Open problems in the marginal-stability theory of thermoacoustic oscillations in closed and looped tubes with pore stacks

When a gas-filled tube contains a stack of narrow pores between hot and cold heat exchangers, steady heat flow can destabilize small acoustic disturbances into self-excited thermoacoustic oscillations. Sugimoto (2018) computed marginal conditions for this onset in a closed tube and a looped tube by solving Rott's linear narrow-tube eigenvalue problem numerically and checking two reduced approximations — the thin-diffusion-layer expansion in $\delta_e$ and the thick-layer (diffusion-wave) expansion in $\zeta_e$ — each of which reproduces one branch of the marginal curve but fails near its minimum, where the local $|\delta_e|$ is no longer small. That paper left a bundle of open issues toward a fuller onset theory: deriving the quadratic-and-higher-order terms of both expansions so the reduced equations track Rott's solution through the minimum; settling the existence, precise value and physical origin of an apparent lower bound of pore radius on the closed-tube left branch (numerical termination near $R_s/\sqrt{\nu_0/\omega}\approx2.2$ as $T_H/T_0\to\infty$); constructing and validating nonlinear generalizations of the reduced equations for fully developed post-onset oscillations; and quantifying sensitivity to non-ideal effects (buffer thermal recovery, wall heat capacity, heat leaks, pore shape, non-uniform stack temperature) behind the substantial theory–experiment discrepancy. Follow-up work through 2024 advanced the nonlinear simulations and identified cross heat flow as a discrepancy source, but the analytical cores remain open.

Background

A gas in a tube whose wall or insert maintains a steep axial temperature gradient can turn steady heat flow into acoustic energy: small disturbances then grow into self-excited, thermally driven oscillations, the phenomenon first analyzed systematically by Rott for the Taconis oscillations spontaneously arising in liquid-helium cryostat tubes (Taconis et al., 1949). Practical devices exploit it by inserting a stack — a solid block pierced by many narrow, parallel pores — between a hot and a cold heat exchanger, so that each pore carries a controlled temperature gradient. In a closed straight tube the resulting oscillation is standing-wave-like; in a closed loop with one or more stacks it is travelling-wave-like, the working principle of thermoacoustic engines and refrigerators. The porosity $\phi$ (fraction of cross-sectional area open to gas) modifies mass-flux continuity through the stack and hence the instability balance.

The linear theory of onset is Rott's narrow-tube formulation (Rott, 1969, 1973). Assuming a time-harmonic disturbance of angular frequency $\omega$ about a given axial equilibrium temperature distribution, it reduces the linearized Navier–Stokes and energy equations to a second-order ordinary differential equation for the complex pressure amplitude, with coefficients depending on the pore radius, the frequency and the local temperature through the thermoviscous diffusion layers — the regions adjacent to the pore wall over which viscosity and thermal conduction diffuse momentum and heat. A marginal condition is a parameter combination for which a neutral oscillation exists with real $\omega$: the temperature ratio $T_H/T_0$ between hot heat exchanger and ambient gas at which acoustic energy production by heat-flow-driven expansion exactly balances viscous and thermal dissipation, computed as an eigenvalue problem (solved numerically, e.g. by shooting). The key parameter is $\delta_e=(\nu_e/\mathrm{i}\omega)^{1/2}/R$, the ratio of the viscous diffusion-layer thickness to the pore radius $R$ ($\nu_e$ is the local kinematic viscosity).

Sugimoto's programme built reduced, asymptotically derived alternatives to Rott's equation: a marginal-condition analysis for a gas in a tube (Sugimoto and Yoshida, 2007), thermoacoustic-wave equations for gas in channels and tubes under a temperature gradient (2010), and a nonlinear theory for thermoacoustic waves in a narrow channel and pore (2016). In the source paper (Sugimoto, 2018), the marginal eigenvalue problem of Rott's equation is solved by shooting (Runge–Kutta integration with Newton iteration) for two experimental-scale geometries: a closed tube of length 3.63 m with one stack ($\phi=0.49$) and a looped one-wavelength tube of length 13.4 m with two stacks ($\phi=0.86$). The marginal curves of $T_H/T_0$ versus the pore radius relative to the ambient viscous diffusion-layer thickness, $R_s/\sqrt{\nu_0/\omega}$ (with $R_s$ the stack pore radius and $\nu_0$ the ambient kinematic viscosity; note that $|\delta_e|$ is the reciprocal of this abscissa only at ambient temperature, because $\delta_e$ is evaluated with the local viscosity $\nu_e$, which grows with temperature and therefore makes $|\delta_e|$ larger along the hot part of the stack), have single minima near $T_H/T_0\approx1.47$ at $R_s/\sqrt{\nu_0/\omega}\approx3.63$ (closed tube) and $T_H/T_0\approx1.49$ at $\approx3.01$ (looped tube, giving $T_H/T_0\approx1.7$ at the experimental pore radius 0.48 mm). The thin-layer expansion, derived for $|\delta_e|\ll1$ (diffusion layer thin compared with the pore), reproduces the right branch of the curve; the thick-layer or diffusion-wave expansion, derived in a small parameter $\zeta_e$ for the inverse regime in which the diffusion layer fills the pore ($|\delta_e|\gg1$), reproduces the left branch; both agree with Rott's solution to within a few percent except near the minimum, where the local $|\delta_e|$ is no longer small (of order $1/3$ at the minimum, close to the reciprocal of the abscissa there) and both deteriorate. Porosity sweeps, higher-mode marginal curves and axial amplitude profiles complete the results. Related stability analyses of thermally induced gas oscillations include Ueda-type eigenvalue studies and the complex-frequency analysis of Hyodo, Muraoka and Biwa (2017).

Problem Statement

The problem, as posed by the open issues of Sugimoto (2018), is to complete the quantitative theory of the onset of thermoacoustic oscillations in gas-filled closed and looped tubes containing pore stacks. The four items below are independent open sub-problems of that programme: each can be attacked and resolved on its own terms, and none of them subsumes another. They nevertheless share a single completion standard: a resolution of the task requires all of items (1)–(3) together with at least one of items (4a) and (4b), and a full treatment of a single item is a valid partial contribution that does not by itself resolve the task. Specifically:

(1) Asymptotic completion near the minimum. Derive the neglected higher-order (quadratic and beyond) terms of the thin-layer expansion in the small parameter $\delta_e$ and of the thick-layer expansion in the small parameter $\zeta_e$, and incorporate them into the reduced equations, so that their marginal conditions and neutral frequencies track Rott's solution through the region of the marginal-curve minimum, where the local $|\delta_e|$ is no longer small and both first-order approximations deteriorate — that is, so that through that region the reduced marginal conditions and neutral frequencies attain the few-percent agreement with Rott's solution that the first-order expansions already achieve away from the minimum.

(2) The small-pore-radius limit. Settle whether the left branch of the closed-tube marginal curve terminates at a finite lower bound of pore radius — suggested numerically near $R_s/\sqrt{\nu_0/\omega}\approx2.2$ as $T_H/T_0\to\infty$ — and if so determine its precise value and physical origin: establish rigorously, by analysis or certified computation, whether below such a radius oscillations cannot be excited for any finite temperature ratio $T_H/T_0$, or disprove this.

(3) Nonlinear post-onset models. Construct nonlinear generalisations of the reduced thin-/thick-layer equations for fully developed, post-onset thermoacoustic oscillations and validate them against experiment.

(4) Robustness and discrepancy. This item separates into two independently addressable parts.

(4a) Sensitivity of marginal thresholds and mode selection. Quantify how marginal thresholds and mode selection respond to non-ideal conditions such as buffer-region temperature recovery and geometric tolerances (for example the buffer length $L_b$ in looped-tube computations), finite wall heat capacity, heat leaks, pore cross-section shape, and non-uniform stack temperature giving rise to cross heat flow.

(4b) Sources of the theory–experiment discrepancy. Identify and quantify the sources of the remaining discrepancy between theoretical and observed marginal conditions, which experiments have shown to be substantial (measured onset temperatures considerably higher, and growth rates much smaller, than linear-theory predictions).

References

1. Nobumasa Sugimoto (2018). Marginal conditions for the onset of thermoacoustic oscillations due to instability of heat flow. IMA Journal of Applied Mathematics 84(1), 118–144. DOI: 10.1093/imamat/hxy051. https://doi.org/10.1093/imamat/hxy051
2. Nikolaus Rott (1969). Damped and thermally driven acoustic oscillations in wide and narrow tubes. Zeitschrift für angewandte Mathematik und Physik 20(2), 230–243. DOI: 10.1007/BF01595562. https://doi.org/10.1007/BF01595562
3. Nikolaus Rott (1973). Thermally driven acoustic oscillations. Part II: Stability limit for helium. Zeitschrift für angewandte Mathematik und Physik 24(1), 54–72. DOI: 10.1007/BF01593998. https://doi.org/10.1007/BF01593998
4. N. Sugimoto and M. Yoshida (2007). Marginal condition for the onset of thermoacoustic oscillations of a gas in a tube. Physics of Fluids 19(7), 074101. DOI: 10.1063/1.2742422. https://doi.org/10.1063/1.2742422
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9. Nobumasa Sugimoto, Keisuke Minamigawa (2018). Experiments on Self-excited Thermoacoustic Oscillations in an Air-filled Looped Tube with a Pair of Stacks. Journal of the Physical Society of Japan 87(10), 104401. DOI: 10.7566/jpsj.87.104401. https://doi.org/10.7566/jpsj.87.104401
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11. Nobumasa Sugimoto, Dai Shimizu (2019). Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube Subject to a Temperature Gradient. In: Mathematics of Planet Earth, Springer, pp. 187–204. DOI: 10.1007/978-3-030-29951-4_9. https://doi.org/10.1007/978-3-030-29951-4_9
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13. Nobumasa Sugimoto (2021). Effects of a Cross Heat Flow on Thermoacoustic Instability in a Gas-Filled Channel. Journal of the Physical Society of Japan 90(1), 014402. DOI: 10.7566/jpsj.90.014402. https://doi.org/10.7566/jpsj.90.014402
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15. Nobumasa Sugimoto, Dai Shimizu (2024). A computational method for a thermoacoustic boundary layer in a gas-filled tube. The Journal of the Acoustical Society of America 155(1), 98–113. DOI: 10.1121/10.0023970. https://doi.org/10.1121/10.0023970
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