Fishbone-like instability in a looped-tube thermoacoustic engine Open Access

Thermally generated spontaneous acoustic oscillations of gas such as those present in a Sondhauss or Rijke tube^1,2 and the so-called Taconis oscillations³ have attracted a lot of research attention. They belong to a group of “thermoacoustic effects” that have also been utilized in recent years as a means of energy conversion in a range of thermoacoustic engines.^4–6 Great variety of nonlinear phenomena has been observed in thermoacoustic oscillations, in both simple arrangements such as Taconis tube, or more complex thermoacoustic engine designs. Yazaki et al. observed that two different acoustic modes with incommensurate frequencies could be induced simultaneously in a Taconis tube.^3,7 The coupling and competition between these modes led to rich nonlinear phenomena, such as quasi-periodicity, frequency locking, and onset of chaos.

Swift⁴ observed a periodic onset and quenching phenomena in a standing wave engine. The engine turned on and off with a period of one hour or more when the heating power of the engine was fixed. Atchley et al.⁸ reported a quasi-periodicity in a standing wave thermoacoustic engine. The fundamental mode and the second mode were induced simultaneously. However, these two modes were not harmonic, and their frequencies were incommensurate. The combination of them showed an amplitude modulation which indicates a quasi-periodicity. Biwa et al. observed a transition from a standing wave mode to a traveling wave mode through a quasi-periodic state.⁹ Furthermore, a thermodynamic mode selection rule was proposed: the mode with the minimum increase in entropy flow has the priority to be induced among the permissible modes. Interestingly, Yu et al.¹⁰ also observed similar mode transition phenomena in a thermoacoustic engine with a similar configuration. However, the transitions between the two modes happened in an opposite way compared with those shown in Ref. 9. Additionally, the harmonics and amplitude saturation effects were also observed in thermoacoustic devices.^4,5 Most of these nonlinear phenomena can be attributed to the interaction between different acoustic modes.

More recently, a double-threshold effect has been observed in a looped-tube thermoacoustic engine by Penelet et al.¹¹ It was characterized by two separate exponential growths of acoustic amplitude. Furthermore, the influence of the acoustic field on temperature field has also been investigated theoretically.^12–14 It was reported that the acoustic amplification effect depend not only on the temperature difference between the two ends of the regenerators, but also on the temperature profiles along the regenerator and the thermal buffer tube.¹⁴ 

This letter reports a new nonlinear phenomenon observed in a looped-tube thermoacoustic engine. Quasi-periodic bursts of acoustic oscillations were observed during the engine start-up process, between the onset and the quasi-steady state oscillations. It is referred to as a fishbone-like instability, because of the shape of the envelope of acoustic oscillations. It cannot be classified within the known nonlinear phenomena observed in acoustic systems. The significance of the observations presented is twofold: On the fundamental level, they provide an interesting insight into the nonlinear nature of the thermoacoustic effect and complex systems in general. On the practical level they concern a critical issue in the design of thermoacoustic engines, i.e., a rapid start-up leading to stable operation.

The experimental apparatus is essentially a looped-tube thermoacoustic engine⁵ (cf. Fig. 1). Its resonator is made of stainless steel duct with the total length of 306 cm and I/D of 56 mm. The HHX (25 mm long) is made of Nickel-Chromium resistance wire, connected to a variable power DC supply. The CHX (42 mm long) is made using six stainless steel tubes (O/D 6 mm) protruding through the main engine duct wall and passing through a car radiator matrix fitted on the inside. Tap water is used for cooling. The regenerator (100 mm long) is made of multi-layered stainless steel mesh (30 wires per linear inch, wire diameter 0.28 mm). It has a hydraulic radius of $196μm$ and 72.7% porosity. The working gas is compressed air.

Seven type-K thermocouples are installed to measure the gas temperature along the center of the regenerator (T1 to T5) and within the HHX and CHX (Th and Tc). T1 and T5 are attached to hot and cold ends of the regenerator. T2, T3 and T4 are 33, 50 and 75 mm away from T1, respectively (the uneven spread for practical reasons). Tc is in the middle of the cold heat exchanger, while Th is about 10 mm away from T1. A microphone (PCB PIEZOTRONICS model 112A22, shown as P in Fig. 1) is used to measure the acoustic pressure. It is 147.5 cm away from the CHX along coordinate $x$⁠.

A variety of transient processes was observed, as heating power $Q$ or mean pressure $Pm$ varied. For example at $Pm=8bars$⁠, the minimum $Q$ required to start the oscillation (onset condition) was 181 W. The engine showed an onset and quenching behavior⁴ when $181<Q<225W$⁠. The engine would undergo a typical smooth onset process⁴ when $Q>381W$⁠. The fishbone-like instability was observed for $225<Q<381W$⁠. Furthermore, there was a critical $Pm$ around 5.7 bars below which it would not occur regardless of the value of $Q$⁠. A typical case of a fishbone-like instability (⁠$Q=308W$⁠, $Pm=8bars$⁠) is shown in Fig. 2.

Figures 2(a) and 2(b) show the evolution of the acoustic pressure $p1$ and temperatures, respectively, after the power supply has been turned on. The sampling rate is 2 kHz. The envelope of the acoustic oscillation resembles a fishbone structure. There appear to be three stages of transient processes: (I) from the onset until the occurrence of the bursts; (II) quasi-periodic amplitude bursts; and (III) quasi-steady state.

In Fig. 2(a), the gas oscillations start around 120 s. The acoustic amplitude grows for a short while and then is followed by quasi-periodic amplitude bursts (over 60 individual “spikes” in the fishbone structure). The pressure oscillations do not vanish completely at the end of each burst, but reach an amplitude “trough” of around 1 kPa instead. The height of the amplitude peaks grows with time, and reaches the maximum at around 1460 s, followed by an attenuation phase which ends at around 1620 s. The height of the amplitude troughs decreases slightly, and reaches a minimum at about 800 s, and then gradually increases. Finally, at around 1620 s a quasi-steady state is reached (slow amplitude growth).

A higher sampling rate of 8 kHz was used to study frequency spectra of acoustic oscillation at particular instants of interest (onset, amplitude peaks and troughs, and the quasi-steady state). The acoustic oscillation has a constant frequency of 111 Hz, corresponding to one-wavelength mode of the loop, while the harmonics have trivial power levels compared to the fundamental mode.

Furthermore, by connecting the peaks of the bursts in Fig. 2(a), one can find that the obtained line smoothly follows the envelope of the acoustic oscillation of the first stage. This implies that the first and second stages are two different variants of the same process of acoustic amplitude growth. The bursts in the second stage show a “discretized” form of the amplitude growth.

Figure 2(b) shows that quasi-periodic temperature fluctuations were recorded from thermocouples Th, T1 and T5. Comparing Fig. 2(a), their occurrence coincides with the pressure amplitude bursts. The temperature fluctuations disappear after the bursts die down. At this scale of the graph, no temperature fluctuations can be seen from thermocouples T2-T4. These observations suggest a strong interaction between quasi-periodic temperature fluctuations and the acoustic bursts, i.e., between the acoustic and temperature fields.

Figure 2(b) also shows that thermal energy transported from HHX reaches locations T2, T3 and T4 at around 200, 280 and 400 s, respectively. This indicates that before 200 s only a short section of the regenerator has a temperature gradient $∇T$ that contributes to the acoustic amplification effect. The remainder simply works as an acoustic load. As sufficient $∇T$ is set up on longer sections of the regenerator, they gradually start to contribute to the acoustic amplification.

As seen from Fig. 2, after 1620 s a slow growth in the acoustic oscillation amplitude is accompanied by a slow decrease in T1. This means that the temperature difference between the regenerator ends decreases, while the acoustic amplitude increases. This agrees with the findings of Penelet et al.¹³ that the temperature difference is not the only parameter controlling the acoustic amplification.

To study the pressure amplitude bursts in more detail, the data for periods 100–400 s and 1320–1650 s, have been plotted in Fig. 3 using an expanded time scale. Figure 3(a) shows a small exponential amplitude growth after onset (at about 123.6 s), after which the amplitude grows slowly until the first burst occurs at around 210 s. Thereafter, the height of the bursts increases with time, while the height of the amplitude troughs slowly decreases as discussed with reference to Fig. 2. This behavior is clearly different from the onset and quenching phenomena¹² (which cause a periodic stop of acoustic oscillation) and the double-threshold effect¹¹ (which has two successive exponential amplitude growths). The bursts have “quasi-periods,” which slightly increase from 14.6 to 16.8 s in Fig. 3(a). Figure 3(b) shows that Th and T1 keep rising after the onset. The occurrence and intensity of temperature fluctuations of Th and T1 correspond to those of acoustic bursts as already discussed with reference to Fig. 2. Interestingly, the peaks of Th and T1 occur almost simultaneously, but precede the peaks of acoustic bursts.

Figure 3(c) shows that a rapid attenuation of burst amplitudes corresponds to a rapid rise of the amplitude troughs. The quasi-period of bursts increases from 23.9 to 25.3 s. Figure 3(d) clearly shows temperature fluctuations in T4 and T5 during this period (in addition to Th and T1 fluctuations). Furthermore, the peaks of T1 almost correspond to the peaks of the bursts, while the peaks of Th occur earlier. The fluctuations of Th and T1 are around 60 and $20°C$⁠, respectively, while they are only around 3 and $1°C$ in T5 and T4, respectively. There is also a very small but visible fluctuation in T3, but no visible fluctuation in T2.

From Fig. 2, the quasi-period of bursts increases from 14.6 to 25.3 s in stage II. Thus their duration is three orders of magnitude higher than the acoustic oscillation period. Clearly, they are neither super-harmonics^4,5 nor sub-harmonics¹⁵ (the latter cause a “cockscomb” like oscillation in Faraday experiment).¹⁵ Therefore, the fishbone-like instability cannot be explained by the interaction of different acoustic modes.

Furthermore, the quasi-period of bursts and the time span of Stage II (or in other words, the number of spikes) both depend on the level of input heat $Q$ and mean pressure $pm$⁠. For a given $pm$⁠, the increase of $Q$ leads to a shorter stage II, and a shorter quasi-period of burst. For a given $Q$⁠, the increase of $pm$ leads to a longer stage II, and a slightly longer quasi-period of burst.

The fishbone-like instability shows an effect of a peculiar “mutual feedback mechanism” between the acoustic bursts and temperature fluctuations. The fluctuations in T1 indicate a quasi-periodic “accumulation” and “release” of thermal energy at the regenerator hot end. The thermoacoustic core involves several mechanisms of heat transfer: radiation, natural convection, heat conduction, acoustic streaming forced convection,¹³ and acoustically enhanced heat transport (AEHT).^13,16 The latter two are possible reasons for the complex interactions observed because they depend on the acoustic oscillation amplitude, unlike the others.

A very thin latex membrane (cf. a dotted line in Fig. 1) was installed near the velocity node, to suppress the Gedeon streaming.⁶ The presence of the membrane only increased the height of the first burst slightly, but did not affect the occurrence and evolution of the acoustic bursts and temperature fluctuations. Therefore a link to Gedeon Streaming was excluded.

Hence, it can be inferred that AEHT plays the key role in the apparent interaction between the acoustic and temperature fields. AEHT can be expressed as^13,16

where $A$ is gas area and $ka$ is an effective acoustically enhanced thermal conductivity. Furthermore, $ka$ is proportional to $|p1|2$ and can be expressed as¹³ 

In Eq. (2), $Cp$ is the heat capacity of working gas, $ω$ is the angular frequency, $ρm$ is the density of the working gas, $c$ is the sound speed, $σ$ is Prandtl number, $D$ is the equivalent geometrical diameter of the pores, and $δν$ is the viscous penetration depth, and it is defined as⁶ $δυ=2υ/ω$⁠. Here, $υ$ is the kinematic viscosity of working gas.

Calculations according to Eq. (2) show that $ka∼0.1kr$ at amplitude trough, $ka∼0.4kr$ at the moment when the first burst occurs, and $ka∼10kr$ at amplitude peaks. Here, $kr$ is the effective thermal conductivity of the mesh screen, and is around tenth of the thermal conductivity of stainless steel due to point contact between individual disks.⁶ It can be inferred that AEHT fluctuates in a large range during each acoustic burst, and that AEHT dominates the heat transport along the regenerator at high pressure amplitude during the amplitude peaks.

Before the onset of oscillation, the natural convection, radiation and heat conduction in air bring the heat from HHX to the regenerator, while only the heat conduction in air and regenerator material transport heat from the hot to the cold end. As a result, T1 increases rapidly until the onset of oscillation. After the onset, AEHT intensifies the heat transport from the HHX to the regenerator, as well as the heat transport from the hot to the cold end of the regenerator. At around 150 s, T1 starts to fall, while Th keeps rising until about 210 s. Interestingly, at this moment, (Th-T1) reaches the maximum, and the first amplitude burst occurs. It seems plausible that the effect can be explained in the following manner: Usually, a temperature discontinuity exists between the HHX and regenerator,¹⁷ which causes a higher local $∇T$ than that within the regenerator. According to Eq. (1), this means that the heat transport rate from HHX to the hot end of regenerator may be higher than the rate at which heat leaves the regenerator hot end. Thus a thermal “accumulation” may occur within the regenerator hot end. The local temperature gradient around the hot end of the regenerator quickly rises, and $|p1|$ grows accordingly. The higher $|p1|$⁠, the higher is this thermal accumulation. As a result, the exponential growth of the acoustic amplitude occurs. However, the input heat to the HHX is not large enough to maintain this exponential growth. Before $|p1|$ reaches the peak, the heat transport from HHX starts to drop, while the AEHT within the regenerator remains at a high rate due to a high value of local temperature gradient, which flattens the temperature profile. Shortly thereafter the acoustic amplitude starts to drop, which further decreases the heat transport from HHX to the regenerator. This results in a rapid decrease of the acoustic amplitude.

However, interestingly this rapid decrease leads to a low-amplitude oscillation, but not a complete decay. This means that $∇T$ along the regenerator at the end of the amplitude decrease is still enough to maintain this low-amplitude oscillation. Additionally, the overall trend to increase the amplitude of consecutive bursts can be attributed to the fact that the temperature profile along the regenerator tends to be more linear and thus each time a longer section of the regenerator contributes to the acoustic amplification.

Furthermore, Fig. 3(d) also shows that the temperature fluctuations are prominent at the two regenerator ends, but seem to “attenuate” toward the regenerator center. The regenerator’s large heat capacity (compared with that of air) is likely to damp the temperature fluctuations near its center.

The fishbone-like instability reported in this letter is the first observation of this kind in a thermoacoustic system. It shows a new mode of the acoustic amplitude growth in the thermoacoustic engine, i.e., a series of “discretized” bursts compared with the classical continuous exponential growth of the acoustic amplitude. The instability is attributed to the complex interactions between the acoustic and temperature fields, which are beyond current linear acoustic models. A qualitative explanation of this instability was proposed in this letter. However, an appropriate nonlinear model, which can account for these complex interactions and deal with the different time scales due to the different heat transfer mechanisms, has to be developed to fully understand this new type of instability.

EPSRC is acknowledged for funding this work under grants GR/S26842/01, GR/T04502/01 (EPSRC ARF), GR/T04519/01 and EP/E044379/1.