Dai Shimizu

Boundary-layer theory for Taconis oscillations in a helium-filled tube

This paper develops a one-dimensional formulation to simulate Taconis oscillations in a helium-filled, quarter-wavelength tube in cryogenics within a framework of the boundary-layer theory. Dividing an acoustic field in the tube into a boundary layer on the wall and a main-flow region outside of it, the fluid-dynamical equations are averaged over the whole cross section of the tube, from which the equations averaged over the main-flow region are derived by using the boundary-layer solutions. Nonlinear theory is employed for the main-flow region, whereas the boundary layer is assumed to be described by the linear theory. Resultant equations for the main-flow region are posed in the form of integrodifferential equations due to memory effects by the boundary layer. An initial- and boundary-value problem is solved numerically for the evolution of a small disturbance. It is demonstrated that for temperature distribution of a smooth, step function, a transient behavior leading to emergence of self-excited Tacon...

Numerical study of thermoacoustic Taconis oscillations

This paper studies quantitatively a thermoacoustic field by the Taconis oscillations with finite amplitude in a helium-filled, quarter-wavelength tube. Numerical simulations are performed based on the one-dimensional theory in the boundary-layer approximation developed in a previous paper [N. Sugimoto and D. Shimizu, Phys. Fluids 20, 104102 (2008)] by solving initial- and boundary-value problems for a smooth step temperature distribution. It is found that the variations in the density, temperature, and entropy are so significant that the mean values deviate from the respective values in quiescent state. The mean acoustic energy flux and mean heat flux are calculated not only in the main-flow region but also in the boundary layer. The mean convective heat flux appears locally in the main-flow region due to higher-order nonlinear effects. While the total heat flux into the gas vanishes per one period, the local heat flux flows into the gas over a middle part of the tube.

Physical Mechanisms of Thermoacoustic Taconis Oscillations

This paper clarifies quantitatively physical mechanisms of thermoacoustic Taconis oscillations in a helium-filled, quarter-wavelength tube by performing numerical simulations based on the one-dimensional nonlinear theory in the boundary-layer approximation developed in a previous paper. Thermoviscous effects appear in the form of memory integrals and affect the acoustic main-flow region through the radial velocity at the edge of the boundary layer. Solving an initial- and boundary-value problem for a smooth temperature distribution on the tube wall, it is revealed that the boundary layer in the cold part is active to pump energy into the main-flow region and the initial instability of the gas is brought about by a subtle unbalance in the axial distribution of net power input by the boundary layer. Also checked is the validity of Rayleighs criterion based on the heat flux into the gas through the tube wall. It is also revealed that Rayleighs criterion is similar to the present one based on the power inpu...

Evaluation of mean energy fluxes in thermoacoustic oscillations of a gas in a tube

In a previous paper [N. Sugimoto and M. Yoshida, Phys. Fluids 19, 074101 (2007)], the marginal condition for the onset of thermoacoustic oscillations of a gas in a tube subjected to the parabolic temperature distribution was derived in the framework of the linear and first-order theory in the boundary-layer thickness. This paper examines the marginal oscillations from a viewpoint of mean energy fluxes averaged over one period of oscillations, aiming at understanding an action of the boundary layer under finite temperature gradient. Using the nonlinear energy equation, formulas for the acoustic energy flux and the convective heat flux (entropy flux times temperature) are derived in the main-flow region and in the boundary layer within the lowest, quadratic order in the pressure amplitude. These fluxes may be evaluated in terms of the linearized solutions and their axial distributions are displayed graphically. The boundary layer occupying nearly half of the side wall near the open end plays an active role ...

Determination of Marginal Conditions for Thermoacoustic Oscillations in a Looped Tube by Evolution of an Initial Disturbance Based on the Boundary-Layer Theory

Marginal conditions are determined for the onset of thermoacoustic oscillations of a gas in a looped tube with a so-called stack sandwiched by hot and cold heat exchangers on the basis of the boundary-layer theory so far developed. Given a couple of impulses applied initially to a quiescent gas, the evolution of an infinitesimally small disturbance is studied by solving an initial-value problem to the linearized equations. Marginal states correspond to those in which the initial disturbance neither decays nor grows with time. In the case studied by Ueda and Kato [J. Acoust. Soc. Am. 124, 851 (2008)], the marginal conditions are obtained in two cases of the temperature distribution in the thermal buffer tube and compared with their results. To identify the marginal curves, the porosity of the stack or one of the subsidiary parameters such as the wall thickness of the stack is required in addition to the squared ratio of the hydraulic radius to the typical thickness of the thermal diffusion layer in a flow ...

Autonomous generation of a thermoacoustic solitary wave in an air-filled tube

Experiments are performed to demonstrate the autonomous generation of an acoustic solitary wave in an air-filled, looped tube with an array of Helmholtz resonators. The solitary wave is generated spontaneously due to thermoacoustic instability by a pair of stacks installed in the tube and subject to a temperature gradient axially. No external drivers are used to create initial disturbances. Once the solitary wave is generated, it keeps on propagating to circulate along the loop endlessly. The stacks, which are made of ceramics and of many pores of square cross section, are placed in the tube diametrically on exactly the opposite side of the loop, and they are sandwiched by hot and cold (ambient) heat exchangers. When the temperature gradient along both stacks is appropriate, pulses of smooth profiles are generated and propagated in both directions of the tube. From good agreements of not only the pressure profile measured but also the propagation speed with the theory, the pulse is identified as the acous...

Numerical simulations of thermoacoustic oscillations in a looped tube by asymptotic theories for thickness of diffusion layers

Thermoacoustic oscillations in an air-filled, looped tube with a stack inserted are simulated numerically by using asymptotic theories for the ratio of a radius of flow passage to a typical thickness of the thermoviscous diffusion layer. The stack is composed of many pores axially and is sandwiched by hot and cold heat exchangers to impose a temperature gradient on the air in each pore. Weakly nonlinear wave equations based on the boundary-layer theory are used for a section in the outside of the stack. In each pore, the diffusion-wave (advection) equation is employed. Matching conditions at both ends of the stack require the conservations of mass, momentum and energy fluxes. An initial-value problem is solved from a disturbance of a pulsed axial velocity along the loop. When the temperature ratio is below a certain value, the initial disturbance is decayed out. However when the ratio exceeds it, it becomes unstable to grow in amplitude. Between the stable and unstable regimes, there exists a marginal sta...

Experiments on the acoustic solitary wave generated thermoacoustically in a looped tube

Emergence of an acoustic solitary wave is demonstrated in a gas-filled, looped tube with an array of Helmholtz resonators connected. The solitary wave is generated thermoacoustically and spontaneously by a pair of stacks positioned diametrically on exactly the opposite side of the loop. The temperature gradient is imposed on both stacks in the same sense along the tube. The stacks made of ceramics and of many square pores are sandwiched by hot and cold heat exchangers. The pressure profile measured and the propagation speed show good agreements with the theoretical ones of the acoustic solitary wave obtained by Sugimoto (J. Acoust. Soc. Am., 99, 1971–1976 (1996)).

Numerical simulations of a transient behavior in the onset of thermoacoustic marginal oscillations in a looped tube

This paper simulates a transient behavior in the onset of spontaneous, thermoacoustic oscillations of a gas in a looped tube with a so-called stack sandwiched by hot and cold heat exchangers. Numerical simulations are performed to solve an initial-value problem by employing the linearized boundary-layer theory. Initial conditions are chosen in such a way that the gas under a uniform pressure is given impulses at two locations in the outside of the stack to cancel with each other. Because of installation of the stack and heat exchangers, account is taken of discontinuity in the temperature gradient, the cross-sectional area and the wetted perimeter of the gas passages by imposing continuity of mass and energy fluxes. Except for a special value of the temperature ratio of the two heat exchangers, the pressure fluctuates significantly around the initial value transiently, but it eventually tends to grow indefinitely or decay out. At a marginal case, it is observed that a travelling wave tends to emerge spontaneously. The traveling wave always propagates in the sense from the hot to cold heat exchangers in the tube outside of the stack. It is shown qualitatively that the traveling wave is enhanced as the porosity lowers.

Numerical simulations of thermoacoustic oscillations in a looped tube

This paper attempts to simulate thermoacoustic oscillations in an air-filled, looped tube with a so-called stack inserted and subjected to a temperature gradient. The boundary-layer theory is applied, focusing on initial processes in instability. While the theory can describe the emergence of self-excited Taconis oscillations in a helium-filled, quarter-wavelength tube, it is open whether or not the theory is applicable to the case of the looped tube because the stack consisting of many narrow tubes is usually employed. Marginal conditions of instability are obtained by solving an initial-value problem for evolution of a small disturbance of velocity. It is found that the marginal conditions for the temperature ratios agree qualitatively with the experimental results by Ueda & Kato. It is unveiled that, as the porosity of the stack decrease, the traveling waves tend to appear in the course of time.