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 cut-off 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.
Wide Ducts
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
(2-50)
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 2-50 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 dvisc 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)
Very Narrow Circular Ducts (Isothermal)
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 dtherm 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
(2-51)
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)
Slits, Circular Ducts, Rectangular Ducts, and Equilateral triangular Ducts
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 cut-off frequency).
In a narrow waveguide the complex wave number, kc, and complex specific acoustic impedance, Zc, are given by
where Ψv and Ψh 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 “h” 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 Ψj 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:
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The viscous and thermal wave numbers must be much larger than the acoustic wave number (the acoustic wavelength must be much larger than the boundary layer thickness):
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The cross section of the waveguide must be much smaller than the acoustic wavelength (the model is below the cut-off frequency).
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The cross section of the waveguide must be constant or at most slowly varying in the propagation direction.
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The length of the waveguide in the propagation direction should be larger than the boundary layer thickness.
The resulting analytical expressions, for the viscous and thermal Ψ functions, are for the given geometry (these results are reviewed in Ref. 19):
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Slit of height h
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Circular duct of radius a
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Rectangular duct of side lengths W and H
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Equilateral triangular duct of side length d
User Defined
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.
To determine the complex propagation constants for a waveguide, of arbitrary cross section, use the The Thermoviscous Acoustics, Boundary Mode Interface. Apply it on the cross-section geometry of the waveguide. The interface solves for the propagating modes and includes all losses in detail. The complex wave number kc is then given by the plane wave mode solved for. This is the variable tabm.kn. The predefined variable tabm.Zc gives the (lumped) complex characteristic impedance Zc. Search for the mode nearest to the (lossless) plane wave mode.