1 Introduction

Quantitative planar measurements of the turbulent-jet mixture fraction are of particular interest for studies of fuel–air mixing and have been accomplished using Rayleigh scattering [1] and tracer laser-induced fluorescence (LIF) [2]. Recently, research has been focused on extending these traditional planar diagnostics from 10’s of Hz to kHz rates to temporally resolve turbulent flow behavior [3]. In general, three types of laser systems have been used to accomplish this task. The earliest of these were clusters of multiple high-power lasers (100’s mJ/pulse at 532 nm), which thus far have been limited to 8 high-repetition-rate pulses [4]. More recently, continuously pulsed diode-pumped solid-state (DPSS) lasers (~10 mJ/pulse at 532 nm) have been utilized for capturing image sequences as long as ~10,000 frames at rates up to 10 kHz [5]. However, the pulse energy available from continuously pulsed lasers is low for mixture-fraction imaging using Rayleigh scattering in the visible range or tracer LIF in the ultraviolet (UV) [6]. Hence, the use of clustered or DPSS lasers severely restricts either the number of frames recorded in the image sequence or the accuracy of mixture-fraction measurements, respectively. As an alternative, burst-mode laser systems enable pulse sequences that are longer in duration than clustered laser systems and are more powerful than DPSS lasers [7, 8]. Recent work by the authors in this area has been focused on increasing the laser burst duration utilized for formaldehyde planar laser-induced fluorescence (PLIF) sequences of 200 images or more, representing a 20-fold increase in the record length over previous measurements [9, 10].

The current work focuses on several innovations related to high-speed, burst-mode imaging of mixture fraction. This includes significantly increasing the record length, image area, and signal-to-noise ratio (SNR) as compared to prior mixture-fraction measurements that used burst-mode imaging [11] and significantly increasing the laser pulse energy, spatial resolution, and accuracy as compared to prior work that employed continuously pulsed DPSS lasers [6]. The corresponding increase in SNR enables quantitative measurements in turbulent flows without the use of intensified detectors that would otherwise blur the spatial gradients. Furthermore, by utilizing the fourth-harmonic output of the burst-mode Nd:YAG laser at 266 nm for acetone PLIF mixture-fraction measurements, it is possible to separate the red-shifted fluorescence from particle scattering for simultaneous particle-image velocimetry (PIV). This is in contrast to other techniques for burst-mode mixture-fraction imaging, such as Rayleigh scattering, which are sensitive to interference from surface and particle scattering [11]. Finally, to understand the effects of transient absorption and thermally induced phase mismatch, which can limit burst-mode frequency conversion to the UV, a semiquantitative model is employed for comparison with experimental output energies over a wide range of repetition rates. This model can be used to predict the UV output energy of next-generation burst-mode laser systems.

2 Burst-mode fourth-harmonic generation

The quasi-continuous burst-mode (QCBM) laser used in this work has been previously described in detail and will only be briefly discussed here [9]. A Yb-doped pulsed fiber oscillator and electro-optic modulator (EOM) produce a 10-kHz train of 10-μJ pulses, which is amplified by 16,800× through three diode-pumped Nd:YAG amplifiers (2-, 2-, and 5-mm rod diameters) and by 2.5× through one flashlamp-pumped Nd:YAG amplifier (9.5-mm-diameter rod). The burst duration in this work is 10 ms, although at least 30 ms is feasible [10]. Excellent beam quality and suppression of amplified spontaneous emission (ASE) are maintained by optical relay imaging and spatial filtering between each amplifier. Under typical operating conditions at 10 kHz, the fundamental pulse energy is >150 mJ. The 5-mm-diameter beam (1/e2 width) propagates with a measured M2 factor of 1.85, nearly Gaussian distribution of energy, and excellent stability over the 10-ms burst duration.

The fundamental output of the QCBM laser is converted to 532 nm by second-harmonic generation (SHG) in an 8 × 8 × 20-mm Type I lithium triborate (LBO) crystal with noncritical phase matching, heated to 149.7 °C to optimize conversion. Fourth-harmonic generation (4HG) is achieved by frequency doubling the 532-nm light (>75 mJ) in a 7 × 7 × 6-mm Type I BBO crystal that is anti-reflection coated at 532 and 266 nm. BBO is chosen because of its high nonlinear coefficient and damage threshold. Standard 266-nm mirrors are used to separate the 4HG (>15 mJ) from the SHG and fundamental wavelengths, allowing ~300 μJ per pulse of the residual 532-nm beam to propagate with the 4HG radiation to the test article for particle scattering.

To consider the effects of transient absorption and thermally induced phase mismatch, which can severely limit frequency conversion to the UV at high repetition rates, a finite-difference solution to a coupled set of differential equations is employed. The SHG and 4HG pulse energies can be calculated as a function of the laser intensity, I i=1,2,4, where i represents the fundamental, second-, or fourth-harmonic field, respectively, and z represents the distance along the crystal in the propagation direction. We consider 4HG in BBO, ω 2 + ω 2 = ω 4, assuming a plane wave with uniform spatial distribution of energy and square temporal laser pulse with peak intensity, I i=2,4, and duration, τ. The simplified equations governing the depletion of 532-nm light, E 2, and generation of 266-nm light, E 4 , are given as [12]

$$ \frac{{{\text{d}}E_{2} \left( {z,t} \right)}}{{{\text{d}}z}} = - KE_{4} E_{2}^{*} \cos \left[ {\varDelta k(z)} \right] - \frac{1}{2}\alpha_{2} E_{2} $$
(1)
$$ \frac{{{\text{d}}E_{4} \left( {z,t} \right)}}{{{\text{d}}z}} = KE_{2} E_{2} \cos \left[ {\varDelta k(z)} \right] - \frac{1}{2}\left[ {\alpha_{4} + \sigma_{4} N\left( {z,t} \right)} \right]E_{4} - \frac{1}{2}\beta_{4} E_{4}^{*} E_{4} E_{4} $$
(2)
$$ K = \sqrt {\frac{2}{{\varepsilon_{0} cn_{4} }}} \frac{{2\omega_{4} d_{\text{eff}} }}{{n_{2} c}}\sin (\theta ) $$
(3)

where K is the coupling coefficient in 1/√W, E i is the square root of the field intensity in √W/m, Δk is the phase mismatch in rad, α i is the linear absorption coefficient in m−1, σ i is the color-center absorption cross section in m2, N is the color-center density in m−3, and β i is the two-photon absorption coefficient in m/W. The phase mismatch, Δk(z) = Δk o  + δk T (z), contains an initial phase-mismatch component, Δk o , due to angular detuning and a thermal phase mismatch, δk T , driven primarily by absorption of E 4 . The coupling coefficient is constant for a given set of crystal and wavelength parameters, where ε o is the permittivity of free space in C2/J-m, c is the speed of light in a vacuum in m/s, n i is the index of refraction, ω i is the angular frequency defined as 2πc/λ in s−1, d eff is the effective nonlinear coefficient for BBO in m/V, and θ is the angle between the beam-propagation vector and the optic axis of the crystal.

The measured 4HG performance curves for 10 Hz and ≥10 kHz in Fig. 1 are modeled using Eqs. (1–3), with conversion efficiency η conv = I 4HG /I SHG . The effects of high pulse energy are visible even at 10-Hz repetition rates, where saturation and a slight decrease in conversion efficiency are observed at the highest input intensities. These effects, typically attributed to transient absorption [13] and thermal phase mismatch [14], are even more significant for burst-mode operation because of the nearly 100× higher energy imposed within the 10-ms burst duration as compared to a single 10-Hz pulse. The bars represent a standard 5 % uncertainty determined from 100 laser bursts. The computed results, which include both thermal phase mismatch and transient absorption, are scaled (0.32×) at all conditions to the measured 10-Hz pulse energies. The scaling factor is attributed to losses at surfaces and assumptions of spatial and temporal uniformity within the model.

Fig. 1
figure 1

Fourth-harmonic generation conversion efficiency for individual pulses at 10 Hz, 10 kHz, 20 kHz, and 33 kHz. Lines are modeled results including thermal phase mismatch and transient absorption

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The linear, α 4 (2 % per cm at 266 nm), transient, σ 4 N (σ 4  = 8 × 10−21 m2), and two-photon absorption, β 4 (0.93 × 10−12 m/W), of E 4 alone produce only a 1 % drop in conversion efficiency under the conditions reported here, insufficient to explain the reduction observed in Fig. 1. Instead, the absorption of E 4 causes thermal heating of the crystal, driving a nonlinear reduction in conversion through thermal phase mismatch [13, 14]. At high repetition rates, transient absorption is the primary mode for depositing heat in the crystal. The color-center density, N, is modeled according to Marshall et al. [12] for a temporally square pulse with intensity, (E i )2, and duration, τ, including temporal decay due to thermal bleaching between laser pulses. Since no data were available for BBO, color-center formation and bleaching efficiencies for KDP were used (η f  = 1 and η b  = 0.065) [12], producing comparable absorption values to those published for 266 nm in BBO [15].

The differential rise in temperature and associated thermal phase mismatch is a function of crystal distance, z, given by [14]

$$ \delta k_{T} (z) = \frac{b}{\varDelta T}\left\{ {T_{1} \varDelta z + \frac{{A\left[ {\alpha_{4} + \sigma_{4} N\left( {t,z} \right)} \right]f\tau E_{4}^{*} E_{4} }}{2K}} \right\} $$
(4)

where ΔT is the temperature acceptance bandwidth of BBO in K, T 1 is the initial temperature gradient at the crystal entrance in K/m, A is the area of the laser beam in m2, N(t,z) is the color-center density of 266 nm in m−3, K is the thermal conductivity in W/m–K, and f is the repetition rate of the laser system or the number of pulses per second in the case of burst-mode systems. Time-dependent heat-transfer effects were included to calculate the temperature decrease in the beam-propagation direction between each laser pulse. While thermal relaxation at 10 Hz can influence 4HG significantly, it is negligible for high-repetition-rate systems. An effective heat-transfer coefficient, b = 0.01, is introduced to best fit the experimental data, resulting in a maximum calculated temperature increase of ~5 K over the 6-mm crystal. This provides reasonable agreement at all conditions between the experimental and computed conversion efficiencies, which are in general agreement with previously published results in BBO [14, 15].

The relatively long-time interval between bursts is beneficial for color-center decay and reduces pure absorption losses in the crystal as compared to continuously pulsed systems. However, the cyclical thermal loading also prevents compensation of thermal phase mismatch with angular detuning of the crystal. The counteraction of these two processes results in a reduction in conversion efficiency for higher repetition rates, as shown in Fig. 1. Hence, careful design of the frequency conversion system (e.g., beam size, crystal dimensions, and pump energy) is required at very high repetition rates.

3 Mixture-fraction imaging and particle-image velocimetry

The suitability of 4HG burst-mode laser systems for quantitative acetone-based mixture-fraction imaging was investigated in an 8-mm-inner-diameter jet surrounded by a 10.2 × 10.2-cm coflow designed to produce a nearly uniform velocity profile. The coflow velocity was held constant at 0.4 m/s and 3 % acetone seeding by volume, while the jet was seeded at 16 % acetone by volume with a jet diameter Reynolds number (ReD) of 15,000. Coflow seeding is important for the correction of shot-to-shot laser energy and beam profile, quantifying acetone absorption, and for potential two-line thermometry techniques in heated flows [16]. For PIV, the acetone-seeded jet was mixed with an oil-droplet-seeded flow while maintaining constant ReD and acetone seed concentration.

A laser light sheet was formed using a 1-m UV spherical lens and −75-mm UV cylindrical lens, with only the nearly uniform center section used for imaging. Fluorescence was collected using a high-speed nonintensified complementary metal–oxide–semiconductor (CMOS) camera (Photron SA-X) with a 55-mm f/1.2 Nikkor lens, 12-mm extension ring, and a 512 × 1,064 image acquisition size at 10 kHz. The image resolution was measured with a 1951 USAF target. The measured in-plane resolution is 280 μm at 50 % contrast with minimum resolvable features of 140 μm. Absolute pixel resolution is 109 μm. The in-plane resolution of 140 μm is sufficient to capture the finest dissipative length scale, λ D , of typical fuel mixtures (CH4–H2–N2) at ReD = 15,000 for distances greater than seven diameters from the jet (λ D  >142 μm). The dissipative length scale was computed as a function of downstream distance using known scaling laws for turbulent mixing in axisymmetric jets and is representative of the smallest length scales where scalar mixing dominates molecular diffusion [11, 17]. Representative images from a 10-ms sequence of quantitative mixture fraction centered at x/D = 8 are shown in Fig. 2 with a physical image size of 38 by 40 mm (full sequence in supplemental material). The images are background subtracted, and the coflow PLIF signal (I CF ) is used to normalize the image, correct for pulse-to-pulse variations in energy and laser-sheet profile, and quantify and correct for 266-nm absorption across the jet and coflow regions. A linear relationship between acetone concentration and PLIF signal was verified in the jet core (I J ) at x/D = 1 and used to convert the normalized signal (I x,y ) to mixture fraction, ξ x,y  = (I x,y  − I CF )/(I J  − I CF ), since both the jet and coflow acetone concentrations are known.

Fig. 2
figure 2

Consecutive images from 10-kHz, 10-ms sequence of mixture fraction at ReD = 15,000 and x/D = 8 with scaling from white (ξ = 0) to black (ξ = 1). Entire series in supplemental material. Inset shows detailed view of mixing layer

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The raw image signal-to-noise ratio (SNR) is 90:1 in the jet and 18:1 in the coflow, leading to measurement noise in the computed jet mixture fraction of ~2 %. In addition to a tenfold increase in burst duration, this is a twofold increase in single-shot SNR for twice the image height over previous high-speed mixture-fraction measurements using Rayleigh scattering in a propane jet [11]. To verify the accuracy of the mixture-fraction measurements, the mean, <ξ>, and fluctuating, ξ′, components along the centerline from x/D = 1–10 are compared with known scaling laws [17] and previous measurements using Rayleigh scattering [18], as shown in Fig. 3. The mean mixture fraction is calculated from all images in the burst, and scaling is given by <ξ(x)> = 5.4(x/D*)−1, where x is the distance from the nozzle along the centerline, and D* = (ρ J /ρ CF )1/2 D is the jet diameter (D) corrected for density differences between the jet and the coflow [19]. The root-mean-square of the fluctuating mixture-fraction component within a single burst is normalized by the mean mixture fraction and compared with line Rayleigh measurements made in a propane diffusion jet at a ReD = 68,000 [18]. Temporal noise of ~2 % in the derived measurement was removed from the fluctuating component, ξ′/<ξ>, producing a near-zero baseline in the jet core. A jet core with <ξ> near unity persists for nearly 5 diameters before falling as 5.4(x/D*)−1. The fluctuations compare favorably with the Rayleigh scattering results for x/D > 3. Near the jet outlet (x/D < 4), an increase in noise was observed due to shot-to-shot changes at the edges of the laser sheet.

Fig. 3
figure 3

Mean centerline mixture fraction, <ξ> , and mixture-fraction fluctuation, ξ’/<ξ> , at ReD = 15,000. Solid line is mean scaling law [17], and dashed line is data from single-shot Rayleigh scattering [18]

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Simultaneous mixture-fraction imaging and planar velocimetry were demonstrated at 10 kHz and a jet Reynolds number of 5,000. PIV images were recorded using residual 532-nm light passing through a UV mirror after 4HG. The 532-nm residual was located in plane with the 266-nm radiation with a slight vertical shift due to the “walk-off” in the 4HG crystal. A transmission filter from 370–450 nm was placed in front of the acetone PLIF camera to remove 532-nm Mie scattering from the seed oil droplets. The glass collection optics and filter also effectively blocked any residual 266-nm scatter. PIV detection was performed with a Photron SA-5 CMOS camera, 55-mm f/1.2 Nikkor lens, 20-mm extension ring, and narrow bandpass filter at 532 nm. The image size of both cameras was 512 × 1,064 at 10 kHz. Even at high particle density, 532-nm Mie scattering was not observed in the acetone detection system. Additionally, at acetone saturation levels, no PLIF signal was observed in the PIV detection system. PIV vectors were computed using PIVLab software with an 8 × 8-pixel interrogation region corresponding to 0.7 mm2 [20]. Overlap of mixture-fraction and PIV data was performed using a grid chart, imaged by each camera. A snapshot of simultaneous mixture fraction and overlaid PIV vectors is shown in Fig. 4 (full sequence in supplementary material). To highlight the shear layer, the PIV data have been shown with half of the centerline mean velocity subtracted. Although the coflow contained no particles in these experiments, velocity vectors can be extracted in the densely seeded core of the jet. While this is currently only a demonstration, the mean velocity of 4.8 m/s is in fair agreement with mixing theory at x/D = 8, highlighting the utility of this technique for simultaneous mixture-fraction and velocity measurements. It should be noted for a single burst-mode laser that the interpulse spacing for velocity measurements is fixed by the pulse repetition rate unlike traditional PIV techniques, which utilize pulse pairs. In flows where this interpulse spacing is insufficient to capture high-velocity motion, the repetition rate can be increased or a second time-delayed burst-mode laser can be used to generate the second pulse in the PIV pair with variable interpulse spacing.

Fig. 4
figure 4

Sample image from 10-kHz, 10-ms sequence of mixture fraction and overlaid velocity vectors at ReD = 5,000. Entire series in supplemental material

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4 Summary

High-speed imaging of mixture fraction and velocity was achieved using a quasi-continuous burst-mode laser for a tenfold increase in record length over previous burst-mode mixture-fraction measurements and a twofold increase in signal-to-noise ratio for twice the image area. Modeling of fourth-harmonic conversion revealed the limitations due to thermal phase mismatch and transient absorption when using high-repetition-rate burst-mode laser systems. Nonetheless, the pulse energy was 16-fold higher than previous acetone PLIF measurements at 10 kHz, allowing quantitative mixture-fraction imaging without the need for intensified detectors. High-speed simultaneous particle-image velocimetry performed along with quantitative mixture-fraction imaging shows great promise for high-speed multi-parameter measurements in turbulent flows.