where the subscript “1” variables represent the acoustic perturbations (first-order perturbation) and subscript “0” the background mean flow quantities. Assuming zero mean flow u0 = 0 and after inserting into the governing
Equation 6-3, the steady-state equations can be subtracted from the system, which is subsequently linearized to first order by ignoring terms quadratic in the acoustic variables. Dropping the subscript “1” for readability yields the thermoviscous acoustic equations:
The density ρ is expressed in terms of the pressure and the temperature variations using the density differential (Taylor expansion about the steady quiescent values)
where Ks is the isentropic bulk modulus (sometimes named
K0),
KT the isothermal bulk modulus,
Cv is the heat capacity at constant volume (per unit mass), c is the (isentropic) speed of sound, and
γ is the ratio of specific heats (the adiabatic index). The isothermal compressibility
βT is related to the isentropic (or adiabatic) compressibility
βs (sometimes named
β0) and the coefficient of thermal expansion
αp via the thermodynamic relations
From Equation 6-6 and
Equation 6-7 the isothermal compressibility and the isobaric coefficient of thermal expansion can be expressed in terms of the speed of sound as
The equations presented in Equation 6-4 and
Equation 6-5 are the ones solved in the time domain in
The Thermoviscous Acoustics, Transient Interface. Assuming small harmonic oscillations about a steady-state solution, the dependent variables and sources can be assumed to take on the following form
When the Adiabatic formulation option under the
Thermoviscous Acoustics Equation Settings section the solved equations reduce to:
where the temperature T depends directly on the acoustic pressure
p.
For an ideal gas, the equation of state p = ρRT, where
R is the specific gas constant, leads to
where we have assumed constant background properties. Defining the speed of sound c in analogy with the standard assumptions for linear acoustics (term in front of the pressure in the continuity equation), it is found that
where K0 is the adiabatic bulk modulus,
KT the isothermal bulk modulus, and
α0 the coefficient of thermal expansion.