Thermoacoustic oscillations at a cycle-steady state in a tube with an isothermal outer wall, and with one end closed and the other end connected to a wave generator, is analyzed based on a linearized theory. From the global mass conservation, an analytical solution has been obtained for the cross-sectional and cycle averaged axial velocity. It is shown that this averaged velocity is non-vanishing due to the mass streaming effect. By analyzing the global momentum balance, it is found that the cycle-averaged pressure depends on the momentum streaming and friction force, and a conservation relationship exists between the momentum streaming and the cycle-averaged pressure for the flow oscillating at a high frequency in a wide tube. An investigation of the global energy balance leads to an expression for thermoacoustic energy streaming. Furthermore, it is shown that the refrigeration effect is mainly caused by the non-vanishing mean velocity, and therefore the mass streaming and the energy streaming are intimately connected.

An experimental setup has been constructed to study the performance of a single-stage double-inlet Gifford-McMahon-type pulse tube refrigerator, where the oscillating amplitudes of the physical quantities are large and oscillating frequencies are low in the system. Temperature distributions on the surface of the regenerator and the pulse tube, as well as the refrigeration capacities at different refrigeration temperatures under optimal operation conditions, were measured. A transient one-dimensional numerical simulator has been developed to verify experimental data and to study the nonlinear dynamic characteristics in the double-inlet pulse tube refrigerator. In this numerical simulator, the state equation and the conservation equations of mass and momentum in the fluid phase, as well as the energy equations for the fluid and the solid, are spatially and temporally conjugated. The boundary conditions and the initial conditions for these governing equations are specified, and finite difference solutions at cycle steady states were obtained. The assumption that the refrigeration temperature at the cold-end heat exchanger is kept at a constant and known value during a cycle in the existing simulations is relaxed in our simulator. Instead, the refrigeration capacity is given, whereas the refrigeration temperature is determined from the numerical solution. Fluctuations of all physical quantities at cycle steady states are discussed. When the cycle-averaged values of the physical quantities are analyzed, it is shown that there is a dc flow through the double-inlet valve at a cycle steady state, and the cycle-averaged velocity in the system is negative due to the mass streaming. At cycle steady states, the cycle-averaged pressure of the compressible flow oscillating at a low frequency is shown to have a positive gradient along the axial location with its largest value in the reservoir. Numerical results are shown to be in good agreement with experimental data.

A 1-D transient numerical model has been developed to predict the performance and reveal the nonlinear dynamical characteristics of a G-M type double-inlet pulse tube refrigerator, where the oscillating amplitudes of the physical quantities are large. In this numerical simulator, governing equations consisting of the state equation, the conservation of mass and momentum in the fluid phase, as well as the energy equations for the fluid and the solid, are spatially and temporally conjugated. The boundary conditions and the initial conditions for these governing equations are discussed. The methods for the numerical discretizations of these governing equations are given. The assumption, that the refrigeration temperature at the cold-end heat exchanger is kept at a constant and known value during a cycle in the existing simulations, is released in our simulator. Instead, the refrigeration capacity is prescribed while the refrigeration temperature is determined from the numerical solution. Numerical results, such as cycle-averaged temperature distribution and fluctuations of the physical quantities in a single-stage G-M type pulse tube refrigerator, are analyzed. These numerical results are shown in good agreement with experimental data. This numerical simulator can be used not only to predict dynamical performance of a pulse tube refrigerator, but it can also be used to design a pulse tube refrigerator for optimal performance.

An experimental investigation has been carried out on dynamical pressures of the viscous compressible flow oscillating at different locations in a Gifford–McMahon (G–M) type pulse tube refrigerator operating at cycle-steady states. Measurements show that the oscillating amplitude of the pressure was largest at the hot end of the regenerator while the cycle-averaged pressure was the largest in the reservoir. The latter characteristics can be explained based on a cycle-averaged and cross-sectional averaged of the governing equations for a compressible viscous oscillating flow. The reason why the cycle-averaged pressure of the compressible flow oscillating at low frequencies in a tube increases from the wave generator toward the reservoir is analyzed. In addition, the effect of the cycle-averaged pressure on the refrigeration performance is discussed, which can be used to explain why the system with proper asymmetric charging and discharging periods has a better performance than a symmetric one in a G–M type pulse tube refrigerator.

In order to understand the refrigeration mechanism in a pulse tube refrigerator and a thermoacoustic refrigerator, we study a simplified model of a compressible flow oscillating in an isothermal tube with closed end at a cycle steady state. Based on a thermoacoustic theory and from global conservation of mass, momentum and energy, it is found that the refrigeration effect is essentially caused by the non-vanishing mean velocity of the compressible oscillating flow during the cycle-state steady condition. If the tube is connected by a capillary tube to a reservoir, it is also found that the increase of the refrigeration capacity is due to the large pressure drop across the capillary tube.

The transport phenomena for a viscous compressible oscillating flow (with a zero mean velocity) in a tube subjected to a prescribed cycle-steady axial temperature gradient are analyzed. The governing equations are linearized under the conditions of high oscillating frequencies, small amplitudes, and in a tube with a high length-to-radius ratio. Based on a linearized theory, an analytical expression is obtained for the local friction factor, which depends on the prescribed cycle-steady axial temperature and independent of time. The local friction factor is shown to be a complex number indicating a phase shift between the cross-sectional averaged velocity and the local pressure gradient. Closed-form analytical expressions are also obtained for the temperature distribution and for the Nusselt number of solid/fluid interfacial heat transfer by solving the energy equations of the fluid and solid phases. The Nusselt number is also a complex number indicating a phase shift between the heat flux and the temperature difference between the wall and the oscillating fluid. The magnitude of the Nusselt number is also dependent on the prescribed axial cycle-steady temperature and independent of time, and is related to dimensionless thermal property parameters, dimensionless geometrical parameter, and dimensionless operation conditions parameters. To understand the momentum transport and energy transport characteristics, the radial distributions of axial velocity and temperature of the fluid are presented for different ratios of the inner radius with respect to fluid's viscous penetration depth. Particular attention is given to the transport phenomena in the following two limiting cases: 1) a viscous compressible oscillating flow in a porous medium based on a capillary-tube model, and 2) a viscous compressible oscillating flow in a resonant tube of a thermoacoustic refrigerator or in a pulse tube of a Stirling-type pulse-tube refrigerator.

Flow characteristics through a metering valve, a component in orifice pulse tube refrigerators and double-inlet pulse tube refrigerators, are discussed in this paper. Under the assumption that the inertial and convective terms in the momentum equation for a compressible flow can be neglected compared to the pressure gradient term, an expression is obtained for the transient mass flow rate in terms of the local temperature and pressure as well as pressures across the valve. Depending on where the local pressure and temperature are evaluated, two algebraic expressions for the mass flow rates in terms of the effective area are obtained. The values of the effective areas in these two expressions for different openings of the valve at an oscillating frequency of 1 Hz were evaluated based on pressure and temperature measurements. The results are useful for the numerical simulation of the performance of an orifice pulse tube refrigerator or a double-inlet pulse tube refrigerator.