Mechanics of the influx phase in the jet regurgitant mode of a powered resonance tube

Abstract

Time-resolved visualisation of shock wave motion within a powered resonant tube (PRT) is presented for the regurgitant mode of operation. Shock position and velocity are measured as functions of both time and space from ultra-high-speed schlieren visualisations. The shock wave velocity is seen to vary across the resonator length for both the incident and reflected waves. Three mechanisms are explored as explanations for the variation in velocity: change in local fluid velocity, variation in shock strength and variations in local temperature. For the incident wave, local fluid velocity and shock strength are extracted from the data and both are demonstrated to contribute to the observed variation, with a non-trivial remainder likely explained by variation in temperature.

Keywords

Hartmann-Sprenger tube cavity resonance powered resonance tube shock wave velocity jet regurgitant mode

Introduction

A powered resonance tube (PRT), in its simplest form, is a closed-end cavity which is excited by an air jet. This cavity then acts as a resonator, which may produce tones at high amplitudes over a wide range of frequencies.^1,2 Due to their lack of moving parts, and broad range of available resonance-frequencies, PRTs have been proposed as a novel form of active flow control.^3–5 Suggested applications of the PRT include cavity tone suppression,⁶ rocket engine ignition using resonance^7,8 and noise suppression from jet-ground impingement tones in STOVL.⁹ A lack of understanding regarding the fundamental mechanisms that govern the resonance process remains a barrier to such applications.

The first discovery of PRTs is generally credited to Hartmann in 1918.^10–12 The device was known as the Hartmann whistle until 1954 when Sprenger¹³ discovered heating effects and the device became known as a Hartmann-Sprenger tube.¹⁴ The resonance mechanics of a PRT were first discerned by Hartmann¹¹ who found that when a supersonic jet was used to excite the tube, the resonance was associated with an oscillatory motion in the detached shock at the mouth of the resonator tube. Later, Smith and Powell¹⁵ showed that this resonance occurred when the mouth of the resonator was placed in the ‘unstable’ region of the axial jet, defined by the regions where the pressure in the jet is rising.¹⁶ The resonance associated with stable and unstable regions of the jet were found by Sarohia and Back¹⁷ to occur in three distinct modes of operation. The three resonance modes were the jet instability mode, the jet regurgitant mode^13,16–18 and the jet screech mode.^3,17,19 Transition between each of the three modes is known to be governed by experimental parameters.^19,20

The regurgitant mode of the PRT involves the periodic filling and emptying of the resonator cavity. The influx phase involves a series of compression waves travelling down the resonator tube, coalescing to form a shock wave.²¹ The shock reflects off the end wall of the resonator and travels back towards the resonator mouth. Upon exiting the resonator, an expansion wave travels downstream within the resonator, marking the end of the influx phase. During the efflux phase, fluid within the tube begins to exit the resonator, driven by the reduced pressure behind the travelling expansion wave. This results in an interface forming between the flow from the axial jet and the flow from the resonator tube, forcing both to travel radially away from the resonator mouth. Figure 1 is a schematic of influx phase and transition to efflux phase of the regurgitant mode. Figure 2 is an example of the PRT in the influx phase and efflux phase of the regurgitant mode, taken with a low-speed, high-resolution camera in the absence of Perspex shielding.

The frequency of a PRT is broadly determined by the position of the shock cells (and associated unstable regions) in the supersonic jet, which changes as the NPR is increased.³ When the mouth of the resonator is located in the region of increasing pressure in the jet, the lower frequency jet regurgitant mode dominates.^16,18 When the mouth of the resonator is in a region of decreasing pressure in the jet, then the higher frequency jet screech mode will dominate.¹⁹

The mechanics of the internal motions within the tube have been known to be non-linear for some time.²² Murugappan and Gutmark¹⁹ studied the wave propagation in the regurgitant mode and found that the time spent in expansion and compression phases was not equal. This indicated that an incomplete evacuation and filling of the cavity occurred during the regurgitant mode of the PRT. Sarohia and Back¹⁷ measured the average speed of the downstream compression wave and upstream compression wave using the deviation of two laser beams. They found that average reflected shock speed was lower than the incident shock wave speed. Beyond this, a detailed understanding of the internal wave motion in the PRT resonator remains elusive.

This paper presents the mechanics of the regurgitant mode of the PRT, particularly the dynamics of the shock wave in the influx phase. We use time-resolved schlieren images to examine the PRT regurgitant mode operation. We measure the velocity of the shock wave within the resonator of the PRT during the influx phase. We attempt to explain the observed shock velocities in the PRT resonator by examining the shock strength, fluid velocity and the temperature.

Methodology

Experimental facility

Experiments were performed in the Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC) supersonic jet facility. A schematic of the facility is shown in Figure 3. Air enters the plenum chamber below the nozzle from a continuous high pressure air source supplied to the jet rig. Inside the plenum, a series of screens and honeycomb are used to remove large-scale turbulent structures from the flow before entering the nozzle. A converging circular nozzle was utilised in this investigation. The nozzle had a diameter (D) of 15 mm with a lip thickness of 5 mm and contraction ratio of 240. The stagnation pressure of the plenum chamber was measured using an RS-461 pressure transducer with a range of 0–10 bar and an accuracy of ±0.25%. For the present setup, the nozzle pressure ratio (NPR) is defined as the ratio between the pressure of the plenum chamber P₀ and the pressure of the ambient atmosphere P_a and related by

NPR = \frac{P_{0}}{P_{a}}

(1)

The PRT experimental setup is shown in Figure 3(right). The PRT resonator consisted of a square-bore resonator chamber. The resonator is mounted at a stand-off distance (X_r) of 2D from the nozzle exit. The resonator square cavity bore D_c is equal to the nozzle exit diameter D. The length of the resonator cavity L is 5.20D, whilst the viewable resonator cavity length is 5.00D. The exit of the resonator contains a stainless steel plate to hold the optical glass in place, which restricts optical access. The experimental parameters are summarised in Table 1.

Table 1.

Experimental parameters for the present investigation.

Parameter	Symbol	Value	Non-dimensional
Resonator stand-off distance	X_r	30 mm	2.00D
Resonator cavity diameter	D_c	15 mm	1.00D
Converging nozzle diameter	D	15 mm	1.00D
Resonator cavity length	L	78 mm	5.20D
Resonator viewable cavity length	L_v	75 mm	5.00D
Microphone distance	L_m	830 mm	55.33D
Nozzle pressure ratio range	NPR	2.25–4.00	–
Laboratory ambient temperature	$T_{\infty}$	288 K	–

Note: Values are non-dimensionalised based on the nozzle exit diameter D.

The PRT assembly consists of two stainless steel plate walls and two quartz optical glass walls surrounded by a stainless steel base plate and end plate. The glass walls had channels cut behind them and filled with Bostik RTV-936 silicon. This was to ensure that the glass walls were not able to move and maintained an airtight seal inside the tube during the intense vibrations experienced by the resonator. The entire assembly was mounted on a plate suspended between two support structures located on either side of the jet facility frame. The labelled facility diagram is shown in Figure 4. To protect the surrounding equipment and operator from the high amplitude acoustic tones, MDF boards and perspex sheets were installed around the jet facility.

Experimental measurement techniques

Schlieren measurements were taken using a Toepler Z-Type schlieren system^24,25 with mirrors of focal length 2032 mm. Images with high temporal resolution were obtained using a Shimadzu HPV-1 camera, illuminated by a Metz Mecablitz flash. The camera has a resolution of 312 × 260 pixels and captures a set of 102 images. The present study captured images at an acquisition speed of 125,000 frames per second with an exposure time of 0.25 µs. Density gradients in both x and y, $\partial ρ / \partial x$ and $\partial ρ / \partial y$ respectively, were examined by varying the orientation of a standard razor blade used to provide a spatial cut-off. The extreme amplitude of the acoustic tones necessitated enclosing the facility in thick sheets of perspex to avoid damage to camera and computer equipment; the perspex dramatically reduced the quality of the schlieren images. The dark band associated with the shock wave visible in the images presented here likely has contributions from both schlieren and shadowgraph effects.

Far-field acoustic measurements were obtained using a G.R.A.S. Type 46BE 1/4 inch microphone with a frequency range of 4 Hz–80 kHz. The microphone was located 55.3D from the resonator entrance and mounted to the jet rig frame. The microphone was suspended on a magnetic holder and pointed directly at the resonator entrance. The location of the microphone was the centre of the jet rig frame, in the same plane as the nozzle and resonator. This is shown in Figure 4. All acoustic measurements obtained a recording size of $5 \times 10^{5}$ samples at a rate of 250 kHz. The pressure signals were recorded on a 12-bit National Instruments DAQ. The environment cannot be considered anechoic, and hence acoustic measurements are only used to identify frequencies and make qualitative comparisons. Acoustic measurements given throughout this work are non-dimensionalised using the Strouhal number. The Strouhal number is defined as

St = \frac{fD}{U_{j}}

(2)

where D and U_j are the circular jet exit diameter and the ideally expanded jet exit velocity, and f is the tone frequency.

Results

Acoustic characterisation

A far-field acoustic sweep was performed through the NPR range and at the experimental conditions given in Table 1. Acoustic spectra were taken at an NPR spacing of 0.05 through the range. Results for the sweep are shown in a waterfall plot in Figure 5(left). Figure 5(left) indicates that as expected, there is minimal variation in tonal frequency with changes in NPR, except when a switch between regurgitant and screech mode occurs. High-intensity tones in the NPR range from 2.25 to 3.3 at Strouhal numbers of $St = 0.4, 0.53$ , corresponding to tones associated with the screech mode of resonance. For $3.4 < NPR < 4.0$ , the highest amplitude tone has a Strouhal number $St \approx 0.03$ . From the schlieren visualisations presented later, it is clear that this tone is associated with the regurgitant mode of operation, consistent with earlier findings.¹⁹ Figure 5(left) suggests that there are also a large number of harmonics of this fundamental tone. The optical component of the study considers a single case in the regurgitant mode, at NPR = 3.4.

A short-time Fourier transform for the selected case at NPR = 3.4 is provided as a waterfall plot in Figure 5(right). Figure 5(left) shows the highest amplitude tone occurred at a Strouhal number of $St \approx 0.03$ . The tone amplitude and frequency are observed to vary intermittently as a function of time in Figure 5(right), indicating a degree of unsteadiness in the fundamental resonance process. The tones at higher frequencies ( $St \approx 0.18, 0.21$ ), which may be harmonics of the fundamental, or indicative of a resonance loop closed outside the tube, are even more intermittent than the fundamental.

High-speed imaging

For the regurgitant mode at NPR = 3.4, 20 image sets of 102 images each were obtained, totalling 2040 images. Images were taken of the density gradient $\partial ρ / \partial x$ . An examplar image sequence is shown in Figure 6, showing the influx phase of the resonance cycle. The time-series can be viewed in full as Supplementary Video 1. Progression of a shock wave downstream through the resonator is shown in Figure 6(a) and (b). This wave reflects off the end wall of the resonator and travels back upstream towards the open end (Figure 6(c) and (d)). As the wave moves towards the resonator mouth, a progression of reflected compression waves travels toward and coalesces with the leading wave (Figure 6(e) and (f)). Finally, the influx phase of the PRT resonator ends with the emission of the shock wave from the open cavity (Figure 6(g) and (h)). Also evident are the flow features at the resonator mouth such as turbulent structures and the shock cell, previously identified by Sarohia and Back.¹⁷

In Figure 6(a), at the beginning of the influx phase, radial flow away from the resonator mouth is evident through the turbulent flow structures in the Y direction at approximately $X / D = 2$ . Radial flow continues for some time until the structures reduce in size and gradually decay (Figure 6(b) to (d)). As the incident shock wave reaches the wall of the resonator, radial flow evidence at the mouth reduces significantly (Figure 6(e) and (f)). Little to no radial flow is seen, as the reflected wave travels back to the resonator mouth (Figure 6(g) and (h)).

Due to limitations in the camera architecture, only 102 images can be obtained in a sequence, insufficient to capture a full resonance cycle. Figure 7 presents an image sequence whose beginning overlaps with the end of Figure 6. The time-series shown in Figure 7 can be viewed in full as Supplementary Video 2. Figure 7(a) continues where Figure 6(g) and (h) ceased. The shock exits the resonator in Figure 7(b) and interacts with the jet flow from the supersonic jet. The exiting shock wave causes a reflected expansion wave to travel downstream in the resonator. The expansion wave reduces pressure and flow exits the resonator. This leads to two jet flows impinging on each other (Figure 7(c)), travelling radially away from the resonator mouth (Figure 7(d)). Further images in the sequence (Figure 7(e) and (f)) show evidence of greater radial flow away from the resonator mouth. This is indicated by the large turbulent structures seen moving away from the resonator mouth.

The formation of coherent structures through receptivity at the nozzle lip is seen in Figure 7. This process has been known to be an integral part of the feedback loops in both screeching and impinging supersonic jets.^25–27 The process can be seen to occur in Figure 7(c) and (d). The process is seen more clearly in Supplementary Video 2. In Figure 7(c), evidence of an upstream wave created by the emission of the shock from the resonator mouth can be seen at approximately $X / D = 0.6$ . The formation of coherent structures as a result of the impingement and subsequent receptivity of the wave on the flow from the nozzle lip can be seen in Figure 7(d). Coherent structures can be seen forming in the shear layer of the impinging supersonic jet at $X / D = 0.6$ and $X / D = 0.9$ . These appear to be larger than the structures forming from the nozzle lip in any of the other cases in either Figure 6 or 7.

Shock position–time distributions

The shock appears in the images as a dark vertical line; a consideration of the trajectory of this line in X–t space allows for the extraction of both wave position and velocity. The image intensity is averaged in the Y direction across the width of the tube, and this intensity is plotted as a function of both axial position and time, six examples of which are provided in Figure 8. That the process is highly unsteady is immediately apparent from this selection of X–t plots; while there are clear similarities between the cases, no two are exactly alike and some exhibit significantly different behaviour to the rest. The variation seen in Figure 8 may be due to the intermittency in the resonant process, as demonstrated in Figure 5(right). Figure 8 shows that the wave motion associated with the influx phase of the PRT regurgitant mode is highly non-linear. As expected, the incident shock wave (positive gradient) travels down the resonator before reflecting off the end wall and travelling towards the resonator mouth (negative gradient). The curvature of the trajectories of both the incident and reflected waves makes clear that wave velocity is a function of position within the tube. Wave coalescence is observed not only to happen for the incident wave but also for the reflected wave; a train of compression waves (visible as weaker lines), are observed to catch up to the reflected wave. The degree and position of this coalescence are observed to vary significantly between the different examples. Figure 8(a), (b) and (f) appears to show similar shock trajectories, despite having different coalescence patterns. Figure 8(c) to (e) highlights the variance of the phenomenon. Iwamoto and Deckker²⁸ studied the non-linear aspects of a PRT using a hydraulic analogy. They showed that the wave diagrams for different resonator mouth locations can change significantly. Figure 8 indicates that the wave diagrams for a fixed resonator mouth position can also show significant shot-to-shot variation.

In addition to the motion of the shock, Figure 8 also yields the trajectory of small turbulent eddies within the tube. These trajectories are generally of constant gradient prior to interaction with the shock, indicating that the Lagrangrian velocity of a fluid packet within the tube is largely invariant with space other than the gas-dynamic effect of the shock waves. Following a line of constant gradient (such as the one beginning at approximately $t / T = 0.075$ in Figure 8(b)), the interaction of turbulent flow structures with the shock and compression waves is seen. These structures were first noted by Sarohia and Back.¹⁷ After interacting with the shock wave, the gradient of the line decreases, indicating that the velocity of the turbulent structures decreases in the laboratory reference frame. The gradient further decreases after interaction with the train of reflected compression waves behind the leading shock. The fluid then approaches a state of rest until the reflected expansion wave travels downstream and induces upstream flow in the tube.

This reflected expansion wave is most clearly visible in Figure 8(b), evident from the moment the shock wave exits the resonator cavity at $t / T = 0.4$ . This line travels back towards the resonator wall and is seen to induce an upstream motion in the fluid within the tube between $t / T = 0.5$ and 0.6 at approximately $L_{v} / D = 3.5$ .

Shock velocity variation

Based on the position and time distributions, an edge-detection method was used to track the maximum intensity of the shock wave front along the resonator. Considering the inter-frame timing, the shock velocity was then computed. A cross-correlation technique was excluded, since the eddies in the surrounding fluid move at different velocities to the shock, potentially creating multiple correlation peaks. The error associated with this method is a function of the uncertainty in the position of the shock wave front and the camera frame rate. The uncertainty in the velocity is given by

Δ (v) = v \times \sqrt{{(\frac{ϵ_{s}}{Δ (s)})}^{2} + {(\frac{ϵ_{t}}{Δ (t)})}^{2}}

(3)

where v is the velocity of the shock wave,

Δ (s) = s_{2} - s_{1}

is the measured change in the position of the shock wave from s₁ to s₂,

Δ (t)

is the change in inter-frame timing,

ϵ_{s} = \sqrt{{(u (s_{2}))}^{2} + {(u (s_{1}))}^{2}}

is the uncertainty in the measurement of the change in position of the shock wave and ϵ_t is the uncertainty in the measurement of the change in the inter-frame timing. The position of the shock wave front, denoted by s₁ and s₂, could be measured to ±0.5 pixels

(\approx 0.25 mm)

. To determine the uncertainty in inter-frame timing, the clock rate of the camera is required. Vollmer and Möllmann²⁹ state clock frequencies can reach 800 MHz in modern high-speed cameras. A conservative estimate of 100 MHz is used, giving a timing uncertainty ϵ_t of 10 ns.

Figure 9 presents the shock velocity v_p across the resonator for four datasets, where the image sequence contains the full propagation of both the incident and reflected wave. The incident wave, travelling downstream in the resonator tube, is represented by black circles (o) whilst the reflected wave, travelling upstream is given as blue crosses (x). The shock is seen to enter the tube at the lowest velocity, monotonically accelerating during its propagation down the tube, reaching maximum velocity prior to reflection from the closed end wall. Variation is evident in the shape of the velocity distribution of the incident wave in Figure 9; this is particularly evident when comparing Figure 9(a) with Figure 9(b) to (d). The initial acceleration of the incident shock wave in Figure 9(b) to (d) is variable, whereas for Figure 9(a), the initial acceleration appears constant. Upon reflection from the end wall of the resonator, the shock immediately loses velocity as seen in Figure 9. This is consistent with the behaviour expected of a shock wave reflecting from the closed end of a shock tube in the laboratory reference frame.³⁰

The reflected wave in Figure 9 initially decreases in velocity, before increasing velocity again. This increase continues until the shock approaches the resonator mouth at which point the velocity begins to decrease before the wave exits the resonator. This behaviour is seen in Figure 9(b) to (d) but not seen in Figure 9(a). Figure 9 also shows that the velocity of the shock exiting the resonator tube is higher than the velocity of the wave that initially enters the tube. This is consistent with the result found by Sarohia and Back.¹⁷ The reflected wave was shown in Figure 6 and previous work²² to include the coalescence of other compression waves that reflect from the end wall of the resonator. It is thus surprising to see that the velocity of the reflected wave decreases between the end wall of the resonator and the mouth of the resonator. Error estimates, given as an envelope in Figure 9, were computed using equation (3). These represent the upper and lower bounds of the velocity values based on the experimental measurements. Determining the position of the shock can be difficult when it is close to the wall or mouth. Thus, error bounds for the end points, near the resonator mouth and PRT end wall, have been excluded. An average velocity distribution for all the measured image sets is now considered to see if the velocity behaviour persists.

The average velocity distribution for the regurgitant PRT phenomenon is shown in Figure 10. The distribution shown in Figure 10 is a composite of all 2040 images taken in this study, averaged based on the position of the shock wave within the PRT resonator. Error bars, representing the 95% t-based confidence interval for the data, were constructed using the following expression

C I_{95} = \bar{X} \pm t (S / \sqrt{n})

(4)

where

\bar{X}

is the mean of the sample, t is the critical value of the two-tailed t distribution for a 95% confidence interval, S is the standard error of the sample with a size n. The varying number of samples and the non-linear nature of the phenomenon has led to the varying sizes in the error bars. However, the phenomenon still demonstrates remarkably consistent behaviour; the greatest variation is seen close to the resonator end wall (represented by the red dashed vertical line). Variation is also considerably greater in the reflected wave when compared to the incident wave, consistent with the observations in Figure 8. The vast majority of the features captured in the raw distributions in Figure 9 are still shown in Figure 10, including the consistent shape of the incident and reflected waves, the slowing of the reflected wave at reflection and the deceleration (

3 < L_{V} / D < 5

) and subsequent acceleration (

1 < L_{V} / D < 3

) of the reflected shock wave in the resonator tube. The velocity of the reflected wave at the exit of the resonator is greater than the velocity of the incident wave as it enters the tube. Finally, the unusual behaviour involving the slowing of the reflected wave travelling upstream in the resonator tube (

0 < L_{V} / D < 1

) is preserved in the average data.

Discussion

While variation in the wave velocity within the PRT would generally be expected, the degree is perhaps surprising, as is the non-monotonic nature of the reflected wave velocity. To determine the cause of the variation in the shock wave speed through the PRT resonator requires a consideration of the mechanisms by which the shock wave speed can change. There are three possible mechanisms to explain the variation in the shock velocity as measured in the laboratory reference frame:

Variation in velocity of the fluid through which the shock is propagating

Variation in shock strength, i.e. shock Mach number

Variation in the speed of sound due to variation in the fluid temperature

In the following sections, we consider one exemplary case to estimate the relative contribution of these three mechanisms. At the outset, it should be emphasised that these calculations require quantities estimated from the schlieren image sequences, which will have a significant (and unquantifiable) uncertainty associated with them. The purpose of this section is simply to provide first estimates of proportionality between these mechanisms.

Variation in local fluid velocity

The velocity of the fluid ahead and behind the shock wave can be estimated based on gradient of lines in the position and time distributions. These lines represent small-scale turbulent eddies in the flow; these eddies appear to be small in size and they are assumed to approximate the fluid velocity within the tube.³¹ The gradient of the lines were measured by estimating the angle between the line signifying the path of the turbulent eddies and the horizontal. This was measured directly ahead and behind the shock wave front and used to determine the velocity of the flow. To assist with the measurement of velocity, an unsharp mask filter was applied to the raw images. Figure 11(left) shows an example of the raw position and time distribution and the same distribution with an unsharp mask filter applied Figure 11(right). The unsharp mask was applied with a radius of 3 pixels and a normalised strength of 0.6. This enhanced the contrast of the image and allowed for more accurate measurement of the angle at which the fluid approached and travelled away from the shock wave.

Measurement of the velocity was only possible at locations where eddies were imaged in the tube. The measurement of velocity ahead and behind the reflected wave was not possible due to the trailing compression waves. The measured velocity of the fluid v_f ahead (o) and behind (□) the incident shock wave in the laboratory reference frame is given in Figure 12(a). The incident wave velocity is transformed to the reference frame of the shock wave by subtracting the velocity of the shock from the fluid velocity. The shock reference frame velocity is given in Figure 12(b). The velocity ahead of the incident shock increases by $0.15 c_{0}$ across the measured region. No error bars have been included in Figure 12, as the degree of uncertainty in this crude velocity estimation technique is unknown. If all the variation in the Eulerian shock velocity could be explained by variations in the fluid velocity, the values in Figure 12(b) should be invariant with spatial position. Thus, variation in the fluid velocity alone is not able to explain the variation in the incident shock wave, although it may contribute a component to it.

To determine the proportion which the fluid velocity variation contributes to the overall shock velocity changes, a measurement of the total velocity change for the shock is required. The shock velocity and position diagram corresponding to the experimental data in Figure 11 are provided in Figure 9(c); as shown in Figure 9(c), the velocity change is $0.29 c_{0}$ across the region $1.7 < L_{v} / D < 3.7$ , where the schlieren images permit the inference of fluid velocity. This suggests that the variation of the fluid velocity ahead of the incident shock wave is responsible for $\approx$ 50% of the velocity change in this particular case.

Variation in shock strength

To determine if there is any variation in the shock strength across the resonator tube, shock Mach number must be extracted from the measured fluid velocity. The characteristic Mach number $M^{*}$ can be found from the fluid velocity ahead u₁ and behind u₂ the shock wave through

M^{* 2} = \frac{u_{1}}{u_{2}}

(5)

The shock Mach number can then be found from the characteristic Mach number as

M^{2} = \frac{2}{(γ + 1) / M^{* 2} - (γ - 1)}

(6)

where γ is the ratio of specific heats. These relations arise from the energy expressions for steady, inviscid, adiabatic flow. They are derived from the conservation of enthalpy across a normal shock and assume a calorically perfect gas. The shock here is treated as quasi-steady, although this is clearly only an approximation; it is important to keep these limitations in mind when interpreting the results of the Mach number distribution in the PRT resonator tube.

Figure 13 gives the Mach number distribution for the incident shock wave in the PRT resonator. A linear fit to the data is given (-.) and the end wall of the resonator is also indicated (-). The general trend is for the shock Mach number to increase as the shock wave travels further down the resonator tube. This is consistent with observations of the thickness and darkness of the shock wave in the schlieren time sequence (Figure 6), both of which appear to increase (the line gets both thicker and darker) as the wave travels inside the resonator tube. This change in the apparent nature of the wave is most evident at the mouth of the tube, and based on the schlieren visualisations, it is suggested that the trendline in Figure 13 does not capture the true non-linearity of the initial wave formation process. The monotonically increasing Mach number indicates that the coalescence of compression waves plays a key role in the incident wave as well as the reflected wave.

The Mach number increase is relatively small, changing by 0.07 across the measured range. Directly estimating how much this change in Mach number contributes to the observed change in Eulerian shock velocity requires an assumption regarding the local soundspeed. The present experimental setup does not allow for the measurement of temperature within the tube; however, the sensitivity of the contribution of shock strength variation to temperature T can be expressed as

Shock Strength Contribution (%) = 1.43 \sqrt{T}

(7)

Assuming that the tube is unlikely to be cooler than the ambient temperature in steady-state operation, this suggests the minimum contribution to the velocity change from the change in Mach number is $\approx$ 25%.

Measurement of the reflected shock Mach number is not possible due to the inability to obtain velocity measurements. Qualitative observations of the schlieren images indicate that the Mach number may continue to increase, as the reflected shock wave travels towards the resonator mouth. However, further work is required to confirm this.

Variation in temperature

The third possible mechanism by which the velocity can vary within the resonator tube is the variation in the local temperature in the fluid ahead of the shock. Given the inability to measure temperature in this facility, and the additional assumptions regarding the nature of the flow required to calculate it from the quantities inferred from the schlieren, no direct estimation is presented here. However, there are only three likely mechanisms by which the Eulerian velocity of the incident shock wave could change, and thus the relative contribution due to variation in temperature effects can be estimated. Within the large uncertainty bounds associated with estimation from schlieren, the variation of the fluid velocity ahead of the shock accounted for $\approx$ 50% of the overall velocity change; the remaining contribution is a function of the variation in shock strength and variation in temperature. Given the minimum contribution from variation in the strength of the shock is $\approx$ 25%, and this would require no variation in temperature, it can be assumed that variation in fluid temperature likely is responsible for less than 25% of the overall change in Eulerian shock velocity. These are of course only initial approximations, and direct measurement of temperature or fluid velocity (or detailed numerical simulation) would be required to make a definitive statement.

Conclusions

Time-resolved visualisation of the regurgitant mode of a PRT was conducted at NPR = 3.4. Ultra high-speed schlieren imaging was used to measure the position and velocity of the shock wave through the resonator tube. The velocity of the shock wave varies as a function of position in the resonator tube. The incident shock wave increased in velocity throughout the tube. The reflected shock wave decreased in velocity immediately after reflecting off the end wall, continued to decrease for some distance back towards the open end of the tube, then increased prior to exit. The exit velocity of the reflected shock wave was lower than the initial reflected shock velocity but significantly higher than the initial incident wave velocity.

To explain the variation in the Eulerian shock wave velocity, three mechanisms were explored – variation in the local fluid velocity, variation in the shock strength and the variation in the local temperature. The variation in the local fluid velocity was measured from position and time distributions of the schlieren images. It was found that the velocity ahead of the incident shock increased by $0.15 c_{0}$ as the shock travelled down the tube. This corresponded to $\approx$ 50% of the velocity change in the incident shock over the measured range. Using the measured fluid velocity, the shock Mach number was shown to increase by 0.07, as it travelled down the resonator tube. The change in shock strength contributed to at least $\approx$ 25% of the velocity change in the incident shock. This suggests that the coalescence of compression waves plays an important role in the regurgitant mode of the PRT. The remaining velocity contribution was suggested to be due to the variation in the temperature, which appears to make the weakest contribution of the three. Further work is required to confirm the influence of the variation in the local temperature ahead of the shock wave on the measured velocity distributions.

Footnotes

Declaration of conflicting interests

The author(s) declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the Australian Research Council. This research is supported by an Australian Government Research Training Program (RTP) Scholarship.

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