2 The experiment set-up

The principal components of the ASOS shock tube, driver, double-diaphragm bursting device and expansion section, can be seen in the overall perspective of figure 1, and with dimensions in the schematic of figure 2. The expansion section consists of a short transition piece, taking the flow smoothly from the circular cross-section of the driver to the square one, $200\times 200~\text{mm}^{2}$ , of the development section, and the test section proper, which connects to the catch tank. All of these components are long enough to yield the design objective of steady blast duration as described below. Polycarbonate construction of the test section allows for nearly uninhibited visual access. The structural design envelope is defined by the release of 10 MPa Helium in the driver: it yields Mach 3.6 shocks and gas-dynamic pressures of 1.5 MPa. In the present experiments we are limited by the strong reflection from the curtain to 4 MPa Helium in the driver. The double-diaphragm bursting device, equipped with home-made diaphragms from Mylar stock, allows on-demand release of the driver pressure with a maximum uncertainty of $\pm$ 2.5 ms. The operating range of the shock tube is illustrated in figure 3, which also demonstrates the theoretical support of the measured shock wave amplitudes. The scatter in data points is covered by accuracy of the pressure sensors (2 %) and gauge used to charge the driver (1 %). Of course helium is more effective (than nitrogen) in creating stronger shocks, but it is quite expensive, so we choose to use it only when necessary by design considerations.

Figure 4. The instrumentation schema. All dimensions in centimetres. The high-speed video cameras are marked by HS1, HS2, HS3. Pressure transducer locations are marked by P1, P2, P3. Top view.

The instrumentation schema is illustrated in figure 4. For visualization we employ the shadowgraphic method with high-speed digital cameras from Vision Research. Camera HS1 is a Phantom V12, capable of 1 megapixel at up to 6.2 kHz, and HS2/HS3 are Phantom 7.1 cameras allowing 0.5 megapixel at up to 6.7 kHz. In this application we operate the cameras at 10 kHz, thus sacrificing about half of the pixel capability in favour of obtaining adequate time resolution at the high-end of the pressure ratios. Exposure times are limited by camera sensitivity to $5~{\rm\mu}\text{s}$ , and with appropriate optics we obtain resolutions of ${\sim}$ 10 pixels per particle in a field of view of the full curtain at HS1. The lights for shadowgraphic imaging are provided by four white-light lamps (500T3/Q/CL 300 W). The pressure transducers are of the piezoelectric type, Kistler models 601A at position P1 and 211B3 at positions P2, and P3, with ranges 0–25 MPa and 0–3.4 MPa, respectively. The manufacturer calibrations were checked independently in our laboratory and found to be within the specified uncertainty ( $2\,\%$ ). In a run, as the curtain enters the path to the light detector (ThorLabs PDA8A), a trigger signal is generated to sequence, with appropriate delays, diaphragm rupture and initiation of data acquisition by the cameras. The pressure transducer outputs are routed through signal amplifiers (Kistler 5010) to LeCroy Wave Surfer 424 digital oscilloscope, which triggers by incident shock front from P1 and records also the camera trigger for synchronization with the pressure transients.

In preliminary testing (Chang et al. Reference Chang, Sushchikh, Mitkin and Theofanous2011), we worked with particle curtains of low enough volume fraction to be considered dilute ( ${\it\alpha}_{d}\sim 0.4\,\%$ ). These curtains were created by releasing particles through vibrating grids. In the present setting, the particles are allowed to fall from a supply box, connected through a slight taper to a chute (figure 5 a), so as to focus the particles into a curtain with well-defined boundaries (figure 5 b). The release, by a quick-action shutter, is timed with driver-diaphragm rupture to yield shock impact coincident ( $\pm$ 2.5 ms) with first formation of a full curtain. Based on extensive testing, we determined that this timing must involve a slight delay after the very first particles touched the channel floor, to exclude some non-uniformities associated with opening of the shutter. This created some ‘floating’ particles (bounced off the floor) before and after the curtain. Their effect is negligible. This also created a slight accumulation on the floor (a layer about 5 mm thick), which did eventually affect the interaction over the lowermost ${\sim}$ 10 % of the curtain, while also revealing some interesting particle-lifting phenomena that are of interest in their own right.

Figure 5. Schematic of the supply box and taper into the chute (a), visualization of a curtain at the time of shock impact (b) and size distribution of the particles utilized in the experiments (c). All dimensions in millimetres.

The glass particles were bought commercially (MO-SCI Specialty Products) and sieved carefully to isolate the particle-diameter range, $0.85<d<1.0$  mm, used in these experiments. The size distribution was determined by microscopic analysis of a random sample of 600 particles, yielding $0.86<d<0.96$  mm. The distribution is shown in figure 5. Sphericity was measured with the help of high-resolution visualizations of multiple, randomly sampled particles, and found to be accurate to within 2 %. The ideal soda-lime glass density is ${\it\rho}_{d}=2.5\times 10^{3}~\text{kg}~\text{m}^{-3}$ . We determined the density independently by measuring the free volume of a certain mass of glass spheres packed inside a graduated cylinder. This gave $2.46\times 10^{3}~\text{kg}~\text{m}^{-3}$ , which is 98.4 % of the ideal value, a finding supported by our microscopic examination that revealed no significant gas intrusions.

In addition to ensuring a reproducible, optically high-quality curtain, we also paid special attention to the characterization of the particle volume fractions in it. To this end, the particle-supply device was fitted to an outside construction that matched exactly the actual experimental channel. In this way the cloud could be approached freely, and videos could be made from all sides simultaneously, including the upper interface inside the particle-supply box (see figure 5). From the latter, we determined the mass flow rate ( $7.79~\text{kg}~\text{s}^{-1}\pm 3$  %) history over the useful/complete emptying transient ( ${\sim}125/225$  ms events), and this, under free-fall acceleration (which was in agreement with direct velocity measurements), could be translated to the volume fraction distribution. As a diverse method, we placed blackened-particle marks at regular intervals in the supply box, and by visualizing their relative distances at the time the main front touched the channel floor we could determine the volume fraction distribution as a function of height. The results of both of these quantifications are consistent as represented in figure 6. This figure also provides an overall measure of reproducibility and uncertainty.

Figure 6. (a) The vertical volume fraction distribution in the ASOS curtain. The line is calculated from the mass flow rate and velocity measurements, shown in (b), and the data points are from direct measurements of volume expansion between particle markers in the supply load. The line in (b) is calculated assuming free-fall. (c) An overall measure of uncertainty in determination of volume fractions in the curtain; seven independent runs. The volume fraction distribution (a), was integrated to find the volume of particles between two extreme markers in the curtain (d), and this was compared with the actual volume known from the placement of these markers inside the supply box: both volumes converted to per unit area of the supply box; true value $dH=6$  cm. All measurements taken at the time corresponding ( $\pm$ 2.5 ms) to shock impact in an actual test run. Notable is the large vertical gradient of volume fraction in the curtain; imposed by the free-fall condition, most of it is contained in the upper 20 % of the curtain’s height.

Figure 7. Illustration of the curtain geometry, pressure waves, contact waves, and related nomenclature. Here $(p,u,T)$ is the gas flow state (pressure, velocity, temperature). The $u_{s}$ are shock velocities. (UF, DF) designate the (upstream/downstream) fronts of the curtain.

The curtain fits exactly over the whole of the channel cross-sectional area as illustrated in figure 7. This figure also shows the nomenclature employed throughout. The test conditions were selected to span the attainable flow Mach numbers, which, as indicated in table 1, are used to identify individual runs. In this table, the flow Mach number is $M_{IS}\equiv u_{IS}/c_{IS}$ , where ( $u_{IS}$ , $c_{IS}$ ) are (gas, sound) speeds behind the shock; the shock Mach number is $M_{s}\equiv u_{s,IS}/c_{0}$ , where $c_{0}$ is the speed of sound in the air ahead of the shock ( $346~\text{m}~\text{s}^{-1}$ in this case); $P_{IS}/P_{W}$ are the incident/reflected shock pressures, the latter from a hypothetical rigid wall at the position of the curtain (the relevance of the latter is discussed at the beginning of § 3), both normalized by $p_{0}$ ; and $t_{ub}^{\ast }\equiv t_{ub}/{\it\tau}$ , where $t_{ub}$ is the duration of uniform blast (constant flow condition behind the shock) at the curtain position and ${\it\tau}$ is a characteristic time to be defined shortly. The time of uniform blast is limited by the reflected shock reflecting back to the curtain off the incident contact. The proximity of this contact with the incident shock wave increases with shock strength, and this results in diminishing $t_{ub}$ as indicated in the table 1. The maximum Reynolds number ( $Re_{max}$ ) is based on particle diameter and free-stream velocity behind the shock: it does not exist as such anywhere, but it may be useful as a basis for comparison between runs. It can be seen that the flow speeds in this set of runs vary from ${\sim}$ 100 to ${\sim}\!700~\text{m}~\text{s}^{-1}$ .

Table 1. Summary of the test conditions investigated. All tests were conducted at standard atmospheric condition ( $p_{0}=0.1$  MPa,  $T_{0}=298.15$  K). A number of duplicate runs made to test reproducibility are not included in this table.

3 Experimental results

We have organized the presentation of results in two subsections covering respectively: pressure transients and 2D wave shapes (§ 3.1), and curtain displacements and cloud morphologies (§ 3.2). The cross-flow particle motion (‘lifting’) from the small amounts found on the channel floor is illustrated in appendix B.

Due to the very high density ratio, particle motion on the acoustic time at the particle scale is negligible, so the characteristic length is the initial curtain thickness (it is used to scale distances and displacements throughout), the actual value as found in the respective video is used in each case. In seeking the characteristic time, we recognize the importance of the pressure gradient across the curtain in the momentum exchange process. Moreover, we anticipate, and subsequently verify, that, for the high-speed flows of concern here, the pressure reflection off the upstream face of the curtain would be quite similar to that off a rigid/impermeable wall (the $p_{W}$ quantity introduced earlier). Thus, in the early, all-too-crucial portion of the transient the pressure gradient should be $(p_{W}-p_{0})/l_{0}$ , which together with the characteristic length, and mass (involving the density of the disperse phase), reveals (theory of dimensions) the characteristic time scale of the process as

(3.1) $$\begin{eqnarray}{\it\tau}=l_{0}\sqrt{{\it\rho}_{d}/p_{0}(P_{W}-1)}.\end{eqnarray}$$

This $P_{W}\equiv p_{W}/p_{0}$ can be found from the positive root of the quadratic

(3.2) $$\begin{eqnarray}{\it\xi}^{2}-\left(2+{\textstyle \frac{1}{2}}{\it\gamma}({\it\gamma}+1)M_{IS}^{2}\right){\it\xi}+1-{\textstyle \frac{1}{2}}{\it\gamma}({\it\gamma}-1)M_{IS}^{2}=0,\end{eqnarray}$$

where ${\it\xi}=p_{W}/p_{IS}$ , $M_{IS}$ can be calculated from the applicable $P_{IS}$ , and ${\it\gamma}$ is the ratio of specific heats. Thus, we have the scaled quantities: $t^{\ast }\equiv t/{\it\tau}$ , $X\equiv x/l_{0}$ , and $L\equiv l/l_{0}$ where $l$ is the curtain thickness at any time $t$ after shock impact. For pressures, of principal importance are the incident and reflected shock pressure ratios, $P_{IS}\equiv p_{IS}/p_{0}$ and $P_{RS}\equiv p_{RS}/p_{0}$ respectively. The complete pressure transients we present in terms of $P\equiv p/p_{0}$ . All displacements are relative to the initial UF of the curtain, and all times are referred to the instant of shock impact.

3.1 Pressure transients and 2D wave shapes

We use representative pressure transients (figure 8) to illustrate the main types of event in the curtain–shock interaction. These are: (i) reflection/transmission of shock waves following impact on the curtain; (ii) decay/surge of blast pressures upstream/downstream of the curtain as the curtain’s permeability increases due to dispersal; (iii) cloud passage by the downstream pressure-measuring stations; and (iv) shaping and speeds of the reflected-shock and transmitted-contact waves, both generated upon shock impact as a result of rapid change in acoustic impedance. The following explanations are to be understood with the help of figures 8–10.

Figure 8. A sample of complete pressure transients: (a) M0.44, (b) M0.74, (c) M0.92, (d) M1.23. The marked ‘uniform blast’ period refers to the flow condition at the curtain as discussed in the text. ‘CA’ stands for cloud arrival. Red/blue/green refer to sensor positions P1, P2, P3, whose locations are shown in figure 4.

Figure 9. Variation of reflected shock pressure ratios with strength of the incident shock. Here $P_{RS1}$ is reflected shock as measured on P1 and $P_{RS}$ is reflected shock at the curtain position (see table 2 and associated text). The uncertainty bounds on $P_{RS}$ correspond to the shock velocity values obtained by measuring translation from the left/right boundaries of the finite-thickness shock to the middle of the much sharpened shock on the next frame (see figure 10).

Figure 10. A sample visualization of reflected shock and transmitted contact waves in the early stages of shock–cloud interaction: (a–c) M0.74, $t=0$ ; 0.3; 0.5 ms; (d–f) M0.92, $t=0$ ; 0.2; 0.3 ms; (g–i) M1.19, $t=0;0.2;0.3$  ms; (j–l) M1.23, $t=0;0.1;0.2$  ms. All images show 15 cm from top to bottom. There is an absolute uncertainty in time ( $\pm 50~{\rm\mu}\text{s}$ ) due to the camera frame rate, but the relative uncertainty between successive frames, for computing speeds, is negligible.

1. (i) Reflection/transmission of shock waves. Arrival times at the upstream measuring station (P1), embody the competition between compressive transmission to the upstream and convection by the flow to the downstream. This competition is expressed by the net speed of the reflected wave: $u_{RS}$ – $u_{IC}$ . We can see that in scaled time the arrival of the reflected shock is delayed with increasing flow Mach number (figure 8), although in real time (see characteristic times in table 1) the delay is not quite as much. In the transonic run M0.92, the net speed becomes small enough that the reflected shock is not seen at P1 even well past the constant-blast time (figure 8 c). At the supersonic condition, run M1.23, the incident contact follows closely the incident shock and gets to P1 before the reflected shock; in this case, the high-amplitude reflected shock (see below) is able to reach P1, but only after it is partially quenched, as it has to pass through the incident contact (with very cold helium behind it) before reaching P1. As we demonstrate below, as a result of this quenching, the registered pressure at P1 (figure 8 d) is much lower than the actual reflected pressure at the curtain position.

   The amplitudes of reflected waves (figure 9) increase with increasing strength of the incident shock, in ‘parallel’ with what would be expected in a reflection from a solid wall. The approach is within 85–75 % as the $P_{IS}$ varies from 1.5 to 3. For the supersonic (helium-driven) runs M1.19 and M1.23 the reflected shock is significantly quenched by the contact as noted above. Finally, in comparison with the reflected shock, the strength of transmitted waves is seen to be rather modest and increase rather slowly with flow Mach number (figure 8).

2. (ii) Decay/surge of blast pressures. As we will see in the next subsection, the curtains expand rapidly. The so-obtained increase in permeability is evidenced by the decay in the succession of reflected pressure waves and surge of transmitted waves as seen in figure 8. Note that in the scaled time of this figure these expansion processes and their effects on pressure responses both upstream and downstream of the curtain appear to be similar over a time frame of $t^{\ast }<1.5$ . In the next subsection, we will see that this is the time frame for the main (accelerated) expansion regime. These decay/surge transients provide a sensitive measure of how well curtain dispersal is reflected into the curtain permeability in numerical simulations of such events.

3. (iii) Cloud passage by the pressure-measuring stations. Run M0.74 shows clearly the cloud arrival at both positions, P2 and P3 (figure 8 b). At P3, the cloud is more dispersed and this results to more violent oscillations of the pressure signal. At P2, we can see the whole event, as evidenced by the ‘passing by’ of the pressure gradient found within the cloud. We can see these same signatures, albeit more intense, at the higher Mach number run M0.92 (figure 8 c), while at the still higher M1.23, by the time the cloud arrives at P2 and P3, it is significantly dispersed and sustains no pressure gradient. However the noise associated with cloud passage is quite visible. In scaled time, these (cloud-passage) events are reasonably coincident. In the next subsection we explain why this is so.

4. (iv) Shaping and speed of the contact and reflected-shock waves. The vertical volume fraction gradient, and associated acoustic impedance as well as permeability variations were directly visualized through the reflected shock and transmitted contact waves. A sample of such results is provided in figure 10. The reflected shock is stronger in the upper parts, and therefore it travels faster, while the contact permeates faster through the lower parts. The transmitted shocks are of much lower strength, rather elusive in that they transit through the field of view very rapidly, and not as interesting for purposes of a general discussion. However, they are crucial in providing ground-truth in assessing numerical simulations; these data are shown in figure 8.

   The measured speeds of shocks and contacts as obtained from the images of figure 10 are summarized in table 2, which also shows the translation of these speeds to shock amplitudes. The method of estimating uncertainty is explained in the caption of figure 9. In run M1.19, the shock appeared in three frames, which allows us to understand the direction and roughly the magnitude of changes with time. In table 2 we see that the shock amplitude tends to decrease with time, while the contact speed tends to increase. Likely, they reflect changes of cloud permeability with time. These results of relatively small, but clear variations with time provide important benchmarks for numerical simulations. In figure 9 we can see that the reflected shock at the curtain position is quite close (typically within 10–20 %) from the value resulting from reflection off a rigid wall.

Table 2. Measured RS and TC wave speeds from the visualizations of figure 10. For run M1.19 the reflected shock appears in three consecutive frames (the third frame is not included in figure 10); the two entries correspond to the two velocity measurements between successive images. Transmitted contact velocities $u_{TC}$ for runs (M0.57, M0.74) are $(85,136)~\text{m}~\text{s}^{-1}$ .

3.2 Curtain displacements and cloud morphologies

The topological evolutions of the dispersing clouds are quite self-similar across all flow conditions: as illustrated in figure 11, only the rates of expansion vary with incident shock strength. Note that the UF front tilts, but only slightly, while the DF exhibits a differential expansion, from top to bottom by about 20 %. Superficially, this might suggest that the rate of expansion is favoured (in a quasi-1D view) by decreasing particle volume fraction, but this would be in conflict at the limit of dilute dispersions that yield no significant dynamic expansion. Rather, we expect that cloud permeability variations, as noted already, and associated cross-flow, pressure-rebalancing phenomena, would impose effectively higher/lower pressures at the bottom/top elevations of the upstream boundary of the cloud, thus biasing the accelerations (and expansions) in the manner seen in figure 11. As an interesting aside, we note the small waviness at the DF. It is limited, and oblique visualizations did not reveal anything else remarkable about it.

Figure 11. Cloud morphologies at position HS1, at near full coverage of the available field of view, which is smaller for the high-pressure runs due to additional structural supports needed for the channel walls: (a–d) runs M0.29, M0.44, M0.57, M0.74, at 10.0, 7.4, 5.6, 4.0 ms, respectively; (e,f) runs M0.92, M1.19, M1.23 at 1.6, 1.1, 1.0 ms, respectively. The white line on the first frame represents a length scale of 1 cm, which is the same for all frames (all images show 16 cm from top to bottom). The full videos can be found as supplemental material available at http://dx.doi.org/10.1017/jfm.2016.97.

Figure 12. Trajectories of the upstream/downstream fronts (at curtain mid-height) in dimensional (a) and scaled (b) coordinates.

The UF/DF positions with time were quantified visually, for the mid-height elevation, by two experienced individuals working independently. At position HS1, at early times, the fronts are sharp (as in figure 11), and a front position was recognized, and marked, as the point where no light could be seen coming through the cloud (from Monte Carlo simulations we find that a 2 % volume fraction is sufficient to block all light). At later times, when the fronts become somewhat diffuse, and we can count particles, we mark fronts so that they leave 1 % of the volume on the outside; that is, on a horizontal slit 1 cm high, total ${\sim}$ 6000 particles, we leave 30 particles each on the outside of the two fronts. The so-obtained data are summarized in figure 12. Yet more revealing is a plot of curtain thickness against time squared (figure 13 a): a constant acceleration period is observed, which in scaled time is seen to last for up to $t^{\ast }\sim 1.0$ (figure 13 b). During this time the cloud expands with a constant scaled acceleration of ${\sim}$ 3, and after a rather quick transition, it expands with a constant scaled velocity of 3.5 (figure 14). Thus, we have in general the cloud-expansion laws:

(3.3) $$\begin{eqnarray}\displaystyle \text{Constant acceleration regime (CAR):}\quad L\sim 1+1.5(t^{\ast })^{2}\quad (t^{\ast }<1.4). & & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle \text{Constant velocity regime (CVR):}\quad L\sim 2.5+3.5(t^{\ast }-1)\quad (t^{\ast }>1). & & \displaystyle\end{eqnarray}$$

In the transition interval $1<t^{\ast }<1.4$ both cloud lengths and expansion velocities are nearly the same. The longer-term data obtained by reconstructing whole clouds from the video images in stations HS2 and HS3 (see appendix A) further support (3.4). They are not shown in figure 14 because the much larger scale required for them would diminish clarity in the important transition period. The method of assembling these composite clouds is described in appendix A, and it has been verified by comparison with a case that the whole cloud at HS2 happened to be captured in just one frame.

Figure 13. The constant-acceleration regime of cloud expansion, in dimensional (a) and scaled (b) coordinates.

Figure 14. The linear-growth regime of cloud expansion.

Returning to a couple of remarks we made in § 3.1, we can see the consistency, in scaled time, between cloud-expansion processes and pressure gradients sustained therein: ‘in the scaled time of this (figure 8) these expansion processes and their effects on pressure responses both upstream and downstream of the curtain appear to be similar over a time frame of $t^{\ast }<1.5$ ’ and ‘in scaled time these (cloud-passage) events are reasonably coincident’.