About the Narrow Region Acoustics Models

The Narrow Region Acoustics fluid models are used to mimic the thermal and viscous losses that exist in narrow tubes where the tube cross-section length-scale is comparable to the thermal and viscous boundary layer thickness (boundary-layer absorption). It is essential to include these losses in order to get correct results.

These models are commonly used in situations where solving a full detailed thermoviscous acoustic model is computationally costly; for example when analyzing long narrow ducts/tubes of constant cross section. Here it is possible to add or smear the losses associated with the boundary layer onto the bulk of the fluid: an equivalent fluid model. For many geometries, analytical expressions exist for the losses associated with the acoustic boundary layers. The models can be applied under different assumptions. The models and assumptions are discussed in this section.

	In more complex geometries where thermal and viscous losses are important, see The Thermoviscous Acoustics, Frequency Domain Interface, which is more fundamental and detailed.

	Narrow Region Acoustics

Several fluid models exist:

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The wide duct approximation can be used for any duct cross section in the limit where the duct width is significantly larger than the acoustic boundary layer thickness. See Wide Ducts.

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The very narrow circular ducts (isothermal) can only be used when the duct width is so small that isothermal conditions apply. This is when the duct width is much smaller than the acoustic thermal boundary layer thickness. See Very Narrow Circular Ducts (Isothermal).

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The slit, circular duct, rectangular duct, and equilateral triangular duct models are based on an analytical solution of the thermoviscous acoustic equations in the limit where the acoustic wavelength is much larger than both the duct cross section (below the cutoff frequency) and the boundary layer thickness. This is the case in most engineering applications. See Slits, Circular Ducts, Rectangular Ducts, and Equilateral triangular Ducts.

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Finally, selecting the user defined option enables you to enter expressions for the complex wave number and the complex acoustic impedance. These may be analytical expressions, interpolated values, or values extracted from a detailed boundary mode analysis using the full formulation of The Thermoviscous Acoustics, Boundary Mode Interface.

For a relatively wide duct, the losses introduced in the acoustic boundary layer may be studied by adding these as an effective wall shear force. This approach is used in Blackstock (Ref. 6) and results in equivalent fluid complex wave number kc defined by

where Hd is the hydraulic diameter of the duct, S is the duct cross-section area, C is the duct circumference, μ is the dynamic viscosity, ρ is the fluid density, γ is the ratio of specific heats, Cp is the specific heat at constant pressure, k is the fluid thermal conductivity, and Pr is the Prandtl number. For a cylindrical duct, Hd = 2a where a is the radius.

The approximation in Equation 2-52 is only valid for systems where the effective radius Hd/2 is larger than the boundary layer, but not so small that mainstream thermal and viscous losses are important. Thus requiring

where δv is the characteristic thickness of the viscous boundary layer (the viscous penetration depth), c is the speed of sound, and ω is the angular frequency. The complex wave number is related to the complex density and speed of sound by the equation (assuming a real valued bulk modulus)

In the other limit where the duct diameter is sufficiently small or the frequency sufficiently low, the thermal boundary layer thickness becomes much larger than the duct cross section a. This is the case when

where δth is the characteristic thickness of the thermal boundary layer (thermal penetration depth), ρ is the density, Cp is the heat capacity at constant pressure, and k is the fluid thermal conductivity. In this case see Pierce (Ref. 5); the system may be seen as isothermal and the acoustic temperature variation is zero everywhere in the duct T = 0. The fluid complex wave number kc is then defined by

where cT is the isothermal speed of sound, a is the duct radius, µ is the dynamic viscosity, and ω is the angular frequency. The theory is derived for ducts of circular cross section — the model is therefore only applicable for systems with small variations away from a circular cross section. The complex wave number is related to the complex density and speed of sound by the equation (here the bulk modulus is defined in the isothermal limit)

The slit, circular duct, rectangular duct, and equilateral triangular duct models are based on the so-called low reduced frequency (LRF) model that describes the propagation of acoustic waves in small waveguides (ducts and slits) including thermal and viscous losses. Details about these models are in Ref. 19, Ref. 20, and Ref. 21. The models cover the range from fully isothermal conditions (very low frequencies or very narrow tubes) to large ducts where the boundary layer only represents a fraction of the duct size. The models apply as long as the cross section of the duct is much smaller than the acoustic wavelength (the model is below the cutoff frequency) and when the cross section is constant or only very slowly varying.

In a narrow waveguide the complex wave number, kc, and complex specific acoustic impedance, Zc, are given by

where Υv and Υth are the mean value (cross section averaged) of the scalar viscous and thermal field functions, respectively. The functions are geometry and material-dependent functions (specified below) and γ is the ratio of specific heats. The fluid density ρ, the speed of sound is c, and the angular frequency ω define the free space wave number k0 and the specific acoustic impedance Z0. The subscripts “v” and “th” stand for viscous and thermal (heat) fields, respectively. Once these are known, the complex speed of sound and complex density are given by

The values of the Υj functions can be derived by solving the full set of linearized Navier-Stokes equations (the equations solved by the thermoviscous acoustics interfaces, see Theory Background for the Thermoviscous Acoustics Branch) by splitting these into an isentropic (adiabatic), a viscous, and a thermal part. Doing this introduces the viscous and the thermal wave numbers for the system

The equations may be solved analytically under the following assumptions used for the LRF models:

The resulting analytical expressions, for the viscous and thermal Υ functions, are for the given geometry (these results are reviewed in Ref. 19):

The user defined option in the Narrow Region Acoustics domain feature can be used to define a LRF model for a waveguide of an arbitrary cross sections. Enter values for the complex wave number kc and the characteristic complex impedance Zc. This can be a user defined analytical expression or values derived from a mode analysis study.