Linearized Navier-Stokes Model
The Linearized Navier-Stokes Model sets up the governing equations, defines the background mean flow, fluid properties, and the compressibility and thermal expansion properties of the fluid. The governing equations solved are the continuity, momentum, and energy equations:
(4-2)
where pt, ut, and Tt are the acoustic perturbations to the pressure, velocity, and temperature, respectively. The subscript “t” refers to the fact that the acoustic variables are the total fields, that is, the sum of possible Background Acoustic Fields and the scattered fields.
In the frequency domain, the time derivatives of the dependent variables are replaced by multiplication with iω. The stress tensor is σ and Φ is the viscous dissipation function. The right-hand-side source terms M, F, and Q are initially zero; they can be defined using the Domain Sources feature. The variables with a zero subscript are the background mean flow values. The material parameters are defined below. Details about the physics and references are found in the Theory Background for the Aeroacoustics Branch section.
The constitutive equations are the stress tensor and the linearized equation of state, while the Fourier heat conduction law is readily included in the above energy equation,
(4-3)
The linearized viscous dissipation function is defined as
(4-4)
Model Inputs
In order to model the influence of the background mean flow on the propagation of the acoustic waves in the fluid, the background mean flow temperature T0, absolute pressure p0, and velocity field u0 need to be defined. The density is defined in the Fluid Properties section below, and is per default taken from the material. It is thus a function of the model inputs, that is, the background pressure and temperature. Enter User defined values for the:
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Background mean flow temperature T0 (SI unit K). The default is 293.15 K.
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Background mean flow pressure p0 (SI unit: Pa). The default is 1 atm.
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Background mean flow velocity u0 (SI unit: m/s). The defaults are 0 m/s.
When modeling aeroacoustics it is important how the Mapping Between CFD and Acoustics Mesh is done from a numerical perspective. Physically the Coupling to Turbulent Flows (Eddy Viscosity) is also important to model the attenuation of acoustics waves due to turbulence.
Fluid Properties
The defaults for the following are taken From material. For User defined edit the default values:
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Background mean flow density ρ0 (p0, T0) (SI unit: kg/m3).
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Dynamic viscosity μ (SI unit: Pa⋅s).
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Bulk viscosity μB (SI unit: Pa⋅s).
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Thermal conductivity k (SI unit: W/(m⋅K)).
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Heat capacity at constant pressure Cp (SI unit: J/(kg⋅K)).
Thermal Expansion and Compressibility
Select an option from the Coefficient of thermal expansion αp list: From material (the default) or User defined. For User defined enter a value for αp (SI unit: 1/K). The subscript p refers to the fact that this is the isobaric coefficient of thermal expansion.
Select an option from the Isothermal compressibility βT list: From isentropic compressibility (the default), From speed of sound, or User defined.
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For From isentropic compressibility the values for the Isentropic compressibility βs (SI unit: 1/Pa) and Ratio of specific heats γ (dimensionless) are taken From material. For User defined enter a different values or expressions.
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For From speed of sound the values for the Speed of sound c (SI unit: m/s) and Ratio of specific heats γ (dimensionless) are taken From material. For User defined enter a different value or expression.
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For User defined enter a value for βT (SI unit: 1/Pa).
Viscous Dissipation Function
Select the Include viscous dissipation function check box if you wish to include the heat source generated by the viscous losses. The viscous dissipation function Φ is defined in Equation 4-4.