A general predictive law for acoustic pressure losses in woven-screen regenerator matrices

In thermoacoustic engines and Stirling-cycle machines, the regenerator is a stack of woven stainless-steel mesh whose small passages give intimate gas-solid thermal contact but also dissipate acoustic power through viscous pressure loss. Lawn (2014) measured these acoustic losses for 34# and 75# meshes (34 and 75 wires per inch; 28-112 Hz) and correlated them in an impedance-oriented form containing an empirical, mesh-dependent parameter $\theta$, for which he could propose no universal prescription: separate fits were needed for each mesh, the loss enhancement with dimensionless frequency $(r_h/\delta_v)^2$ was not consistent across pooled mesh data, tighter stack packing changed the measured loss by about 30 percent, and other laboratories' published data fell well below the steady-flow correlations. The open problem is to establish and validate a general predictive law for the oscillatory-flow acoustic pressure loss of randomly stacked woven screens, from mesh geometry and packing state alone, over the range of the source measurements (air at ambient conditions, 28-112 Hz, pore velocities of order a few m/s); the documented inter-dataset discrepancies serve as diagnostic context for such a law rather than as separate completion requirements.

Background

A thermoacoustic engine or refrigerator converts between heat and acoustic power; its core component, the regenerator (also called the stack in standing-wave devices), is a porous solid matrix through which gas oscillates while in intimate thermal contact with the solid, so that each gas parcel exchanges heat over every acoustic cycle. To achieve this contact, the passages must be comparable in size to the viscous and thermal penetration depths, $\delta_v=\sqrt{2\nu/\omega}$ and $\delta_k$, where $\nu$ is the kinematic viscosity and $\omega$ the angular frequency; $\delta_v$ is the distance over which oscillatory velocity diffuses into the fluid in one cycle. The most common regenerator is a stack of many layers of woven stainless-steel mesh laid in random registration, characterized by the mesh count, wire diameter, three-dimensional porosity $\phi$ (void fraction), and hydraulic radius $r_h$ (flow area divided by wetted perimeter). The stack itself also has a packing state, separate from these mesh properties, given quantitatively by the achieved layer density — the number of mesh layers per unit stack thickness (for the source's two 75# stacks, 53 layers over 10 mm versus 60 layers over 9.5 mm).

Viscous dissipation in these narrow passages is a first-order term in the loss budget of travelling-wave engines, so designers need to predict it. The traditional route uses steady-flow friction correlations for woven-screen matrices, obtained from the classic Tong-London experiments and interpreted for regenerator use by Swift and Ward in their simple harmonic analysis: the loss is a sum of a viscous term (dominant at low Reynolds number) and a form-drag term (pressure drag from flow separation around the wires), with porosity-dependent coefficients. Whether these steady-flow correlations remain valid in oscillatory flow at real engine frequencies (tens to a few hundred Hz) was historically uncertain, because when $\delta_v$ becomes comparable with $r_h$ the flow in the tortuous passages is no longer quasi-steady. Earlier oscillating-flow measurements at low frequencies (Tanaka et al.; Hsu, Fu and Cheng; Wakeland and Keolian on single screens) gave partly conflicting guidance, and the problem is naturally framed in Rott's linear acoustics of narrow channels, where the averaged viscous response of a channel is described by a complex, frequency-dependent function $f_v$ (Rott's viscous function), which for simple geometries can be computed exactly from the diffusion (Helmholtz-like) equation for the oscillatory velocity field.

Lawn (2014) reported direct acoustic measurements of the pressure loss through 10 mm stacks of 34# and 75# woven mesh (34 and 75 wires per inch; hydraulic radii 0.174 mm and 0.084 mm) in two impedance-tube configurations, using the two-microphone method (inferring complex acoustic velocity from pressures at two axial positions) over 28-112 Hz at pore velocities up to about 3 m/s. He expressed the results through the non-dimensional loss coefficient $LC=\mathrm{Re}\{\Delta p/(\rho c\,u_{pore})\}$, where $\Delta p$ is the complex acoustic pressure drop, $\rho$ the mean density, $c$ the sound speed, and $u_{pore}$ the pore velocity (volumetric flow divided by pore area). Main findings: the velocity dependence is the same as in steady flow (LC linear in velocity amplitude, as expected when form drag dominates); at low frequency and amplitude the measurements agree with the steady-flow correlation within the estimated ±10% experimental accuracy, except for one data set with anomalously low loss attributed to partial layer separation; above roughly 40 Hz the loss is enhanced, correlating within each mesh with the dimensionless frequency $(r_h/\delta_v)^2$; and thermal relaxation contributes negligibly to the dissipation. To express the loss directly as an acoustic impedance (the quantity used in thermoacoustic network calculations), he proposed the second-order representation $f_v=1-j\left(r_h/\delta_v\right)^2\theta$ with an empirical parameter $\theta$ fitted per mesh, for example $\theta^{-1}=1.25+0.0048\,\mathrm{Re}$ for the 34# mesh and $\theta^{-1}=2.08+0.0237\,\mathrm{Re}$ for the 75# mesh, where the Reynolds number is based on pore velocity and an equivalent passage diameter. This works to about ±10% for the fitted data, but the fitted $\theta$ behavior differs between meshes, the frequency enhancement is not captured consistently when data sets are pooled (coarse-mesh data lie 25-30% below the fine-mesh trend), compaction and inter-layer registration changes the loss by about 30%, and no universal prescription for $\theta$ across mesh geometries could be proposed.

Problem Statement

Establish and validate a general predictive law for the acoustic (oscillatory-flow) pressure loss of randomly stacked woven-screen regenerator matrices, expressed in a form directly usable in acoustic impedance calculations (for example the loss coefficient $LC=\mathrm{Re}\{\Delta p/(\rho c\,u_{pore})\}$ or Rott's viscous function $f_v$), such that:

(a) from mesh geometry (mesh count, wire diameter, porosity, hydraulic radius) and stack packing state alone, without per-mesh or per-dataset fitting, the law reproduces both the steady-flow limit and the observed linear dependence of the loss on pore-velocity amplitude; and

(b) the law captures the enhancement of loss with dimensionless frequency $\left(r_h/\delta_v\right)^2$ consistently across different mesh geometries, removing the mesh-specific offsets that currently require separately fitted parameters $\theta$ (for which no universal prescription exists).

The required scope is that of the source measurements: air at ambient conditions, frequencies 28-112 Hz, pore velocities of order a few m/s (Reynolds number based on pore velocity and equivalent passage diameter up to about 60 for the 75# mesh and about 200 for the 34# mesh; acoustic Mach number up to about 0.013), for stacks about 10 mm thick of randomly registered woven stainless-steel mesh comparable to the 34# and 75# meshes studied. Extension to pressurized noble gases or imposed axial temperature gradients is context for later application, not part of the required completion boundary.

The documented inter-dataset discrepancies — the ~30% loss difference between Lawn's two 75# series attributed qualitatively to packing, Yu et al.'s steady-flow data lying 20-35% below the Swift-Ward correlation (their acoustic data lying almost an order of magnitude below), and the 25-30% offset of pooled coarse-mesh data from the fine-mesh frequency-enhancement trend — serve as background motivation and diagnostic checks on any proposed law, not as separate mandatory completion criteria.

References

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