The viscous and thermal boundary layer thickness (thermal and viscous penetration depth) as defined in Theory Background for the Thermoviscous Acoustics Branch can be evaluated in postprocessing for the frequency domain models. The same is the case for the Prandtl number relating the two length scales (available in both frequency and time domain).
Material properties are readily available for postprocessing in plots. In plots, click the Replace Expression icon and browse to the
Material properties list under the thermoviscous acoustics interface. Important parameters to plot are the coefficient of thermal expansion
phys_id.alpha0 and the isothermal compressibility
phys_id.betaT. These should not evaluate to zero.
The energy conservation-dissipation corollary describes the transport and dissipation of energy in a system (see Ref. 1 p. 516 or
Ref. 6). In linear acoustics, this equation is derived by taking the dot product (scalar product) of the momentum and the velocity
v, adding it to the continuity equation, and then adding the entropy. After some manipulation and integration, the use of the divergence theorem yields
Equation 6-1
where w is the disturbance energy of the control volume,
u = |u| is the velocity,
T is the temperature variation,
p is the acoustic pressure variations,
p0 is the background equilibrium pressure,
T0 the background equilibrium temperature,
ρ0 the background density,
c0 the (isentropic) speed of sound,
Cp the heat capacity at constant pressure (per unit mass),
k the coefficient of thermal conduction,
i is the instantaneous intensity (flux of energy out of a control volume),
Δ is the dissipated energy per unit volume and time (SI unit: Pa/s = J/(m
3s) = W/m
3),
s is the entropy,
τ is the viscous stress tensor,
is the viscous dissipation function, and
T indicates transpose of vector.
Δv and
Δt are the viscous and thermal contributions to the dissipation function. In
Equation 6-1 we have made use of
Ref. 6 for the expression for the intensity
I.
where “:” in
Equation 6-2 is the double dot operator (or total inner product) and * is the complex conjugate. In the above expressions, the time averaged expressions for a product in the frequency domain is defined as:
In the Thermoviscous Acoustics, Frequency Domain interface the (time averaged) intensity I is given by averaging the instantaneous intensity vector
i in
Equation 6-1 using the same time averaged products defined above. The intensity and intensity magnitude are defined in
Table 6-5.
In the Thermoviscous Acoustics, Transient interface the instantaneous intensity i variables is available for postprocessing. The instantaneous intensity and instantaneous intensity magnitude are defined in
Table 6-6.
Several dedicated variables exist for The Thermoviscous Acoustics, Boundary Mode Interface where quantities are defined in terms of their in-plane and out-of-plane values. For example, the intensity variable
I has the following derived values
where ip stands for in-plane and
op for out-of-plane. These two variables are named
tabm.Iip and
tabm.Iop (with spatial components
x,
y, and
z). The magnitude of these two variables is given by
tabm.Iip_mag and
tabm.Iop_mag. In the same manner variables exist for the acceleration and the velocity.