Acoustic Perturbation and Linearization
For small perturbations around steady-state solution, the dependent variables and sources can be assumed to take on the following form:
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 and after inserting into the governing Equation 5-3, 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:
(5-4)
where the unprimed variables are now the acoustic deviation from the steady state.
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)
(5-5)
The two thermodynamic quantities (the coefficient terms in square brackets) define the isobaric coefficient of thermal expansion αp (sometimes named α0) and the isothermal compressibility βT, according to the following relations
(5-6)
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
(5-7)
It is derived using the Maxwell relations; see, for example, Ref. 5 and Ref. 7.
From Equation 5-6 and Equation 5-7 the isothermal compressibility and the isobaric coefficient of thermal expansion can be expressed in terms of the speed of sound as
(5-8)
The equations presented in Equation 5-4 and Equation 5-5 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
Inserting this into the governing equations and performing the linearization yield the equations solved in the frequency domain in The Thermoviscous Acoustics, Frequency Domain Interface:
(5-9)
The system of equations implemented in the thermoviscous acoustics interfaces is further given in a scattered field formulation, as described below in Scattered Field Formulation and Background Acoustic Fields.
When the Adiabatic formulation option under the Thermoviscous Acoustics Equation Settings section the solved equations reduce to:
(5-10)
where the temperature T depends directly on the acoustic pressure p.
Ideal Gas
For an ideal gas, the equation of state p = ρRT, where R is the specific gas constant, leads to
and the density
Inserting these expressions and dividing the continuity equation by the reference density, Equation 5-9 in the frequency domain take on the following simplified form
This is, for example, the system of equations implemented in the Thermoviscous Acoustics, Frequency Domain interface when the ideal gas law is selected.
Isentropic (adiabatic) Ideal Gas case
If the process is assumed to be adiabatic and reversible — that is, isentropic — the thermal conductivity is effectively zero. Then also the temperature can be eliminated, giving for the ideal gas case:
(5-11)
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
or
In the case with a general fluid, the corresponding relation is using Equation 5-6 and Equation 5-7:
where K0 is the adiabatic bulk modulus, KT the isothermal bulk modulus, and α0 the coefficient of thermal expansion.
Isothermal case
If, on the other hand, the thermal conductivity is high, or the thermoviscous acoustic waves propagate in a narrow space between highly conductive walls, the temperature can be assumed to be constant (isothermal assumption) and the system of equations for an ideal gas becomes:
which, again comparing to standard assumptions, gives
or equivalently
Therefore, thermal conductivity and/or conducting walls decrease the apparent speed of sound in narrow domains.
The Helmholtz Equation
If the thermodynamic processes in the system are assumed to be adiabatic and viscosity can be neglected Equation 5-11 reduces, for constant background properties, to
Now, taking the divergence of the momentum equations and inserting the expression for the divergence of the velocity, taken from the continuity equations, yields the Helmholtz equation for constant material properties: