1 Introduction

Particle image velocimetry (PIV) is a time-of-flight technique based on particles seeded in a flow field (Adrian and Westerweel 2011). However, it is difficult to apply this technique to transonic and supersonic flows because of unreliable particle traceability. Previous experimental results indicate that even particles with a size of ~0.1 μm cannot trace a supersonic flow (Huffman and Elliott 2009). Despite the difficulties associated with particle traceability, the PIV technique has such a specific benefit that the flow velocities on a plane can be measured nonintrusively with a high spatial resolution. Therefore, to overcome the difficulty related to particle traceability, a method to correct the PIV data was proposed on the basis of the equation of particle motion (Koike et al. 2007). In this method, the drag coefficient of a tracer particle is calculated using Stokes’ drag law. However, the appropriateness of use of Stokes’ drag law is questionable for supersonic flows because the drag of the tracer particle is strongly affected by compressibility and rarefaction of gas. According to Loth (2008), the compressibility and rarefaction effects are remarkable for particle Reynolds numbers lower than 45. These Reynolds numbers coincide with those for typical PIV measurements of supersonic flows.

Molecular tagging velocimetry (MTV) is also a time-of-flight technique (Lempert et al. 2002, 2003; Handa et al. 2012, 2014). In this technique, molecules are seeded in a flow field as tracers. This technique does not have the benefits of the PIV technique because a laser is injected into a flow as a beam. Instead, with MTV we do not have to be concerned about particle traceability for the application of this technique to transonic and supersonic flows. Using MTV, Lempert et al. (2003) measured a highly underexpanded jet issuing from a supersonic nozzle and demonstrated that the experimental results agreed with the results of numerical simulations. Handa et al. (2014) also applied the MTV technique to microjets with a Mach number of 2. Their experimental data suggested that this technique identified a phenomenon of rapid flow deceleration (from a Mach number of 2 to 1) within a short length of ~5 mm, even in a relatively rarefied supersonic flow with a Knudsen number (on the basis of nozzle exit parameters) of ~0.02. The measured supersonic-core length of ~5 mm agreed with that obtained in the numerical simulation.

The final goal of our study is to apply the PIV technique to transonic and supersonic flows in which gas accelerates and decelerates rapidly. To achieve this, the particle traceability in the flows should be understood and the particle drag coefficient should be calculated correctly for the flows where the compressibility and rarefaction effects are remarkable. In the present study, the particle traceability in transonic and supersonic flows is investigated by measuring two types of flows using the PIV and MTV techniques. The drag coefficient for a particle is also estimated from the velocity data measured using the two techniques and then compared with the drag coefficients calculated from the empirical formula proposed by Loth (2008).

2 Experiments

2.1 Tested flows

Two flows are selected for testing. One of the two tested flows is shown schematically in Fig. 1a. This flow is henceforth referred to as “flow 1” in this paper. The flow issues from a rectangular convergent-divergent nozzle that is connected to a stagnation chamber. The nozzle heights at the throat and exit are 5.4 and 6.0 mm, respectively. The nozzle width is 12.0 mm. The gas is accelerated to a Mach number of ~1.4 in the nozzle and then discharged into an expansion chamber. The discharged flow impinges on a cylinder whose diameter is 4.0 mm. The pressures in the stagnation and expansion chambers are set to 80 and 25 kPa, respectively; i.e., the nozzle pressure ratio (NPR) is set to 3.2. In this case, a bow shock wave appears in front of the cylinder. The second flow is an underexpanded jet discharged into an expansion chamber from an orifice placed at the downstream end of the stagnation chamber. The pressures in the stagnation and expansion chambers are set to 81 and 8.1 kPa, respectively. The present jet is shown schematically in Fig. 1b. We henceforth refer to this jet as “flow 2”. The gas in the jet is rapidly expanded through the expansion waves generated from the orifice lip and then abruptly decelerated through a Mach disk. The temperature in the stagnation chamber is maintained at 293 ± 1.5 K for both flows.

Fig. 1
figure 1

Tested flows. a Flow 1, b flow 2

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The coordinate systems used to analyze the results are defined in Fig. 1a, b, where x and y are the stream wise and height coordinates, respectively. The coordinate systems for flows 1 and 2 are placed in the central spanwise plane of the nozzle, and in the central plane of the jet, respectively. The origins for flows 1 and 2 are located at the upstream face of the cylinder on the nozzle centerline, and at the orifice exit on the jet centerline, respectively.

2.2 PIV measurement

Figure 2 shows the measurement system and the experimental setup for PIV. Components (1)–(7), depicted in Fig. 2, correspond to the PIV system. The laser (1) is a double pulsed Nd:YAG laser (Quantel, CFR400) operated with a second harmonic mode of 532 nm. The laser beam is shaped to a sheet through cylindrical lenses (3) and it is then irradiated into the flow from underneath. Scattered light from the particles is detected by a charge-couples device (CCD) camera (5) through a lens (Nikon, f/200 mm), and then recorded by a personal computer (7). One pixel on the digitized image corresponds to the physical size of 5.53 μm × 5.53 μm. A PIV image is captured in synchrony with a laser pulse using a programmable timing unit (Lavision) (6). Correspondingly, the time between two successive laser pulses is set to 110 ns. This value is checked by the signals detected by a photo detector (4). A pair of images is captured at a rate of 4 Hz. 250 pairs of images are captured in a single experimental run. Eight experimental runs are conducted and the image data are recorded on a personal computer (7). The velocity vectors are then calculated based on the image data correlating 32 × 32 pixel2 windows with a 50 % overlap.

Fig. 2
figure 2

Measurement system and experimental setup for PIV: 1 Nd:YAG laser, 2 beam splitter, 3 cylindrical lenses, 4 photo detector, 5 CCD camera, 6 programmable timing unit, 7 personal computer, 8 flow meter, 9 mist generator, 10 valve, 11 balloon, 12 valve, 13 stagnation chamber, 14 expansion chamber, 15 valve

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Uncertainty in the PIV data was estimated on the basis of the method stated in the handbook of particle image velocimetry edited by the Visualization Society of Japan (Kobayashi et al. 2002). Nevertheless, the uncertainty related to the particle traceability is not included in this estimation. The uncertainties in the averaged particle velocities are indicated by error bars in the experimental results of the centerline velocity distribution (shown later). The averaged uncertainties in the centerline velocities were 10.9 and 9.2 m/s for flows 1 and 2, respectively.

Components (8)–(15), shown in Fig. 2, correspond to the experimental setup used for the PIV measurements. Specifically, the experimental setup for flow 2 is representatively shown. A suction-type wind tunnel is used. The tracer particles of dioctyl sebacate (DOS) were produced through the Laskin nozzles in a mist generator (9). The nitrogen gas is introduced into a balloon made of vinyl chloride (11) through the mist generator. The gas reserved in the balloon is supplied into a stagnation chamber (13) and then discharged into an expansion chamber (14) through the orifice. In the measurements of flow 1, the nozzle is connected to the stagnation chamber and the balloon is replaced with a plenum chamber. To maintain a sufficient mass flow rate, the gas is supplied directly into the flow channel from the gas receiver in which high-pressure nitrogen is stored.

2.3 Pitot tube measurement

The centerline velocities of the jet (flow 2) are measured using a pitot tube only in the isentropic expansion region, which is bounded by the orifice exit, barrel shock wave, and Mach disk (see Fig. 1b). The inner and outer diameters of the pitot tube are 0.50 and 0.80 mm, respectively. The pitot pressure is measured using a semiconductor-type pressure transducer (Kulite XCS-190) that is calibrated just before the experimental runs and that has an accuracy of ±0.5 kPa (corresponding to ±2 m/s in velocity). The velocity is calculated from the pitot pressure and the pressure and temperature in the stagnation chamber using the normal shock relations and assuming an isentropic condition; i.e., this calculation is successful only in the isentropic expansion region.

2.4 MTV measurement

The flow velocities are measured using MTV, in which acetone is used as the molecular tracer. Detailed descriptions of the present MTV system are provided elsewhere (Handa et al. 2012, 2014). The laser is a Nd:YAG laser (spectra physics, 130-10) operated with a fourth-harmonic mode of 266 nm. Located in the central plane of the flow, the laser beam is focused by a spherical lens and irradiated into the flow field parallel to the y-axis. The incident laser energy is ~5 mJ. The resulting luminescence is collected by a lens, and then detected by the CCD camera with an image intensifier (Hamamatsu C7772S). The image intensifier gate is set to be open for 20–30 ns. The time delay between the tag and interrogation is controlled by the delay pulse generator (Stanford Research System, DG645) and is varied between 400 and 800 ns. In a single experimental run, 50–200 images are captured and averaged. Eight experimental runs are conducted at each measurement location. The physical size of one pixel on the digitized image is 20.0 μm × 20.0 μm. The data processing methodology used to extract the velocity data from the MTV images is identical to the method employed by Lempert et al. (2003) and Handa et al. (2014). Using the least squares method, the luminescence intensity across a tagged/interrogated line is fitted to a Gaussian profile and the displacement between the tagged and interrogated lines is determined from the peaks of the two profiles. In this study, the luminescence intensity data along the eleven pixels (corresponding to the width of 220 μm) around the peak are fitted to a Gaussian profile.

Acetone is evaporated and mixed with nitrogen gas in the acetone seeder, which is identical to the seeder used by Handa et al. (2011). In the measurement of flow 2, components (10)–(15), shown in Fig. 2, are also used in the MTV measurements. The seeding rate of acetone is set to be less than ~1 % by adding pure nitrogen gas to the mixed nitrogen gas (containing acetone vapor) in the balloon (11). In the measurements of flow 1, similarly to the PIV measurements, the balloon (11) is replaced with a plenum chamber and the gas is supplied directly into the flow channel from the gas receiver. The seeding rate of acetone is set to be less than ~1 % by controlling the flow rate of gas supplied to the plenum chamber through two tubes. In other words, similarly to the method of Handa et al. (2011), nitrogen gas, into which acetone is seeded through the acetone seeder, is supplied into the plenum chamber using one of the tubes, and pure nitrogen gas is added to the nitrogen gas mixture (containing acetone vapor) in the chamber using the other tube.

According to Lempert et al. (2002) and Handa et al. (2014), the accuracy in the velocity measured using the MTV technique is significantly influenced by mass diffusion because the fluorescence intensity profile across the interrogated line spreads over time, between tag and interrogation. They evaluated the statistical uncertainty of the velocity data measured in a supersonic jet issuing from a micronozzle by changing the tag-interrogation time delay. The results of Lempert et al. (2002) revealed that the statistical uncertainty (expressed as the standard deviation measured over several experimental runs) decreased rapidly from 35 to 4 m/s, as the time delay was increased from 50 to 300 ns, reaching a plateau of 4 ± 1 m/s between 300 and 1000 ns for a nozzle exit pressure of 0.13 kPa. The results of Handa et al. also revealed that the statistical uncertainty decreased rapidly from 45 to 6 m/s as the time delay was increased from 50 to 300 ns, reaching a plateau of 6 ± 1 m/s between 300 and 900 ns for a nozzle exit pressure of 12 kPa. This explains why the tag-interrogation time delay is set between 400 and 800 ns in this study.

The data measured by the present MTV technique should contain an error owing to a gradient of flow velocity because the velocity is determined from the peaks of fluorescence intensity across the tagged and interrogated lines. This error increases with the tag-interrogation time delay. We estimated the error considering the acceleration and deceleration of the acetone gas in the time interval between tag and interrogation. As a result, the maximum absolute values of error are estimated to be 2.8 m/s and 2.1 m/s for flows 1 and 2, respectively, except for the locations near the normal shock wave.

In addition to the mass diffusion and velocity gradient, Lempert et al. (2002) and Handa et al. (2014) identified several other potential sources of error, such as flow unsteadiness, an insufficient depth of field in the imaging system, the photodissociation of acetone, and image degradation due to a low acetone number density. However, there is presently no consensus on the method for estimating their respective significance. This is why uncertainties in the MTV velocity data are represented as 2σ, where σ is the standard deviation of eight experimental runs of velocity data. The uncertainties are indicated by the error bars in the experimental results of the centerline velocity distribution (shown later). Reliability of the MTV velocity data was checked by comparing the MTV data with the pitot tube velocity data measured along the centerline of flow 2, between x = 9.0 mm and x = 12.0 mm (in the isentropic region). Both datasets agreed well and were within ±1.2 %.

3 Results and discussion

3.1 Measurement results

Figure 3a, b show the velocity maps for flow 1 obtained using the PIV and MTV techniques, respectively. The velocity data in the figures are measured in the central spanwise plane of the nozzle. The gas discharged from the nozzle impinges on the circular cylinder. As a result of this impingement, a bow shock wave appears in front of the cylinder. Abrupt deceleration across the bow shock wave is clearly visible in both figures, whereby the tracer particles seem to decelerate rapidly across the shock wave. The velocity distributions along the line of y = 0 measured using the two techniques are shown in Fig. 4. Based on the MTV data, it is found that the gas is accelerated to a Mach number of 1.6. The PIV data agree very well with the MTV data with the exception of the data relevant to the airflow just behind the shock wave. Behind the shock wave, the gas velocity measured by the MTV technique decreases rapidly to the theoretical value of 230 m/s that is calculated using the normal shock relations, whereas, the particles decelerate more gradually than the gas. The spatial length over which the particles decelerate to the gas velocity is estimated to be ~0.6 mm.

Fig. 3
figure 3

Velocity maps for flow 1. a PIV, b MTV

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Fig. 4
figure 4

Velocity distributions along the line of y = 0 for flow 1

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Figure 5a, b show the velocity maps for flow 2 obtained using the PIV and MTV techniques, respectively. The velocity data in the figures are measured in the central plane of the jet. The expansion waves are generated from the orifice lip (see Fig. 1b) because the pressure at the orifice exit is higher than the pressure in the expansion chamber. The gas passing through these expansion waves should be rapidly accelerated and then decelerated abruptly across the Mach disk. Such abrupt deceleration is invisible in the PIV data (Fig. 5a) although it is clearly visible in the MTV data (Fig. 5b). This implies that the particles cannot trace the flow behind the Mach disk. Figure 6 shows the velocity distributions along the central axis of the jet measured by the two techniques. As stated above, the MTV data upstream of the Mach disk (isentropic expansion region) agree well with the data measured by the pitot tube. Both data indicate that the gas is accelerated to a Mach number of 3.8. The MTV data just behind the Mach disk (x ~ 12.5 mm) agree very well with the theoretical velocity of 150 m/s that is calculated based on the normal shock relations. It is found from Fig. 6 that in a highly underexpanded jet, such as the present jet, the particles cannot trace the flow either behind a Mach disk or within the expansion region upstream of the Mach disk.

Fig. 5
figure 5

Velocity maps for flow 2. a PIV, b MTV

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Fig. 6
figure 6

Velocity distributions along the line of y = 0 for flow 2

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As described in the following sections, the particle drag coefficient is calculated from the velocity data measured using the PIV and MTV techniques. In this calculation, the flow velocities relevant to the PIV data are necessary, although the MTV data (flow velocity data) are limited. Therefore, we obtain the flow velocity, u f by fitting the MTV data into appropriate equations; i.e., the MTV data upstream and downstream of the Mach disk are fitted respectively into the following equations:

$$u_{f} \; = \;1859.2x^{0.22821} ,$$
(1)
$$u_{f} \; = \;877.52\; - \;100770x\; + \;3513400x^{2} .$$
(2)

The results calculated by Eqs. (1) and (2) are shown by the solid lines in Fig. 6.

3.2 Calculation of drag coefficient

The particle motion is generally expressed by the Basset–Boussinesq–Oseen (BBO) equation (Soo 1967):

$$\frac{4\pi }{3}d_{p}^{3} \rho_{p} \frac{{D{\varvec{u}}_{p} }}{Dt}\; = \,\frac{4\pi }{3}d_{p}^{3} \rho_{p} F\left( {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right)\; - \;\frac{4\pi }{3}d_{p}^{3} \frac{{\partial p_{f} }}{\partial {\varvec{x}}}\; + \;\frac{1}{2}\frac{4\pi }{3}d_{p}^{3} \rho_{f} \frac{{D\left( {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right)}}{Dt}\; + \;3d_{p}^{3} \sqrt {\pi \rho_{f} \mu_{f} } \int_{{t_{0} }}^{t} {\frac{{\left( {D/D\tau } \right)\left( {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right)}}{{\sqrt {t\; - \;\tau } }}d\tau } \; + \;\frac{1}{8}{\varvec{F}}_{e}$$
(3)

In this equation, d p is the particle diameter, ρ is the density, u is the velocity vector, p is the pressure, x is the position vector, t is the time, τ is the relaxation time, μ is the viscosity, and F e is the external force vector. The subscripts f and p denote variables of the flow and particle, respectively. The first to fifth terms in the right hand side of Eq. (3) represent the drag force, pressure gradient, added mass, Basset term, and external force, respectively. The symbol F in the first term is written as,

$$F\; = \;\frac{3}{4}C_{\text{D}} \frac{{\rho_{f} }}{{\rho_{p} }}\frac{1}{{d_{p} }}\left| {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right|,$$
(4)

where C D is the drag coefficient for a particle.

When the flow is steady and there are no external forces acting on the particle, the third, fourth, and fifth terms are neglected. In this condition, C D is given by,

$$C_{\text{D}} \; = \;\frac{4}{3}\left( {u_{pj} \frac{{\partial u_{pi} }}{{\partial {x}_{j} }}\; + \;\frac{1}{{\rho_{p} }}\frac{{\partial p_{f} }}{\partial {\varvec{x}}}} \right)\frac{{\rho_{p} }}{{\rho_{f} }}\frac{{d_{p} }}{{\left| {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right|\left( {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right)}}.$$
(5)

Using the above equation, we calculated values of C D from the measured data. The flow variables in Eq. (5), except for the flow velocity, were estimated by assuming an isentropic condition. In this estimation, the total pressures upstream and downstream of the Mach disk were necessary for flow 2. The total pressure upstream was presumed to be identical to the pressure in the stagnation chamber and the downstream total pressure was estimated using the normal shock relations. The values for C D were not reasonable for flow 1 because the difference between the PIV and MTV data was quite small in the regions where the flow is presumed to be steady. Unreasonably large values of C D were estimated because the difference between flow and particle velocities in the denominator of Eq. (5) becomes quite small.

3.3 Drag coefficient formula

Considering the compressibility and rarefaction effects on the drag of a spherical particle, Loth (2008) recently proposed an empirical formula for calculating the drag coefficient C D for low particle Reynolds numbers as follows:

$$C_{\text{D}} \; = \;\frac{{C_{{{\text{D}},{\text{Kn}},\text{Re} }} }}{{1\; + \;M_{p}^{4} }}\; + \;\frac{{M_{p}^{4} C_{{{\text{D}},{\text{fm}},\text{Re} }} }}{{1\; + \;M_{p}^{4} }}\quad{\text{for}}\quad \text{Re}_{p} \; \le \;45.$$
(6)

The particle Reynolds number, Re p , and the relative Mach number, M p , in Eq. (6) above are defined as,

$$\text{Re}_{p} \; = \;\frac{{\rho_{f} \left| {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right|d_{{p}} }}{{\mu_{f} }},$$
(7)
$$M_{p} \; = \;\frac{{\left| {{\varvec{u}}_{f} \; - \;{\varvec{u}}_{p} } \right|}}{{a_{f} }}.$$
(8)

In Eq. (8), a is the sound speed. The detailed expressions of C D, Kn, Re and C D, fm, Re in Eq. (6) are given by Loth (2008). According to Loth (2008), C D, Kn, Re in Eq. (6) depends on the Knudsen number, Kn p , and on Re p , and C D, fm, Re depends on Re p , M p , and T p /T f (T is the temperature). Herein, the Knudsen number Kn p is defined as,

$${\text{Kn}}_{p} \; = \;\sqrt {\frac{\pi \gamma }{2}} \frac{{M_{p} }}{{\text{Re}_{p} }}.$$
(9)

Equation (6) was validated by Loth using the experimental data and the computational results of direct simulation Monte Calro (DSMC) for M p  > ~2. In this case, almost all the present PIV data were measured for M p  < ~1; i.e., it is necessary to validate Loth’s formula for M p  < ~1, if the PIV data measured in supersonic flows are going to be corrected on the basis of Loth’s formula. In the calculation of C D, the temperature ratio T p /T f is set to 1. The value of C D calculated from Loth’s formula is practically independent of the temperature ratio; i.e., the temperature effect is estimated to be within 4 % in the value of C D for the present experimental conditions.

Kähler et al. (2002) experimentally obtained the size distributions of the particles produced by a Laskin nozzle for two pressure states (50 and 100 kPa). Their experimental results indicated that the distributions had a peak at particle sizes of 1.36 and 1.28 μm for the pressure states of 50 and 100 kPa, respectively. The Laskin nozzle is also used in the present experiments. In the estimation of C D and Re p , the mean value of the above two particle sizes (1.32 μm) are used because the experimental results of Kähler et al. (2002) indicated that both the feed-hole ring of the Laskin nozzle, and the number and diameter of nozzle hole had little effect on the mean particle size. The estimated results are compared with the results calculated from Loth’s formula [Eq. (6)] for each M p , as shown in Fig. 7a, b. It is seen from the figures that the results calculated from Loth’s formula agree reasonably well with the experimental data for Re p  > ~1. However, the formula underestimate the drag coefficient for Re p  < ~1. In constructing the formula, Loth considered the free molecular limit for an infinitesimal Reynolds number on the basis of the analysis of Stadler and Zurick (1951) (Loth corrected the limit empirically). Loth validated the free molecular limit using the DSMC data for M p  > ~3; i.e., the data agreed reasonably well with the free molecular limit for Re p  < ~1. However, we have presently no method for verifying the free molecular limit for M p  < ~3 because the experimental and computational data are limited.

Fig. 7
figure 7

Plots of drag coefficient C D versus particle Reynolds number Re p . a M p  = 0.1, M p  = 0.2, and M p  = 0.3, b M p  = 0.5, M p  = 0.7, and M p  = 0.9

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4 Conclusions

To evaluate the traceability of the particles in high-speed flows, the PIV and MTV techniques were applied to two flows in which a normal shock wave appeared. One of the flows tested was a flow impinging on a circular cylinder in front of which a bow shock wave appeared. The Mach number just upstream of the shock wave was 1.6. In this case, the particles were found to trace the flow over the entire region except in a region just behind the shock wave. The other flow tested was an underexpanded jet in which a Mach disk appeared. The Mach number just upstream of the disk was 3.8. In this case, the particles could not trace the flow behind the Mach disk or in the isentropic expansion region.

The drag coefficient for a particle was estimated from the velocity data measured using the two techniques and the results were compared with the results calculated from the empirical formula proposed by Loth (2008) for particle Mach numbers ranging between 0.1 and 0.9. As a result, it is found that the experimental drag coefficients were estimated reasonably well by Loth’s formula for particles with Reynolds numbers higher than ~1. However, the data for particles with Reynolds numbers lower than ~1 led to large deviations although the free molecular limit was considered in Loth’s formula.