General Governing Equations
The equations governing the physics in any fluid are the foundation for deriving the linearized aeroacoustic equations and in general any acoustics equations. The governing equations for the motion of a compressible fluid are the continuity equation (mass conservation), the Navier-Stokes equation (momentum conservation), and the general heat transfer equation (energy conservation). In order to close the system of equations, constitutive equations are needed, along with the equation of state, and thermodynamic relations. See, for example, Ref. 3, Ref. 4, Ref. 5, Ref. 6, or Ref. 7 for details and further reading.
Conservation equations
The conservation equations are for mass, momentum, and energy:
(4-5)
where u is the velocity field, ρ is the density, T is the temperature, s is the specific entropy, σ is the stress tensor, q is the local heat flux, Φ is the viscous dissipation function, and M, F, and Q are source terms. The operator D/Dt is the material derivative (or advection operator) defined as
Thermodynamic Relations
Some thermodynamic relations are necessary when reformulating the energy equations in terms of other sets of thermodynamic variables, like (pT) or (ρp). They are the density differential, the specific energy relation, a relation due to Helmholtz, and the fundamental entropy relation:
where u is the specific internal energy, αp is the coefficient of thermal expansion (isobaric), and βT is the isothermal compressibility. See, for example, Ref. 5 and Ref. 6 for details. They are defined together with the specific heat at constant pressure Cp and specific heat at constant volume Cv as
Using the above thermodynamic relations, the entropy differential can be expressed as (used for the linearized Navier-Stokes equations)
while for an ideal gas it can be given as (used for the linearized Euler equations)
Constitutive Equations
The constitutive equations are the equations of state (density expressed in terms of any set of thermodynamic variables), the Stokes expression for the stress tensor, and the Fourier heat conduction law
where k is the thermal conduction, μ is the dynamic viscosity, and μB is the bulk viscosity. This then also defines the viscous dissipation function
Perturbation Theory
In the following, the governing equations are linearized and expanded to first order in the small parameters around the average stationary background solution. For details about perturbation theory see Ref. 3, Ref. 4, and Ref. 8. The small parameter variables (1’st order) represent the acoustic variations on top of the stationary background mean (or average) flow (0’th order solution). Note that, when solving the equations, the value of the acoustic field variables can also represent nonacoustic waves like thermal waves (entropy waves) and vorticity waves. In the time domain, these can be linear instabilities and can actually represent the onset of turbulence.
The dependent variables and sources are expanded according to
where A is any of the dependent variables or sources. In the frequency domain, the first order variables are assumed to be harmonic and expanded into Fourier components, such that
The first order variation to material parameters, that are not treated as dependent variables like the density ρ in the Linearized Navier-Stokes interface, is expressed using the density differential and related to perturbations in the temperature and the pressure.
The above perturbation schemes are inserted into the governing equations and the linearized acoustic equations are derived retaining only first order linear terms.