm), or one probe volume diameter, with respect to the bottom wall. Accurate streamwise positioning of the probe volume was achieved by aligning the probe volume with eroding thermocouple probes mounted ¯ush with the surface at predetermined reference locations, and then moving the wind-tunnel to the desired streamwise location. This procedure allowed for streamwise positioning of the probe volume within an uncertainty of about 1 mm. The Reynolds number, based on streamwise distance and centerline velocity, was obtained from hot-wire measurements of the centerline velocity and by evaluating the kinematic viscosity at the local free-stream temperature. The uncertainty in Reynolds number was conservatively estimated at 5%. Temperature pro®les were obtained from CARS spectral signatures which were averaged for 100 laser shots by recording 10 s CCD camera exposures. Use of shot-averaged CARS spectra was desirable in order to decrease the noise level in the spectra and to increase the data acquisition rate. This averaging procedure yields the mean temperature provided that the local rms temperature ¯uctuation is suciently small. In ¯ows with high temperature ¯uctuations, use of a shot-averaged CARS spectrum can yield results which are biased toward low temperatures because the CARS signal
18
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
where the temperature is known, to determine the shape of the broadband dye laser pro®le. Because the broadband dye laser has a ®nite bandwidth, transitions of small Raman shifts are pumped more strongly than those of large Raman shifts. The error introduced by this unequal distribution of dye-laser energy is minimized by normalizing the CARS spectra by an appropriate correction. For DBPR-CARS, the appropriate normalization is the convolution of the dye-laser pro®le with itself. The shape of the dye-laser convolution is determined by dividing the peak signal intensities of the experimentally obtained, room temperature reference spectra by the peak intensities obtained from the theoretical prediction given by the Sandia CARSFT code [18], which does not account for the structure of the dye laser pro®le. A smooth curve is ®t through the correction factors obtained by this process and the resulting curve ®t is used to normalize all experimentally obtained spectra. 6. CARS data reduction ± integrated line intensity technique
Fig. 5. Hot-wire results for characterization of the boundary layer ¯ow. Mean velocity pro®les in the outer region (u/U P 0.6) of the curved plate boundary layer are given in (a). Streamwise turbulence intensity and one-dimensional power spectra are given in (b) and (c), respectively.
strength decays with increasing temperature. For the experiments reported here, the levels of temperature ¯uctuation were suciently low for use of the shot-averaging procedure. A discussion of estimated mean temperature bias is provided within the section on measurement uncertainty in Section 6 of this paper. 5.2. Calibration The CARS system was calibrated using reference spectra obtained outside the thermal boundary layer,
Temperature, pressure, or other quantities of interest are most often deduced by comparing experimentally obtained CARS spectra to theoretically calculated spectra. This comparison is typically performed by ®nding the calculated spectrum which best ®ts the data in the least-squares sense (spectrum ®tting method), or by comparing the experimental and theoretical integrated line intensity distributions (integrated line intensity method). In this work, temperatures are determined from CARS spectra by using an integrated line intensity technique similar to the ones outlined in Refs. [11,12,19]. The application of this integrated intensity technique is facilitated due to the widely spaced transitions associated with pure-rotational spectra and is preferred because it is much faster than the spectral ®tting method [12]. The Sandia CARSFT code [18] is used to generate theoretically calculated spectra in 1 K increments from 220 to 550 K. At each temperature, normalized integrated intensity ratios are calculated from the theoretical spectra by summing the intensity under each of the isolated N2 peaks, highlighted in the spectra provided in Fig. 6, and dividing the results for each transition by the maximum integrated value. This process yields a library of theoretical, normalized integrated-line-intensity distributions for each temperature. Experimental DBPRCARS spectra are reduced by computing the same normalized integrated line intensities from the experimentally obtained spectra and ®nding the library entry which best ®ts the experimental data in the least-squares sense. This least-squares procedure minimizes the quantity K
N exp X S k1
k exp Smax
S th T ÿ thk Smax T
2 ;
1
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
19
Fig. 6. Experimentally obtained, 100-laser-shot-averaged DBPR-CARS spectra at evaluated temperatures of 297 and 452 K. Rotational S-branch transitions are marked in the legend at the top of each plot, with O2 transitions being indicated by asterisks. The lines used in the integrated line intensity procedure for determining temperature are highlighted in bold. Integrated intensity distributions are given below each spectrum. Note the distinct intensity shift toward higher Raman shifts (and higher rotational energies) with increasing temperature.
where N is the total number of rotational lines used in the data reduction and Skexp and Skth are the integrated rotational line intensities of the kth rotational transition in the experimental and theoretical spectra respectively. exp th The normalizing factors, Smax and Smax , are the integrated line intensity values for the transition with the maximum integrated intensity in the experimental and theoretical spectra. By normalizing by the maximum integrated intensity value, the procedure is sensitive only to the shape of the spectrum and not the signal strength, which can vary substantially with temperature and laser beam intensity. Fig. 6 contains the results of the integrated-line-intensity procedure for evaluated temperatures of 297 and 452 K. There is a noticeable shift in the distribution of peak N2 line intensities toward Raman shifts of higher rotational quantum number with increasing temperature. In this manner the CARS spectra exhibit the expected rotational population distribution. 7. Results 7.1. Mean temperature Mean boundary layer temperature distributions for three selected Reynolds numbers are presented in di-
mensional form in Fig. 7. The results show excellent spatial resolution, with temperature data acquired in increments as small as 50 lm normal to the wall. At Rex 3.1 ´ 104 , a thermocouple mounted ¯ush with the heat transfer surface indicates a temperature of 453 K ± consistent with the value of 450 K obtained by extrapolating the DBPR-CARS temperature pro®le to the wall. The results are plotted in dimensionless form in Fig. 8 against the mean temperature pro®les for turbulent, zero-pressure-gradient boundary layers suggested by Reynolds et al. [20] and Kays and Crawford [21]. Here, the dimensionless temperature and distance from the wall are given by T and
Tw ÿ T qCp u ; qw
2
yu : 3 m To compute the dimensionless forms of the data, it is necessary to know both the heat ¯ux and shear stress at the wall. As explained in the next section, the wall heat ¯ux is obtained from a linear ®t to the temperature data in the near-wall region. The heat ¯ux is used to estimate the wall shear from a modi®ed form of the Colburn analogy as suggested by Reynolds et al. [20]. y
20
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
The qualitative agreement is especially good in the conduction sublayer and buer region where free-stream eects should not be signi®cant. In the logarithmic regime, the data deviate more signi®cantly from the zeropressure-gradient pro®les, with the data falling into two categories depending upon the free-stream speed. This deviation from the zero-pressure-gradient curves is to be expected since the temperature ``law of the wall'' is not a truly universal pro®le as is suspected for the case of a velocity boundary layer. The free-stream pressure gradient signi®cantly in¯uences the thermal boundary layer shape in the logarithmic region, with pro®les in favorable pressure gradients exhibiting the usual boundary layer thinning indicated by an upward shift of the T ±y curve ± Refs. [16,21]. Therefore, the upward shift with free-stream speed should be expected as the pressure gradient must increase with the mass ¯ow through the channel. Finally, we must note that the uncertainty in the values of T presented in the logarithmic layer in Fig. 8 is largely dominated by the estimates of u and qw . The heat ¯ux is determined directly from the spatially resolved boundary layer temperature pro®les as described below. The random error contribution to the uncertainty in qw is about 10±15% while the uncertainty in u is likely to be in¯uenced by an ambiguous systematic error associated with the use of Eq. (4) which is valid for 105
Fig. 7. DBPR-CARS temperature data recorded in a low-Reynoldsnumber turbulent boundary layer at 3 representative Reynolds numbers. The estimated uncertainty due to random errors in these 100laser-shot-averaged data is 4 K. Data were captured as close as 50 lm from the wall with a spatial resolution of 50 lm in the wall-normal direction. This excellent spatial resolution allows for a quantitative estimate of the heat ¯ux obtained from the linear ®ts to the data in the near-wall region shown in the plots.
Cf : 4 2 In dimensionless form, the data exhibit the shape of typical turbulent boundary layers suggested by the law of the wall temperature pro®les. This agreement further supports the observation of a bypass transition to boundary layer turbulence at Reynolds numbers well below the usual critical value, and is similar to the results of Feiler [22], who used thermocouples to observe temperature pro®les of comparable shape at Reynolds numbers as low as 2.8 ´ 104 . St Pr0:4
Following the procedure suggested by Qiu et al. [23], an estimate of the heat ¯ux can be obtained from the slope of a least-squares line ®t to the temperature data in the conduction sublayer. Calculation of heat ¯ux from the wall temperature pro®le is useful because the convective heat ¯ux is measured directly without correction for radiative heat transfer. The set of points used for least-squares ®tting were determined using two criteria, y 6 8 and y 6 11. These two criteria provided a sucient number of data for least-squares ®tting while remaining within the upper sublayer limit of y 13.2 quoted in Ref. [21]. The wall heat ¯ux was calculated by multiplying the least-squares slope by the thermal conductivity of air evaluated at the wall temperature extrapolated from the least-squares ®t to the temperature data. Use of this extrapolation procedure instead of an independent measurement of the wall temperature (by a thermocouple or RTD) eliminates any systematic error associated with recording Tw and T1 from two instruments with dierences in their calibrations [23]. The heat ¯ux data obtained using both the y 6 8 and y 6 11 criteria are given in Table 1 along with the number of data
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
21
Fig. 8. Mean temperature data plotted in dimensionless wall units. The data are compared to the predictions of Reynolds [20] and Kays and Crawford [21].
points that were used in each least-squares ®t, the extrapolated wall temperature, and local Nusselt number. At each value of Rex , the heat ¯ux estimates obtained from the y 6 8 and y 6 11 criteria are within 10%, an estimate of the scatter, or precision limit, in the heat ¯ux measurement that is consistent with the 10% uncertainty quoted in Ref. [23]. Based on this agreement, the y 6 11 criterion was chosen for calculation of the Nusselt number, as this provides more data points for estimation of the slope. For comparison, predicted Nusselt numbers are tabulated by extrapolating the correlation of Reynolds et al. [20], 0:6 Nux 0:0296 Re0:8 x Pr
Tw T1
ÿ0:4 5
all properties evaluated at T1 ;
below its lower limit of applicability of Rex 105 . The heat ¯ux data are in fair agreement with the correlation, with Eq. (5) underpredicting the data by 7±43%. 7.3. Potential for single-laser-shot measurements Single-laser-shot CARS data oer the possibility for measurement of rms temperature ¯uctuation in complex ¯ows, as well as non-intrusive point estimates of turbulent heat ¯ux if LDV data are also available. Sets of 500 single-shot DBPR-CARS spectra have been acquired from N2 gas in a room temperature cell at four dierent pressures. Pure nitrogen was used instead of air in these measurements to obtain simpler CARS spectra without overlapping N2 ±O2 transitions. Four 500-realization temperature pdfs, each obtained at a dierent gas cell pressure, are displayed in Fig. 9.
Table 1 Results of heat ¯ux estimates obtained from least-squares ®ts to the linear, near-wall temperature pro®le. Two criteria are presented for choosing the data set used in the least-squares ®t, y 6 8 and y 6 11. The tabulated values include the number of points used in the least-squares ®t, the wall temperature obtained by extrapolating the least-squares line to the wall, the heat ¯ux, Nusselt number data, and predicted Nusselt numbers from the correlation suggested in Ref. [20] Rex
N (y 6 8)
N (y 6 11)
Tw (K) (y 6 8)
Tw (K) (y 6 11)
qw (kW/m2 ) (y 6 8)
qw (kW/m2 ) (y 6 11)
Nux (y 6 11)
Nux Eq. (5)
3.1 ´ 104 7.8 ´ 104 1.1 ´ 105
12 4 4
16 6 6
449.6 393.6 414.9
450.3 393.4 413.2
2.72 4.22 5.48
2.80 4.17 4.96
113 187 263
79.1 175 226
22
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
Fig. 9. Temperature histograms obtained from 500 single-shot DBPR-CARS temperature realizations from room-temperature N2 in a gas cell at various pressures. The results show that the precision of the measurement improves with increasing pressure.
At 1 atm, the standard deviation is 18.3 K, yielding a 2r uncertainty of 36 K due to precision error only. Improved precision is obtained with an increase in gascell pressure. This improvement is due to a decreased noise-to-signal ratio (NSR) in the DBPR-CARS signal at higher pressures. This reduced noise eect is readily seen upon examination of the following equation for the NSR due to Bengtsson et al. [24]. r p ÿ 1=2 NSR 2p=tC2 1 C22 =W 2 XM =2pCR : 6 In Eq. (6), t is the duration of the CARS signal pulse, C2 and CR are the Nd:YAG laser and Raman linewidths, respectively, W is the spectrometer slit width, and XM is the mode spacing of the broadband dye laser. As the pressure increases the second term becomes smaller because CR increases due to pressure broadening eects. The single-shot, DBPR-CARS temperature precision results from this study and from Ref. [11,24] are summarized in Table 2. The results of the present study exhibit the expected trend of decreased uncertainty with increased pressure, as do the results of Refs. [11,24] (obtained by the same group of researchers), where the atmospheric results of Ref. [11] indicate less precision than the data presented in Ref. [24], obtained at pressures of 2.6 atm and higher. Improved single-shot precision at atmospheric pressure is achievable by decreasing the NSR with an increase in the dye-laser oscillator cavity length. An elongated cavity results in a
reduced mode spacing, XM , which will decrease the second term in Eq. (6). 8. Measurement uncertainty 8.1. Uncertainty in shot-averaged CARS temperature measurements The uncertainty in the DBPR-CARS mean temperature data can be estimated in terms of precision (random) and bias (systematic) contributions as described by Moat [25]. The precision error contribution is largely due to dye-laser mode-phase ¯uctuations [11] and CCD camera read noise, while the bias contribution is a result of density weighting eects present in regions of high rms temperature ¯uctuation. Estimates of both contributions to the shot-averaged temperature uncertainty are given here. An estimate of the precision limit in the mean temperature evaluated from 100-laser-shot-averaged DBPR-CARS spectra was made from 237 temperature realizations in room air. A histogram (pdf) constructed from these room-air data is shown in Fig. 10. The mean of these measurements is 294 K, and the standard deviation about 2 K. Using these results we can obtain a 2r (95% con®dence interval) estimate of 4 K for the precision limit contribution to the uncertainty. Since a controlled temperature source was not available for further testing, the pdf at 294 K was the only data set
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
23
Table 2 Comparison of single-shot temperature results to the results of previous work Investigator
Gas
Pressure (atm)
T (K)
TRMS (K)
TRMS =T(%)
Laser dye
Present study
N2
Air
Alden et al. [11]
N2
293 293 294 297 426 490 585 706 294
18.3 13.4 11.9 11.0 4.6 4.9 5.9 7.7 13.8 12.3
6.2 4.5 4.0 3.7 1.1 1.0 1.0 1.1 4.7 4.2
Rhodamine 640
Bengtsson et al. [24]
1.0 3.1 4.5 5.6 2.6 4.1 7.8 16.1 1.0
taken. However, from the scatter of the data in the mean temperature pro®les (see Fig. 7), it seems reasonable to assume that 4 K is a good estimate of the random error over the temperature range of 290±450 K observed in the experiments. With this uncertainty estimate, we required a wall-to-air temperature dierence of 100±150 K so that the precision error was 2±4% of the total temperature dierence. Use of shot-averaged CARS spectra can also cause a bias error toward low temperatures due to densityweighting eects. This eect is signi®cant in regions of high rms temperature ¯uctuation because the CARS signal strength decays with increasing temperature so that contributions to the shot-averaged spectrum from lower temperature gas are weighted more heavily than those from higher temperatures. To estimate the magnitude of this density weighting bias error, several ¯uctuating temperature ®elds with Gaussian probability were numerically generated, and the Sandia CARSFT code [18] was used to compute theoretical CARS spectra at each temperature in these pdfs. Representative, ``shot-averaged'' CARS spectra
DCM
DCM Rhodamine 610
were computed by summing the intensity data from the numerically generated spectra, and the mean temperature was calculated from these averaged spectra using the integrated-line-intensity method. The results for mean temperatures of 300, 350, 400, and 450 K and rms temperature ¯uctuations of 10, 20, 30, and 50 K are given in Table 3. At each value of T, the systematic error, or bias, in mean temperature becomes larger with increasing rms temperature ¯uctuation, so that the maximum bias toward low temperature occurs in the buer layer and in the outer regions of the sublayer where the temperature ¯uctuations are highest [26]. Assuming a 20 K maximum rms temperature ¯uctuation, consistent with the results reported in Ref. [26], the maximum bias error incurred is 4 K toward cooler temperatures ± a value comparable to the precision error contribution. The bias error is reduced to 1 K or less in other regions of the boundary layer where the ¯uctuations are smaller. This 1±4 K bias toward lower temperatures can be combined with the 4 K precision error to yield a composite uncertainty in the shot-averaged mean temperature that varies from (+3,ÿ5) K to (+0,ÿ8) K depending upon the location in the boundary layer. The bias error can be eliminated by abandoning the more-ecient shot-averaging procedure used in this study, and recording a large sample of single-laser-shot CARS spectra and averaging the evaluated temperatures.
Table 3 Results of CARS density weighting study performed by generating a Gaussian pdf with mean temperature T and rms ¯uctuation rT . Mean temperatures calculated from the averaged CARS spectra are presented for several T, rT combinations
Fig. 10. Histogram (pdf) obtained from 237 DBPR-CARS temperature realizations in room air. The results from these 100-laser-shotaveraged spectra and the scatter in the boundary layer temperature pro®les indicate that 4 K is a reasonable estimate of the 2r precision limit in the shot-averaged mean temperature measurements.
T K
rT 10 K
rT 20 K
rT 30 K
rT 50 K
300 350 400 450
299 349 400 449
295 346 396 446
290 343 391 443
272 326 378 428
24
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
8.2. Uncertainty in wall heat ¯ux The uncertainty in the heat ¯ux and Nusselt number data presented in Table 1 can be estimated in terms of precision and bias contributions. The uncertainty in the heat ¯ux measurement is dominated by temperature gradient uncertainty, because errors in jw due to uncertainty in the extrapolated wall temperature are comparatively small. Similarly, it was found that the Nusselt number uncertainty is dominated by the uncertainty in the heat ¯ux measurement, with measurement errors in x, j1 , and (Tw ÿ T1 ) making a minor contribution (<1%). Therefore, the uncertainties in both heat ¯ux and local Nusselt number are essentially equal and are determined by the temperature gradient uncertainty. For each temperature pro®le presented in this paper, the precision limit (random error) in the wall temperature gradient was estimated by plotting the data used in the least-squares ®t alongside the best-®t line and lines with slopes 10 and 20% of the best-®t slope. A representative plot is shown in Fig. 11. For all three data sets, the 10% lines are a much more reasonable estimate of the possible error in slope due to the 4 K scatter in the temperature data. With this in mind, we feel that 10±15% is a conservative estimate of the precision limit the heat ¯ux measurement. Mean temperature bias due to shot averaging introduces a bias error toward higher wall heat ¯uxes. This temperature bias increases with rms temperature ¯uctuation. Published data and DNS calculations [26] show that rT monotonically increases with distance from the wall, peaking near y 18. The eect of this temperature bias on the evaluated heat ¯ux was estimated by assuming a maximum temperature bias of 4 K (rT 20 K) and a linear pro®le of the bias error from y 0 to y 11. This temperature bias pro®le was then added to the temperature data used in the least-squares
Fig. 11. Near wall temperature pro®le obtained at Rex 1.1 ´ 105 . The data are plotted alongside lines with slopes of 10 and 20% in order to obtain an estimate of the random error contribution to the uncertainty in the estimated temperature gradient. The data point closest to the wall is at y 1.9, the third data point is at y 5.9, and the sixth data point is at y 9.9.
®t. This estimate indicates that calculated heat ¯ux could be biased from +5% to +10%. Combining the precision and bias limit estimates, we can obtain an estimate of the composite uncertainty in qw of ÿ5% to +25%, using the worst case bias and precision limits of +10% and 15%, respectively. This uncertainty can be compared to the values for conventional embedded heat ¯ux gauges of the thermopile type, which oer typical uncertainties ranging from 10% in low-speed ¯ows under steady laboratory conditions to 25% in harsh environments such as IC engines or industrial settings [27]. Improved heat ¯ux measurements are possible if a large sample of single-laser-shot CARS spectra are used for determination of the mean temperature at each point, so that any bias in the measurements is eliminated. 9. Practical signi®cance This paper describes an application of a shot-averaged DBPR-CARS technique for non-intrusive measurement of mean temperature in laboratory studies of convective heat transfer. The CARS technique oers the ability for non-intrusive, spatially resolved mean temperature measurements in complex thermal boundary layers. The spatial resolution of the method allows for calculation of the heat ¯ux from the near-wall temperature pro®le. The technique is applicable in boundary layers with overall temperature dierences of 100 K or more and provides a precision error of 4 K with a 1±4 K bias toward low temperatures in ¯ows with low-tomoderate rms temperature ¯uctuations. The potential for non-biased instantaneous temperature measurements in boundary layers exists with increased dye-laser oscillator cavity length and/or high pressure environments. 10. Conclusions Dual-broadband, pure-rotational CARS has been used to map temperature pro®les in a low-Reynoldsnumber turbulent boundary layer subjected to a favorable pressure gradient. A bypass transition to turbulence occurred very close to the leading edge due to a largeamplitude disturbance. When plotted in wall units, the mean temperature data exhibited the shape and trends with increasing pressure gradient that are characteristic of turbulent boundary layers. The estimated precision limit associated with the use of shot-averaged spectra for mean temperature measurement is 4 K (based on 2 standard deviations of a temperature pdf obtained in room air). A bias toward low temperature which can be as high as 4 K in regions of high rms temperature ¯uctuation to 1 K or less in regions of small rT is also present. With this level of uncertainty, the DBPR-CARS system proved useful for spatially resolved characterization of the mean temperature ®eld in boundary layers with wall-to-air temperature dierences in excess of 100 K.
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
The spatial resolution of the CARS measurements normal to the heat transfer surface was 50 lm, and mean temperatures were measured within 50 lm (25 lm) of the wall. This excellent resolution allowed for quantitative estimates of the local wall heat ¯ux with an estimated uncertainty of ÿ5% to +25%. This uncertainty can be reduced by acquiring a large sample of singleshot CARS spectra to determine the mean temperature at each point, in place of the shot-averaged spectra used in the present work. Use of this single-shot sampling procedure eliminates any density-weighting bias error. The precision of single-shot DBPR-CARS temperature measurements in a gas-cell improved signi®cantly with increasing pressure. This observation is consistent with previously published data [11,24]. Future work is needed in the area of instantaneous, single-laser-shot DBPR-CARS temperature measurements, as this procedure will allow for determination of the local mean temperature, rms temperature ¯uctuation, and wall heat ¯ux without bias due to density weighting eects. Singleshot data can also be used in conjunction with simultaneous velocity measurements to map the turbulent heat ¯ux distributions in complex ¯ows. These singleshot, boundary-layer measurements may be feasible with an elongated dye-laser cavity or at higher pressures.
Nomenclature B molecular rotational constant (cmÿ1 ) Cf local skin-friction coecient [2sw /qU2 ] (dimensionless) Cp constant-pressure speci®c heat (J/kg K) J rotational quantum number (dimensionless) Nux local Nusselt number [qw x/jDT] (dimensionless) Pr Prandtl number [lCp /j] (dimensionless) qw wall heat ¯ux (W/m2 ) Reynolds number [Ux/m] Rex St local Stanton number [St Nux /Rex Pr] (dimensionless) t duration of CARS signal pulse (ns) T temperature (K) u local mean velocity (m/s) U mean free-streamp velocity (m/s) u friction velocity ( sw =q) (m/s) v vibrational quantum number (dimensionless) W spectrometer slit function (cmÿ1 ) x streamwise distance from the leading edge of the test section (m) y coordinate normal to the heat transfer surface (m) Greek symbols l dynamic viscosity (N s/m2 ) j thermal conductivity (W/m K) m kinematic viscosity (m2 /s) k wavelength (nm) q density (kg/m3 )
r sw x XM C Subscripts 1 2 CARS S w 1
25
rms or standard deviation (K) wall shear stress (N/m2 ) wave-number frequency [x 1/k] (cmÿ1 ) mode spacing of the broadband dye laser (cmÿ1 ) linewidth (cmÿ1 or sÿ1 ) pump beam #1 pump beam #2 anti-stokes or CARS signal Stokes beam quantity evaluated at the ``wall'' or heat transfer surface quantity evaluated in the free stream
Note: All dimensionless groups are calculated using free-stream thermophysical properties of air. References [1] S.P. Kearney, R.E. Foglesong, R.P. Lucht, A.M. Jacobi, Temperature measurements in a free convection boundary layer using Coherent Anti-Stokes Raman Spectroscopy (CARS), Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics (ExHFT IV), Brussels, Belgium, Edizioni ETS, 1997, pp. 201±208. [2] M. Alden, P.E. Bengtsson, H. Edner, Rotational CARS generation through a multiple four-color interaction, Appl. Opt. 25 (1986) 4493±4499. [3] A.C. Eckbreth, T.J. Anderson, Dual broadband CARS for simultaneous multiple species measurements, Appl. Opt. 24 (1985) 2731±2736. [4] A.C. Eckbreth, T.J. Anderson, Simultaneous rotational coherent anti-Stokes Raman spectroscopy with arbitrary pump-Stokes spectral separation, Opt. Lett. 11 (1986) 496±498. [5] R.J. Goldstein, Optical measurement of temperature, in: E.R.G. Eckert, R.J. Goldstein (Eds.), Measurements in Heat Transfer, Hemisphere, Washington, DC, 1976, pp. 241±294. [6] R. Fehle, J. Klas, F. Mayinger, Investigation of local heat transfer in compact heat exchangers by holographic interferometry, Exp. Thermal Fluid Sci. 10 (1995) 181±191. [7] M.C. Thurber, F. Grisch, R.K. Hanson, Temperature imaging with single- and dual-wavelength acetone PLIF, Presented at the 32nd AIAA, ASME, SAE, ASEE Joint Propulsion Conference, Lake Buena Vista, FL, AIAA paper no. 96-2936, 1996. [8] J. Sakakibara, K. Hishida, M. Maeda, Measurements of thermally strati®ed pipe ¯ow using image-processing techniques, Exp. Fluids 16 (1993) 82±96. [9] S.A.J. Druet, J.P.E. Taran, CARS spectroscopy, Prog. Quant. Electr. 7 (1981) 1±72. [10] A.C. Eckbreth, Laser Diagnostics for Combustion, Temperature and Species, 1st ed., Abacus Press, Cambridge, MA, 1988. [11] M. Alden, P.E. Bengtsson, H. Edner, S. Kr oll, D. Nilsson, Rotational CARS: a comparison of dierent techniques with emphasis on accuracy in temperature determination, Appl. Opt. 28 (1989) 3206±3219. [12] R.E. Foglesong, S.M. Green, R.P. Lucht, J.C. Dutton, Dualpump coherent anti-Stokes Raman scattering for simultaneous pressure/temperature measurement, AIAA J. 36 (1998) 234±240. [13] J.D. Swearingen, R.F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of a wall, J. Fluid Mech. 182 (1987) 255±290.
26
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
[14] P.S. Klebano, Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient, NACA TN 3178, 1954. [15] R.L. Panton, Incompressible Flow, 2nd ed., Wiley-Interscience, New York, 1996. [16] R.J. Volino, T.W. Simon, Velocity and temperature pro®les in turbulent boundary layer ¯ows experiencing streamwise pressure gradients, J. Heat Transfer 119 (1997) 433±439. [17] D.G. Osborne, F.P. Incropera, Experimental study of mixed convection heat transfer for transitional and turbulent ¯ow between horizontal, parallel plates, Int. J. Heat Mass Transfer 28 (1985) 1337±1344. [18] R.E., Palmer, The CARSFT Computer Code for Calculating Coherent Anti-Stokes Raman Spectra: User and Programmer Information, Sandia National Laboratories Report no. SAND898206, 1989. [19] D.V. Murphy, R.K. Chang, Single-pulse broadband rotational coherent anti-Stokes Raman scattering thermometry of cold N2 gas, Opt. Lett. 6 (1981) 233±235. [20] W.C. Reynolds, W.M. Kays, S.J. Kline, Heat Transfer in the Turbulent Incompressible Boundary Layer, NASA Memo No. 12-1-58W, 1958.
[21] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, New York, 1993. [22] C.E. Feiler, Experimental Heat-Transfer and Boundary Layer Behavior with 100-CPS Oscillations, NASA TN No. 2531, 1964. [23] S. Qiu, T.W. Simon, R.J. Volino, Evaluation of local wall temperature, heat ¯ux, and convective heat transfer coecient from the near wall temperature pro®le, presented at the ASME IMECE Symposium on Heat Transfer in Turbulent Flows, ASME-HTD 318 (1995) 45±52. [24] P.E. Bengtsson, L. Martinsson, M. Alden, B. Johansson, B. Lassesson, K. Marforio, G. Lundholm, Dual-broadband rotational CARS measurements in an IC engine, Proceedings of the Twenty-Fifth Symposium (International) on Combustion, Irvine, CA, The Combustion Institute, 1994, pp. 1735±1742. [25] R.J. Moat, Describing the uncertainties in experimental results, Exp. Thermal Fluid Sci. 1 (1988) 3±17. [26] N. Kasagi, Y. Tomita, A. Kuroda, Direct numerical simulation of a passive scalar ®eld in a turbulent channel ¯ow, J. Heat Transfer 114 (1992) 598±606. [27] T.E. Diller, Personal Communication, 1998.
Temperature measurements in convective heat transfer ¯ows using dual-broadband, pure-rotational coherent anti-Stokes Raman spectroscopy (CARS) Sean P. Kearney *, Robert P. Lucht 1, Anthony M. Jacobi Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., MC-244, Urbana, IL 61801, USA Received 20 March 1998; received in revised form 15 November 1998; accepted 29 December 1998
Abstract This paper describes the use of dual-broadband, pure-rotational coherent anti-Strokes Raman spectroscopy (CARS) as a nonintrusive temperature diagnostic for convective-heat-transfer ¯ows. The characteristics of the dual-broadband, pure-rotational CARS technique are discussed, and the technique is compared to other temperature measurement methods. Dual-broadband, purerotational CARS was used to measure mean temperature pro®les in a low-Reynolds-number, turbulent boundary layer. The results are presented in wall units and compared to the thermal law of the wall for zero-pressure-gradient boundary layers. Temperature data were acquired as close as 50 lm (25 lm) to the wall, with a spatial resolution of 50 lm normal to the heat transfer surface, and a 2r precision limit of 4 K. The spatial resolution of this experimental method provides detailed information in complex thermal boundary layers and allows for an estimation of the convective heat ¯ux to within an estimated uncertainty of ÿ5% to +25%. Singlelaser-shot temperature data were acquired in a gas cell, and the potential for measurement of rms temperature ¯uctuations is discussed in terms of the resulting probability density functions. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Laser diagnostics; CARS; Non-intrusive temperature measurements; Boundary layers
1. Introduction Coherent anti-Stokes Raman spectroscopy (CARS) is a well-established technique for probing high-temperature, combustion ¯ows but has been seldom used in ``lower'' (<500 K) temperature convective-heat-transfer and ¯uid-mechanics applications. The ®rst such use of CARS was reported by Kearney et al. [1]. They used a dual-pump (three-laser) CARS technique to map the thermal boundary layer near a vertical cylinder in freeconvection. The present paper presents boundary layer temperature measurements in a forced ¯ow using a dualbroadband, pure-rotational (DBPR) CARS technique, independently suggested by Alden et al. [2] and Eckbreth and Anderson [3,4].
*
Corresponding author. Tel.: +1 217 244 0778; fax: +1 217 244 6534; e-mail: [email protected] 1 Present address: Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA.
CARS oers several advantages over other temperature measurement methods for probing convectiveheat-transfer ¯ows. Thermocouples, thermistors and RTD's, while often highly accurate and simple to use, are intrusive and thereby introduce uncertainties which can be dicult to quantify. CARS is a non-intrusive tool and is free from these types of ambiguities. Conventional optical methods such as Schlieren, shadowgraph photography and interferometry ([5,6]) are non-intrusive and oer the advantage over CARS of ``full-®eld'' views of the boundary layer. However, these more widely used optical methods provide path-averaged data and can be ineective in 3-D ¯ows. CARS oers pointwise measurements with a spatial resolution dictated by a cylindrical probe volume that is typically 50 lm in diameter and 1±2 mm long. Recently, laser-induced ¯uorescence (LIF), another optical technique which, like CARS, has been developed for combustion applications, has been employed in lower temperature ¯ows [7,8]. Temperature information is derived from LIF by exciting a molecule or atom to a higher energy state and detecting the ¯uorescence signal
0894-1777/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 9 ) 0 0 0 0 4 - 7
14
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
as the excited molecules or atoms decay to lower energy states via spontaneous emission. The ¯uorescence signal is proportional to the number density of the initially pumped state, which is a function of temperature. LIF oers the advantage of full-®eld planar imaging without the line-of-sight averaging inherent in the more conventional optical methods discussed above. However, LIF often requires that tracer species be seeded into the ¯ow while CARS does not. This paper begins with an overview of the CARS technique and a discussion of the advantages of DBPRCARS over other CARS implementations for temperature measurement. Following the overview of CARS, the dual-broadband CARS system and wind-tunnel apparatus are described, and the procedures used in conducting the experiments and data reduction are discussed. The results include CARS measurements of mean temperature pro®les, a comparison of the measured mean temperature pro®les to the thermal law of the wall, a determination of the local wall heat ¯ux from the mean temperature pro®les, and a discussion of measurement uncertainty. The potential for single-lasershot CARS measurements for determination of local rms temperature ¯uctuation is also discussed. 2. Description of the CARS technique As shown in Fig. 1(a), a CARS signal is generated by crossing three high-powered, pulsed laser beams, termed pump beam #1, pump beam #2, and the Stokes beam, in a suitable phase-matched geometry. The beam crossing region is at the common focus of the three laser beams and de®nes the CARS probe volume, a 50-lm diameter cylinder that is 1±2 mm in length. The frequency difference x1 ÿ xS is tuned to a Raman-allowed transition of the probed molecule, creating an oscillating polarization which scatters the incident radiation from pump beam #2 to create a CARS signal at frequency xCARS (x1 ÿ xS )+x2 . In practice, broadband laser sources are used to probe several resonances simultaneously, resulting in a broadband CARS signal beam whose spectral signature depends upon temperature, pressure and species concentration. More detailed descriptions of the physics of CARS signal generation and the dependence of CARS spectra on temperature, pressure, and species concentration can be found in the literature. Druet and Taran [9] and Eckbreth [10] present excellent general discussions, while Alden et al. [11], Foglesong et al. [12], and Kearney et al. [1] provide discussions of pure-rotational CARS. 3. Advantages of DBPR-CARS for thermometry The CARS technique presented here diers from usual the implementation in high-temperature environments in two ways. First, two broadband laser sources are used instead of just a single broadband source. Secondly, the Raman resonances probed are pure-rota-
Fig. 1. (a) Shows the arrangement of laser beams for dual-broadband CARS with planar BOXCARS phase matching. The pump #1 and Stokes beams, from the same broadband dye laser source are shown in dark gray, the pump #2 beam, from the spectrally narrow Nd:YAG laser, is shown in light gray, and the CARS signal is in black. A vector diagram for the planar BOXCARS phase matching is given in (b).
tional S-branch (Dv 0, DJ +2) transitions, as opposed to the vibrational Q-branch (Dv 1, DJ 0) resonances typically probed in combustion applications. In many CARS implementations, a single broadband Stokes laser source is typically used so that the frequency dierence x1 ÿ xS is tuned to the range of transitions desired, as shown in Fig. 2(a). When employing a single broadband source, shot-to-shot spectral variations in dye laser output introduce noise in the resulting CARS spectrum because each Raman transition is coupled with only a narrow frequency range of dye laser output. In our study, a dual-broadband technique was used where both the pump #1 and Stokes laser beams originate from the same broadband dye laser. With this dual-broadband method, shown in Fig. 2(b), each Raman resonance is coupled to a large number of frequency combinations in the broadband dye-laser pro®le, resulting in an eective averaging over the spectral noise in the dye-laser output. This spectral averaging minimizes the eect of dye laser spectral noise which is perhaps the largest cause of temperature measurement error. At temperature less than about 500 K, pure-rotational CARS is often preferred over its vibrational
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
15
Fig. 2. Schematic representation of pure-rotational CARS processes. Energy level diagrams and schematic renderings are given for: (a) the widely used broadband CARS technique and (b) dual-broadband CARS. Note the close spectral proximity of the pump #1 and Stokes lasers and the pump #2 and CARS beams characteristic of the small Raman shifts associated with pure-rotational CARS.
CARS counterpart. Both methods have comparable signal strengths at lower temperatures, but pure-rotational CARS spectra are more sensitive to small temperature changes than vibrational CARS spectra; furthermore, pure rotational CARS spectra exhibit wider spectral separation (4B) between neighboring rotational lines. Large spectral separation makes calculation of temperature by comparison to theoretically calculated spectra easier because it is simpler to calculate the theoretical spectra and to use the integrated-lineintensity technique discussed later in this paper. 4. Apparatus 4.1. Lasers and optics A schematic of the DBPR-CARS apparatus is shown in Fig. 3. Eighty percent of the 350 mJ, 10-ns pulse from the second harmonic of a Q-switched, 10 Hz repetition rate, Nd:YAG laser (k2 532 nm, x2 18,797 cmÿ1 ) is used to pump the broadband dye
laser. Using rhodamine 640 dye, the broadband laser output is centered near kS 607 nm (xS 16,474 cmÿ1 ) with a nominal bandwidth of DxS 130 cmÿ1 (FWHM). Ten percent of the Nd:YAG energy is directed to a beam trap and the residual 35 mJ is used to provide the narrowband pump beam #2 in the CARS process. The 20 mJ/pulse output of the broadband dye laser is directed to a 50/50 beamsplitter to provide both the pump #1 and Stokes beams. This beam arrangement results in nominal laser pulse energies of 10, 35, and 10 mJ for the pump #1, pump #2, and Stokes laser beams, respectively. Several mirrors and prisms are used to arrange the three laser beams into the planar BOXCARS con®guration, and a 250-mm spherical focusing lens provides a cylindrical probe-volume that is 50 lm in diameter and 1±2 mm long. The three input laser beams and the CARS signal are recollimated by another 250-mm lens and the Stokes and pump #2 beams are directed into beam traps by prisms. The CARS signal is generated virtually collinear to pump beam #1 and is relayed by a series of eight 532-nm mirrors to a 100-mm spherical
16
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
Fig. 3. Schematic of the dual-broadband CARS system used to measure boundary layer temperature pro®les.
lens, which focuses the signal at the entrance slit of a 1m spectrometer. The eight mirrors are used to attenuate the strength of the remaining pump #1 beam in order to prevent contamination of the signal. This attenuation is possible because the 532-nm mirrors transmit most of the 607-nm light from the pump laser while re¯ecting the majority of the CARS signal, which is centered around 529 nm. The spectrometer diraction grating successfully separates the remainder of the pump beam radiation from the CARS beam and the signal is imaged onto a 14-bit, 512 ´ 512, back-illuminated, unintensi®ed CCD camera through a relay lens pair. 4.2. Wind tunnel A schematic of the low-speed wind-tunnel used for the channel ¯ow experiments is shown in Fig. 4. Air is supplied by a 62-W blower and ¯ows through a ¯exible, 203-mm diameter tube to the 305-mm long settling chamber of 210 ´ 305 mm cross-section. The ¯ow is conditioned by honeycomb ¯ow straightener (75 mm length, 7.5 mm cell dimension) and three ®ne mesh screens. A 4.6 to 1 rectangular nozzle, 450 mm in length, accelerates the ¯ow into the heated test section. The test section is a semi-rectangular channel, consisting of two heated metal plates at the top and bottom, and side windows fabricated from 6.35-mm-thick plate glass. The channel is 146 mm wide and 286 mm long.
The top plate is ¯at and constructed from 10.9-mmthick, 2024-aluminum alloy. The bottom plate, fabricated from 304 stainless steel, is curved so that the CARS laser beams are not clipped as they enter the test section. The channel height is 30 mm at the center-line, where the measurements are made, and 35 mm at the plate glass windows. To ensure that curvature eects are not important, care was taken to guarantee that the local radius of curvature of the steel plate was at least an order of magnitude greater than the half-height of the channel, which represents the maximum attainable boundary layer thickness. A scheme similar to the one used by Swearingen and Blackwelder [13] is used to provide negligible hydrodynamic boundary-layer thickness at the entrance to the test section. Here, the ®rst 25.4 mm of the channel is located inside of the nozzle and a concentric suction slot of 10-mm minimum clearance is provided for the nozzle boundary layer ¯ow to exit into a low-pressure aluminum chamber. Variable-speed axial fans can be used to provide suction from the aluminum chamber to provide a laminar channel ¯ow with a well-de®ned boundary layer origin. In the experiments reported here, the axial fans were not used. With no suction provided, the pressure in the aluminum plenum was not suciently low to remove recirculating ¯uid from the clearance slots. The recirculating ¯uid fed a large-amplitude disturbance into the boundary layer, providing a boundary
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
17
Fig. 4. Wind-tunnel used for CARS boundary layer temperature measurements.
layer ``trip'' at the test section entrance and turbulent boundary layers in the channel ¯ow. The wind-tunnel ¯ow was characterized using hotwire measurements taken in the outer regions (u/ U P 0.6) of an isothermal boundary layer at Reynolds numbers of 4.6 ´ 104 and 8.3 ´ 104 . Mean velocity, streamwise turbulence intensity, and power spectra data are provided in Fig. 5. When plotted in wall units, the mean velocity data exhibit a logarithmic pro®le that is characteristic of turbulent boundary layers. In addition, the turbulence intensity pro®les were found to be similar in shape and magnitude to the results of Klebano [14], as were power spectra obtained at three selected locations in the boundary layer. These hot-wire results and the mean temperature pro®les to be presented indicate that the boundary layer ¯ow was similar in structure to a fully turbulent boundary layer in spite of the low Reynolds number ¯ows, 104 < Rex < 105 , studied. This early transition is a result of the above-mentioned, largeamplitude disturbance present at the leading edge of the channel. A ``bypass'' [15,16] transition of this sort results when a disturbance of sucient magnitude causes the normal instability and transition process to be replaced by a more abrupt transition. 5. Procedure 5.1. Experiment At the beginning of each experiment, the wind-tunnel, plate heaters, and data acquisition system were allowed to warm up for about 1 h so that the temperature of the metal plates forming the top and bottom channel surfaces reached a steady state in a range of 100±150 K above ambient. The criterion suggested by Osborne and Incropera [17] was applied to verify that the free-con-
vection contribution to the heat ¯ux was negligible under the conditions of the experiments to be reported. The CARS probe volume was accurately positioned by keeping the probe volume stationary and moving the wind tunnel, which was ®xed to a three-axis translation stage. The location of the probe volume with respect to the curved plate was determined by viewing the re¯ection of a reduced-power laser beam from the Nd:YAG laser on a white card and moving the wind-tunnel vertically until the image was clipped by the curved plate. This procedure allowed for positioning of the CARS probe volume to within about 50 lm (25 lm), or one probe volume diameter, with respect to the bottom wall. Accurate streamwise positioning of the probe volume was achieved by aligning the probe volume with eroding thermocouple probes mounted ¯ush with the surface at predetermined reference locations, and then moving the wind-tunnel to the desired streamwise location. This procedure allowed for streamwise positioning of the probe volume within an uncertainty of about 1 mm. The Reynolds number, based on streamwise distance and centerline velocity, was obtained from hot-wire measurements of the centerline velocity and by evaluating the kinematic viscosity at the local free-stream temperature. The uncertainty in Reynolds number was conservatively estimated at 5%. Temperature pro®les were obtained from CARS spectral signatures which were averaged for 100 laser shots by recording 10 s CCD camera exposures. Use of shot-averaged CARS spectra was desirable in order to decrease the noise level in the spectra and to increase the data acquisition rate. This averaging procedure yields the mean temperature provided that the local rms temperature ¯uctuation is suciently small. In ¯ows with high temperature ¯uctuations, use of a shot-averaged CARS spectrum can yield results which are biased toward low temperatures because the CARS signal
18
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
where the temperature is known, to determine the shape of the broadband dye laser pro®le. Because the broadband dye laser has a ®nite bandwidth, transitions of small Raman shifts are pumped more strongly than those of large Raman shifts. The error introduced by this unequal distribution of dye-laser energy is minimized by normalizing the CARS spectra by an appropriate correction. For DBPR-CARS, the appropriate normalization is the convolution of the dye-laser pro®le with itself. The shape of the dye-laser convolution is determined by dividing the peak signal intensities of the experimentally obtained, room temperature reference spectra by the peak intensities obtained from the theoretical prediction given by the Sandia CARSFT code [18], which does not account for the structure of the dye laser pro®le. A smooth curve is ®t through the correction factors obtained by this process and the resulting curve ®t is used to normalize all experimentally obtained spectra. 6. CARS data reduction ± integrated line intensity technique
Fig. 5. Hot-wire results for characterization of the boundary layer ¯ow. Mean velocity pro®les in the outer region (u/U P 0.6) of the curved plate boundary layer are given in (a). Streamwise turbulence intensity and one-dimensional power spectra are given in (b) and (c), respectively.
strength decays with increasing temperature. For the experiments reported here, the levels of temperature ¯uctuation were suciently low for use of the shot-averaging procedure. A discussion of estimated mean temperature bias is provided within the section on measurement uncertainty in Section 6 of this paper. 5.2. Calibration The CARS system was calibrated using reference spectra obtained outside the thermal boundary layer,
Temperature, pressure, or other quantities of interest are most often deduced by comparing experimentally obtained CARS spectra to theoretically calculated spectra. This comparison is typically performed by ®nding the calculated spectrum which best ®ts the data in the least-squares sense (spectrum ®tting method), or by comparing the experimental and theoretical integrated line intensity distributions (integrated line intensity method). In this work, temperatures are determined from CARS spectra by using an integrated line intensity technique similar to the ones outlined in Refs. [11,12,19]. The application of this integrated intensity technique is facilitated due to the widely spaced transitions associated with pure-rotational spectra and is preferred because it is much faster than the spectral ®tting method [12]. The Sandia CARSFT code [18] is used to generate theoretically calculated spectra in 1 K increments from 220 to 550 K. At each temperature, normalized integrated intensity ratios are calculated from the theoretical spectra by summing the intensity under each of the isolated N2 peaks, highlighted in the spectra provided in Fig. 6, and dividing the results for each transition by the maximum integrated value. This process yields a library of theoretical, normalized integrated-line-intensity distributions for each temperature. Experimental DBPRCARS spectra are reduced by computing the same normalized integrated line intensities from the experimentally obtained spectra and ®nding the library entry which best ®ts the experimental data in the least-squares sense. This least-squares procedure minimizes the quantity K
N exp X S k1
k exp Smax
S th T ÿ thk Smax T
2 ;
1
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
19
Fig. 6. Experimentally obtained, 100-laser-shot-averaged DBPR-CARS spectra at evaluated temperatures of 297 and 452 K. Rotational S-branch transitions are marked in the legend at the top of each plot, with O2 transitions being indicated by asterisks. The lines used in the integrated line intensity procedure for determining temperature are highlighted in bold. Integrated intensity distributions are given below each spectrum. Note the distinct intensity shift toward higher Raman shifts (and higher rotational energies) with increasing temperature.
where N is the total number of rotational lines used in the data reduction and Skexp and Skth are the integrated rotational line intensities of the kth rotational transition in the experimental and theoretical spectra respectively. exp th The normalizing factors, Smax and Smax , are the integrated line intensity values for the transition with the maximum integrated intensity in the experimental and theoretical spectra. By normalizing by the maximum integrated intensity value, the procedure is sensitive only to the shape of the spectrum and not the signal strength, which can vary substantially with temperature and laser beam intensity. Fig. 6 contains the results of the integrated-line-intensity procedure for evaluated temperatures of 297 and 452 K. There is a noticeable shift in the distribution of peak N2 line intensities toward Raman shifts of higher rotational quantum number with increasing temperature. In this manner the CARS spectra exhibit the expected rotational population distribution. 7. Results 7.1. Mean temperature Mean boundary layer temperature distributions for three selected Reynolds numbers are presented in di-
mensional form in Fig. 7. The results show excellent spatial resolution, with temperature data acquired in increments as small as 50 lm normal to the wall. At Rex 3.1 ´ 104 , a thermocouple mounted ¯ush with the heat transfer surface indicates a temperature of 453 K ± consistent with the value of 450 K obtained by extrapolating the DBPR-CARS temperature pro®le to the wall. The results are plotted in dimensionless form in Fig. 8 against the mean temperature pro®les for turbulent, zero-pressure-gradient boundary layers suggested by Reynolds et al. [20] and Kays and Crawford [21]. Here, the dimensionless temperature and distance from the wall are given by T and
Tw ÿ T qCp u ; qw
2
yu : 3 m To compute the dimensionless forms of the data, it is necessary to know both the heat ¯ux and shear stress at the wall. As explained in the next section, the wall heat ¯ux is obtained from a linear ®t to the temperature data in the near-wall region. The heat ¯ux is used to estimate the wall shear from a modi®ed form of the Colburn analogy as suggested by Reynolds et al. [20]. y
20
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
The qualitative agreement is especially good in the conduction sublayer and buer region where free-stream eects should not be signi®cant. In the logarithmic regime, the data deviate more signi®cantly from the zeropressure-gradient pro®les, with the data falling into two categories depending upon the free-stream speed. This deviation from the zero-pressure-gradient curves is to be expected since the temperature ``law of the wall'' is not a truly universal pro®le as is suspected for the case of a velocity boundary layer. The free-stream pressure gradient signi®cantly in¯uences the thermal boundary layer shape in the logarithmic region, with pro®les in favorable pressure gradients exhibiting the usual boundary layer thinning indicated by an upward shift of the T ±y curve ± Refs. [16,21]. Therefore, the upward shift with free-stream speed should be expected as the pressure gradient must increase with the mass ¯ow through the channel. Finally, we must note that the uncertainty in the values of T presented in the logarithmic layer in Fig. 8 is largely dominated by the estimates of u and qw . The heat ¯ux is determined directly from the spatially resolved boundary layer temperature pro®les as described below. The random error contribution to the uncertainty in qw is about 10±15% while the uncertainty in u is likely to be in¯uenced by an ambiguous systematic error associated with the use of Eq. (4) which is valid for 105
Fig. 7. DBPR-CARS temperature data recorded in a low-Reynoldsnumber turbulent boundary layer at 3 representative Reynolds numbers. The estimated uncertainty due to random errors in these 100laser-shot-averaged data is 4 K. Data were captured as close as 50 lm from the wall with a spatial resolution of 50 lm in the wall-normal direction. This excellent spatial resolution allows for a quantitative estimate of the heat ¯ux obtained from the linear ®ts to the data in the near-wall region shown in the plots.
Cf : 4 2 In dimensionless form, the data exhibit the shape of typical turbulent boundary layers suggested by the law of the wall temperature pro®les. This agreement further supports the observation of a bypass transition to boundary layer turbulence at Reynolds numbers well below the usual critical value, and is similar to the results of Feiler [22], who used thermocouples to observe temperature pro®les of comparable shape at Reynolds numbers as low as 2.8 ´ 104 . St Pr0:4
Following the procedure suggested by Qiu et al. [23], an estimate of the heat ¯ux can be obtained from the slope of a least-squares line ®t to the temperature data in the conduction sublayer. Calculation of heat ¯ux from the wall temperature pro®le is useful because the convective heat ¯ux is measured directly without correction for radiative heat transfer. The set of points used for least-squares ®tting were determined using two criteria, y 6 8 and y 6 11. These two criteria provided a sucient number of data for least-squares ®tting while remaining within the upper sublayer limit of y 13.2 quoted in Ref. [21]. The wall heat ¯ux was calculated by multiplying the least-squares slope by the thermal conductivity of air evaluated at the wall temperature extrapolated from the least-squares ®t to the temperature data. Use of this extrapolation procedure instead of an independent measurement of the wall temperature (by a thermocouple or RTD) eliminates any systematic error associated with recording Tw and T1 from two instruments with dierences in their calibrations [23]. The heat ¯ux data obtained using both the y 6 8 and y 6 11 criteria are given in Table 1 along with the number of data
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
21
Fig. 8. Mean temperature data plotted in dimensionless wall units. The data are compared to the predictions of Reynolds [20] and Kays and Crawford [21].
points that were used in each least-squares ®t, the extrapolated wall temperature, and local Nusselt number. At each value of Rex , the heat ¯ux estimates obtained from the y 6 8 and y 6 11 criteria are within 10%, an estimate of the scatter, or precision limit, in the heat ¯ux measurement that is consistent with the 10% uncertainty quoted in Ref. [23]. Based on this agreement, the y 6 11 criterion was chosen for calculation of the Nusselt number, as this provides more data points for estimation of the slope. For comparison, predicted Nusselt numbers are tabulated by extrapolating the correlation of Reynolds et al. [20], 0:6 Nux 0:0296 Re0:8 x Pr
Tw T1
ÿ0:4 5
all properties evaluated at T1 ;
below its lower limit of applicability of Rex 105 . The heat ¯ux data are in fair agreement with the correlation, with Eq. (5) underpredicting the data by 7±43%. 7.3. Potential for single-laser-shot measurements Single-laser-shot CARS data oer the possibility for measurement of rms temperature ¯uctuation in complex ¯ows, as well as non-intrusive point estimates of turbulent heat ¯ux if LDV data are also available. Sets of 500 single-shot DBPR-CARS spectra have been acquired from N2 gas in a room temperature cell at four dierent pressures. Pure nitrogen was used instead of air in these measurements to obtain simpler CARS spectra without overlapping N2 ±O2 transitions. Four 500-realization temperature pdfs, each obtained at a dierent gas cell pressure, are displayed in Fig. 9.
Table 1 Results of heat ¯ux estimates obtained from least-squares ®ts to the linear, near-wall temperature pro®le. Two criteria are presented for choosing the data set used in the least-squares ®t, y 6 8 and y 6 11. The tabulated values include the number of points used in the least-squares ®t, the wall temperature obtained by extrapolating the least-squares line to the wall, the heat ¯ux, Nusselt number data, and predicted Nusselt numbers from the correlation suggested in Ref. [20] Rex
N (y 6 8)
N (y 6 11)
Tw (K) (y 6 8)
Tw (K) (y 6 11)
qw (kW/m2 ) (y 6 8)
qw (kW/m2 ) (y 6 11)
Nux (y 6 11)
Nux Eq. (5)
3.1 ´ 104 7.8 ´ 104 1.1 ´ 105
12 4 4
16 6 6
449.6 393.6 414.9
450.3 393.4 413.2
2.72 4.22 5.48
2.80 4.17 4.96
113 187 263
79.1 175 226
22
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
Fig. 9. Temperature histograms obtained from 500 single-shot DBPR-CARS temperature realizations from room-temperature N2 in a gas cell at various pressures. The results show that the precision of the measurement improves with increasing pressure.
At 1 atm, the standard deviation is 18.3 K, yielding a 2r uncertainty of 36 K due to precision error only. Improved precision is obtained with an increase in gascell pressure. This improvement is due to a decreased noise-to-signal ratio (NSR) in the DBPR-CARS signal at higher pressures. This reduced noise eect is readily seen upon examination of the following equation for the NSR due to Bengtsson et al. [24]. r p ÿ 1=2 NSR 2p=tC2 1 C22 =W 2 XM =2pCR : 6 In Eq. (6), t is the duration of the CARS signal pulse, C2 and CR are the Nd:YAG laser and Raman linewidths, respectively, W is the spectrometer slit width, and XM is the mode spacing of the broadband dye laser. As the pressure increases the second term becomes smaller because CR increases due to pressure broadening eects. The single-shot, DBPR-CARS temperature precision results from this study and from Ref. [11,24] are summarized in Table 2. The results of the present study exhibit the expected trend of decreased uncertainty with increased pressure, as do the results of Refs. [11,24] (obtained by the same group of researchers), where the atmospheric results of Ref. [11] indicate less precision than the data presented in Ref. [24], obtained at pressures of 2.6 atm and higher. Improved single-shot precision at atmospheric pressure is achievable by decreasing the NSR with an increase in the dye-laser oscillator cavity length. An elongated cavity results in a
reduced mode spacing, XM , which will decrease the second term in Eq. (6). 8. Measurement uncertainty 8.1. Uncertainty in shot-averaged CARS temperature measurements The uncertainty in the DBPR-CARS mean temperature data can be estimated in terms of precision (random) and bias (systematic) contributions as described by Moat [25]. The precision error contribution is largely due to dye-laser mode-phase ¯uctuations [11] and CCD camera read noise, while the bias contribution is a result of density weighting eects present in regions of high rms temperature ¯uctuation. Estimates of both contributions to the shot-averaged temperature uncertainty are given here. An estimate of the precision limit in the mean temperature evaluated from 100-laser-shot-averaged DBPR-CARS spectra was made from 237 temperature realizations in room air. A histogram (pdf) constructed from these room-air data is shown in Fig. 10. The mean of these measurements is 294 K, and the standard deviation about 2 K. Using these results we can obtain a 2r (95% con®dence interval) estimate of 4 K for the precision limit contribution to the uncertainty. Since a controlled temperature source was not available for further testing, the pdf at 294 K was the only data set
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
23
Table 2 Comparison of single-shot temperature results to the results of previous work Investigator
Gas
Pressure (atm)
T (K)
TRMS (K)
TRMS =T(%)
Laser dye
Present study
N2
Air
Alden et al. [11]
N2
293 293 294 297 426 490 585 706 294
18.3 13.4 11.9 11.0 4.6 4.9 5.9 7.7 13.8 12.3
6.2 4.5 4.0 3.7 1.1 1.0 1.0 1.1 4.7 4.2
Rhodamine 640
Bengtsson et al. [24]
1.0 3.1 4.5 5.6 2.6 4.1 7.8 16.1 1.0
taken. However, from the scatter of the data in the mean temperature pro®les (see Fig. 7), it seems reasonable to assume that 4 K is a good estimate of the random error over the temperature range of 290±450 K observed in the experiments. With this uncertainty estimate, we required a wall-to-air temperature dierence of 100±150 K so that the precision error was 2±4% of the total temperature dierence. Use of shot-averaged CARS spectra can also cause a bias error toward low temperatures due to densityweighting eects. This eect is signi®cant in regions of high rms temperature ¯uctuation because the CARS signal strength decays with increasing temperature so that contributions to the shot-averaged spectrum from lower temperature gas are weighted more heavily than those from higher temperatures. To estimate the magnitude of this density weighting bias error, several ¯uctuating temperature ®elds with Gaussian probability were numerically generated, and the Sandia CARSFT code [18] was used to compute theoretical CARS spectra at each temperature in these pdfs. Representative, ``shot-averaged'' CARS spectra
DCM
DCM Rhodamine 610
were computed by summing the intensity data from the numerically generated spectra, and the mean temperature was calculated from these averaged spectra using the integrated-line-intensity method. The results for mean temperatures of 300, 350, 400, and 450 K and rms temperature ¯uctuations of 10, 20, 30, and 50 K are given in Table 3. At each value of T, the systematic error, or bias, in mean temperature becomes larger with increasing rms temperature ¯uctuation, so that the maximum bias toward low temperature occurs in the buer layer and in the outer regions of the sublayer where the temperature ¯uctuations are highest [26]. Assuming a 20 K maximum rms temperature ¯uctuation, consistent with the results reported in Ref. [26], the maximum bias error incurred is 4 K toward cooler temperatures ± a value comparable to the precision error contribution. The bias error is reduced to 1 K or less in other regions of the boundary layer where the ¯uctuations are smaller. This 1±4 K bias toward lower temperatures can be combined with the 4 K precision error to yield a composite uncertainty in the shot-averaged mean temperature that varies from (+3,ÿ5) K to (+0,ÿ8) K depending upon the location in the boundary layer. The bias error can be eliminated by abandoning the more-ecient shot-averaging procedure used in this study, and recording a large sample of single-laser-shot CARS spectra and averaging the evaluated temperatures.
Table 3 Results of CARS density weighting study performed by generating a Gaussian pdf with mean temperature T and rms ¯uctuation rT . Mean temperatures calculated from the averaged CARS spectra are presented for several T, rT combinations
Fig. 10. Histogram (pdf) obtained from 237 DBPR-CARS temperature realizations in room air. The results from these 100-laser-shotaveraged spectra and the scatter in the boundary layer temperature pro®les indicate that 4 K is a reasonable estimate of the 2r precision limit in the shot-averaged mean temperature measurements.
T K
rT 10 K
rT 20 K
rT 30 K
rT 50 K
300 350 400 450
299 349 400 449
295 346 396 446
290 343 391 443
272 326 378 428
24
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
8.2. Uncertainty in wall heat ¯ux The uncertainty in the heat ¯ux and Nusselt number data presented in Table 1 can be estimated in terms of precision and bias contributions. The uncertainty in the heat ¯ux measurement is dominated by temperature gradient uncertainty, because errors in jw due to uncertainty in the extrapolated wall temperature are comparatively small. Similarly, it was found that the Nusselt number uncertainty is dominated by the uncertainty in the heat ¯ux measurement, with measurement errors in x, j1 , and (Tw ÿ T1 ) making a minor contribution (<1%). Therefore, the uncertainties in both heat ¯ux and local Nusselt number are essentially equal and are determined by the temperature gradient uncertainty. For each temperature pro®le presented in this paper, the precision limit (random error) in the wall temperature gradient was estimated by plotting the data used in the least-squares ®t alongside the best-®t line and lines with slopes 10 and 20% of the best-®t slope. A representative plot is shown in Fig. 11. For all three data sets, the 10% lines are a much more reasonable estimate of the possible error in slope due to the 4 K scatter in the temperature data. With this in mind, we feel that 10±15% is a conservative estimate of the precision limit the heat ¯ux measurement. Mean temperature bias due to shot averaging introduces a bias error toward higher wall heat ¯uxes. This temperature bias increases with rms temperature ¯uctuation. Published data and DNS calculations [26] show that rT monotonically increases with distance from the wall, peaking near y 18. The eect of this temperature bias on the evaluated heat ¯ux was estimated by assuming a maximum temperature bias of 4 K (rT 20 K) and a linear pro®le of the bias error from y 0 to y 11. This temperature bias pro®le was then added to the temperature data used in the least-squares
Fig. 11. Near wall temperature pro®le obtained at Rex 1.1 ´ 105 . The data are plotted alongside lines with slopes of 10 and 20% in order to obtain an estimate of the random error contribution to the uncertainty in the estimated temperature gradient. The data point closest to the wall is at y 1.9, the third data point is at y 5.9, and the sixth data point is at y 9.9.
®t. This estimate indicates that calculated heat ¯ux could be biased from +5% to +10%. Combining the precision and bias limit estimates, we can obtain an estimate of the composite uncertainty in qw of ÿ5% to +25%, using the worst case bias and precision limits of +10% and 15%, respectively. This uncertainty can be compared to the values for conventional embedded heat ¯ux gauges of the thermopile type, which oer typical uncertainties ranging from 10% in low-speed ¯ows under steady laboratory conditions to 25% in harsh environments such as IC engines or industrial settings [27]. Improved heat ¯ux measurements are possible if a large sample of single-laser-shot CARS spectra are used for determination of the mean temperature at each point, so that any bias in the measurements is eliminated. 9. Practical signi®cance This paper describes an application of a shot-averaged DBPR-CARS technique for non-intrusive measurement of mean temperature in laboratory studies of convective heat transfer. The CARS technique oers the ability for non-intrusive, spatially resolved mean temperature measurements in complex thermal boundary layers. The spatial resolution of the method allows for calculation of the heat ¯ux from the near-wall temperature pro®le. The technique is applicable in boundary layers with overall temperature dierences of 100 K or more and provides a precision error of 4 K with a 1±4 K bias toward low temperatures in ¯ows with low-tomoderate rms temperature ¯uctuations. The potential for non-biased instantaneous temperature measurements in boundary layers exists with increased dye-laser oscillator cavity length and/or high pressure environments. 10. Conclusions Dual-broadband, pure-rotational CARS has been used to map temperature pro®les in a low-Reynoldsnumber turbulent boundary layer subjected to a favorable pressure gradient. A bypass transition to turbulence occurred very close to the leading edge due to a largeamplitude disturbance. When plotted in wall units, the mean temperature data exhibited the shape and trends with increasing pressure gradient that are characteristic of turbulent boundary layers. The estimated precision limit associated with the use of shot-averaged spectra for mean temperature measurement is 4 K (based on 2 standard deviations of a temperature pdf obtained in room air). A bias toward low temperature which can be as high as 4 K in regions of high rms temperature ¯uctuation to 1 K or less in regions of small rT is also present. With this level of uncertainty, the DBPR-CARS system proved useful for spatially resolved characterization of the mean temperature ®eld in boundary layers with wall-to-air temperature dierences in excess of 100 K.
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
The spatial resolution of the CARS measurements normal to the heat transfer surface was 50 lm, and mean temperatures were measured within 50 lm (25 lm) of the wall. This excellent resolution allowed for quantitative estimates of the local wall heat ¯ux with an estimated uncertainty of ÿ5% to +25%. This uncertainty can be reduced by acquiring a large sample of singleshot CARS spectra to determine the mean temperature at each point, in place of the shot-averaged spectra used in the present work. Use of this single-shot sampling procedure eliminates any density-weighting bias error. The precision of single-shot DBPR-CARS temperature measurements in a gas-cell improved signi®cantly with increasing pressure. This observation is consistent with previously published data [11,24]. Future work is needed in the area of instantaneous, single-laser-shot DBPR-CARS temperature measurements, as this procedure will allow for determination of the local mean temperature, rms temperature ¯uctuation, and wall heat ¯ux without bias due to density weighting eects. Singleshot data can also be used in conjunction with simultaneous velocity measurements to map the turbulent heat ¯ux distributions in complex ¯ows. These singleshot, boundary-layer measurements may be feasible with an elongated dye-laser cavity or at higher pressures.
Nomenclature B molecular rotational constant (cmÿ1 ) Cf local skin-friction coecient [2sw /qU2 ] (dimensionless) Cp constant-pressure speci®c heat (J/kg K) J rotational quantum number (dimensionless) Nux local Nusselt number [qw x/jDT] (dimensionless) Pr Prandtl number [lCp /j] (dimensionless) qw wall heat ¯ux (W/m2 ) Reynolds number [Ux/m] Rex St local Stanton number [St Nux /Rex Pr] (dimensionless) t duration of CARS signal pulse (ns) T temperature (K) u local mean velocity (m/s) U mean free-streamp velocity (m/s) u friction velocity ( sw =q) (m/s) v vibrational quantum number (dimensionless) W spectrometer slit function (cmÿ1 ) x streamwise distance from the leading edge of the test section (m) y coordinate normal to the heat transfer surface (m) Greek symbols l dynamic viscosity (N s/m2 ) j thermal conductivity (W/m K) m kinematic viscosity (m2 /s) k wavelength (nm) q density (kg/m3 )
r sw x XM C Subscripts 1 2 CARS S w 1
25
rms or standard deviation (K) wall shear stress (N/m2 ) wave-number frequency [x 1/k] (cmÿ1 ) mode spacing of the broadband dye laser (cmÿ1 ) linewidth (cmÿ1 or sÿ1 ) pump beam #1 pump beam #2 anti-stokes or CARS signal Stokes beam quantity evaluated at the ``wall'' or heat transfer surface quantity evaluated in the free stream
Note: All dimensionless groups are calculated using free-stream thermophysical properties of air. References [1] S.P. Kearney, R.E. Foglesong, R.P. Lucht, A.M. Jacobi, Temperature measurements in a free convection boundary layer using Coherent Anti-Stokes Raman Spectroscopy (CARS), Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics (ExHFT IV), Brussels, Belgium, Edizioni ETS, 1997, pp. 201±208. [2] M. Alden, P.E. Bengtsson, H. Edner, Rotational CARS generation through a multiple four-color interaction, Appl. Opt. 25 (1986) 4493±4499. [3] A.C. Eckbreth, T.J. Anderson, Dual broadband CARS for simultaneous multiple species measurements, Appl. Opt. 24 (1985) 2731±2736. [4] A.C. Eckbreth, T.J. Anderson, Simultaneous rotational coherent anti-Stokes Raman spectroscopy with arbitrary pump-Stokes spectral separation, Opt. Lett. 11 (1986) 496±498. [5] R.J. Goldstein, Optical measurement of temperature, in: E.R.G. Eckert, R.J. Goldstein (Eds.), Measurements in Heat Transfer, Hemisphere, Washington, DC, 1976, pp. 241±294. [6] R. Fehle, J. Klas, F. Mayinger, Investigation of local heat transfer in compact heat exchangers by holographic interferometry, Exp. Thermal Fluid Sci. 10 (1995) 181±191. [7] M.C. Thurber, F. Grisch, R.K. Hanson, Temperature imaging with single- and dual-wavelength acetone PLIF, Presented at the 32nd AIAA, ASME, SAE, ASEE Joint Propulsion Conference, Lake Buena Vista, FL, AIAA paper no. 96-2936, 1996. [8] J. Sakakibara, K. Hishida, M. Maeda, Measurements of thermally strati®ed pipe ¯ow using image-processing techniques, Exp. Fluids 16 (1993) 82±96. [9] S.A.J. Druet, J.P.E. Taran, CARS spectroscopy, Prog. Quant. Electr. 7 (1981) 1±72. [10] A.C. Eckbreth, Laser Diagnostics for Combustion, Temperature and Species, 1st ed., Abacus Press, Cambridge, MA, 1988. [11] M. Alden, P.E. Bengtsson, H. Edner, S. Kr oll, D. Nilsson, Rotational CARS: a comparison of dierent techniques with emphasis on accuracy in temperature determination, Appl. Opt. 28 (1989) 3206±3219. [12] R.E. Foglesong, S.M. Green, R.P. Lucht, J.C. Dutton, Dualpump coherent anti-Stokes Raman scattering for simultaneous pressure/temperature measurement, AIAA J. 36 (1998) 234±240. [13] J.D. Swearingen, R.F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of a wall, J. Fluid Mech. 182 (1987) 255±290.
26
S.P. Kearney et al. / Experimental Thermal and Fluid Science 19 (1999) 13±26
[14] P.S. Klebano, Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient, NACA TN 3178, 1954. [15] R.L. Panton, Incompressible Flow, 2nd ed., Wiley-Interscience, New York, 1996. [16] R.J. Volino, T.W. Simon, Velocity and temperature pro®les in turbulent boundary layer ¯ows experiencing streamwise pressure gradients, J. Heat Transfer 119 (1997) 433±439. [17] D.G. Osborne, F.P. Incropera, Experimental study of mixed convection heat transfer for transitional and turbulent ¯ow between horizontal, parallel plates, Int. J. Heat Mass Transfer 28 (1985) 1337±1344. [18] R.E., Palmer, The CARSFT Computer Code for Calculating Coherent Anti-Stokes Raman Spectra: User and Programmer Information, Sandia National Laboratories Report no. SAND898206, 1989. [19] D.V. Murphy, R.K. Chang, Single-pulse broadband rotational coherent anti-Stokes Raman scattering thermometry of cold N2 gas, Opt. Lett. 6 (1981) 233±235. [20] W.C. Reynolds, W.M. Kays, S.J. Kline, Heat Transfer in the Turbulent Incompressible Boundary Layer, NASA Memo No. 12-1-58W, 1958.
[21] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, New York, 1993. [22] C.E. Feiler, Experimental Heat-Transfer and Boundary Layer Behavior with 100-CPS Oscillations, NASA TN No. 2531, 1964. [23] S. Qiu, T.W. Simon, R.J. Volino, Evaluation of local wall temperature, heat ¯ux, and convective heat transfer coecient from the near wall temperature pro®le, presented at the ASME IMECE Symposium on Heat Transfer in Turbulent Flows, ASME-HTD 318 (1995) 45±52. [24] P.E. Bengtsson, L. Martinsson, M. Alden, B. Johansson, B. Lassesson, K. Marforio, G. Lundholm, Dual-broadband rotational CARS measurements in an IC engine, Proceedings of the Twenty-Fifth Symposium (International) on Combustion, Irvine, CA, The Combustion Institute, 1994, pp. 1735±1742. [25] R.J. Moat, Describing the uncertainties in experimental results, Exp. Thermal Fluid Sci. 1 (1988) 3±17. [26] N. Kasagi, Y. Tomita, A. Kuroda, Direct numerical simulation of a passive scalar ®eld in a turbulent channel ¯ow, J. Heat Transfer 114 (1992) 598±606. [27] T.E. Diller, Personal Communication, 1998.