Theory of Thermoacoustics: Acoustics with Thermal and Viscous Losses

Mads Herring Jensen | February 27, 2014
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When sound propagates in structures and geometries with small dimensions, the sound waves become attenuated because of thermal and viscous losses. More specifically, the losses occur in the acoustic thermal and viscous boundary layers near the walls. This is a known phenomenon that needs to be included when studying and simulating systems affected by these losses in order to model these systems correctly and to match measurements.

Introduction to Thermoacoustics

It is, for example, necessary to include the thermal and viscous losses when modeling the response of small transducers, like condenser microphones, MEMS microphones, and miniature loudspeakers (i.e. receivers). Other applications include analyzing feedback in hearing aids and in mobile devices, or studying the damped vibrations of MEMS structures.

A good example for us to investigate here, which relates to an engineering application, is the transfer impedance of the standard IEC 60318-4 occluded ear canal simulator (sometimes referred to as the 711-coupler), as depicted in the figure below. In the graph to the right, the response is modeled including and excluding thermoacoustic losses. It is evident that these types of losses need to be included in order to capture the correct behavior when comparing their curves to the standard simulator’s data.

Thermoacoustics pressure
Graph showing the thermoacoustic pressure

The pressure distribution inside an occluded ear canal simulator at 7850 Hz, complying with the IEC 60318-4 standard, is depicted to the left. The modeled transfer impedance of the coupler (in blue, including thermal and viscous losses) is shown together with the prescribed standard curves (in red), and the curve resulting from a pure lossless model (in green).

The thermoacoustic effect is typically seen and is most pronounced at resonances, which are rounded and shift down in frequency. To model these effects, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations, solving the momentum (Navier-Stokes), mass (continuity), and energy conservation equations. This is achieved by solving the thermoacoustics equations in the Thermoacoustic interface, included in the Acoustics Module. The equations are also known as the thermo-viscous acoustics, visco-thermal acoustics, and linearized Navier-Stokes equations.

Here, we will present the physical background for the thermoacoustics equations along with the important boundary layer characteristic, length scale. We will also provide a short description of the material parameters necessary for describing fluid media.

Physics of Thermoacoustics

Acoustic waves are the propagation of small linear fluctuations in pressure on top of a background stationary (atmospheric) pressure. The governing equations for the fluctuations (the wave equation or Helmholtz’s equation) are derived by perturbing, or linearizing, the fundamental governing equations of fluid mechanics — the Navier-Stokes equation, the continuity equation, and the energy equation. Doing this results in the conservation equations for momentum, mass, and energy for any small (acoustic) perturbation.

For many applications simulating acoustics, a series of assumptions are then made to simplify these equations: the system is assumed lossless and isentropic (adiabatic and reversible). Yet, if you retain both the viscous and heat conduction effects, you will end up with the equations for thermoacoustics that solve for the acoustic perturbations in pressure, velocity, and temperature.

Governing Equations

The procedure to derive the governing equations in the frequency domain is to assume small harmonic oscillations about the steady background properties. The dependent variables take the form:

p = p_0+p’e^{i\omega t}, \quad \mathbf{u} = \mathbf{u}_0+\mathbf{u}’ e^{i\omega t}, \quad T = T_0 + T’ e^{i\omega t}

where p is the pressure, \mathbf{u} is the velocity field, T is the temperature, and \omega is the angular frequency. Primed (‘) variables are the acoustic variables, while variables accompanied with the subscript 0 represent the background mean flow.

In thermoacoustics, the background fluid is assumed to be quiescent so that \mathbf{u}_0=\mathbf{0}. The background pressure p_0 and background temperature T_0 need to be specified (they can be functions of space). Inserting the above equation into the governing equations and only retaining terms linear in the first-order variables yields the governing equations for the propagation of acoustic waves including viscous and thermal losses.

Note: Details on this can be found in the User’s Guide of the Acoustics Module in the “Theory Background for the Thermoacoustic Branch” section.

The governing equations in the Thermoacoustic interface, in the frequency domain, are the continuity equation (omitting primes from the acoustic variables):

i\omega\rho =-\rho_0 (\nabla\cdot\mathbf{u})

where \rho_0 is the background density; the momentum equation:

i\omega\rho_0 \mathbf{u} = \nabla\cdot \left(-p\mathbf{I}+\mu ( \nabla\mathbf{u}+(\nabla\mathbf{u})^T )+\left(\mu_\textrm{B}-\frac{2}{3}\mu \right)(\nabla\cdot\mathbf{u})\mathbf{I} \right)

where \mu is the dynamic viscosity and \mu_\textrm{B} is the bulk viscosity, and the term on the right hand side represents the divergence of the stress tensor; the energy conservation equation:

i\omega (\rho_0 C_p T – T_0 \alpha_0 p) = -\nabla\cdot(-\textrm{k}\nabla T) + Q

where C_p is the heat capacity at constant pressure, \textrm{k} is the thermal conductivity, \alpha_0 is the coefficient of thermal expansion (isobaric), and Q is a possible heat source; and finally, the linearized equation of state relating variations in pressure, temperature, and density:

\rho = \rho_0 (\beta_T p – \alpha_0 T)

where \beta_T is the isothermal compressibility.

The left-hand sides of the governing equations represent the conserved quantities: mass, momentum, and energy (actually entropy). In the frequency domain, multiplication with i\omega corresponds to differentiation with respect to time. The terms on the right-hand sides represent the processes that locally change or modify the respective conserved quantity. In two of the equations, diffusive loss terms are present, due to viscous shear and thermal conduction. Viscous losses are present when there are gradients in the velocity field, while thermal losses are present when there are gradients in the temperature.

Viscous and Thermal Boundary Layer

When sound waves propagate in a fluid bounded by walls, so-called viscous and thermal boundary layers are created at the solid surfaces. At the wall, the no-slip condition applies to the velocity field, \mathbf{u} = 0, and an isothermal condition for the temperature, namely T = 0. The isothermal condition is a very good approximation, as thermal conduction is typically orders of magnitude higher in solids than fluids. These two conditions give rise to the acoustic boundary layer, which consists of the viscous and a thermal boundary layers. The flow transforms from the bulk condition of being nearly lossless and described by isentropic (adiabatic) conditions to the conditions in this layer.

The problem of a time-harmonic wave propagating in the horizontal plane along a wall (this could be waves propagating in a small section of a pipe) is illustrated in the figures below. The left figure shows the velocity amplitude and the right figure the fluid’s temperature, from the wall towards the bulk, while the middle figure shows the velocity magnitude as well as an animation indicating the velocity vector over a harmonic period.

Velocity amplitude
Fluid temperature

Thermoacoustics animation
Velocity amplitude (left) and fluid temperature (right), from the wall to the bulk, of an acoustics wave propagating in the horizontal plane (bottom). The viscous and thermal boundary layer thicknesses are indicated by the red dotted lines closest to the wall. The upper dotted lines represent 2 \pi times the boundary layer thickness, in each case. The animation indicates the acoustic velocity components, while the color plot shows velocity amplitude.

The viscous and thermal boundary layers are clearly visible. Because gradients are large in the boundary layer, losses are large here too. This means that in systems of relatively small dimensions, the losses associated with the boundary layer become important. In many engineering applications (miniature transducers, mobile devices, etc.), including the losses associated with the boundary layer is essential in order to model the correct physical behavior and response.

The viscous characteristic length is shown as a red dotted line in the velocity and temperature plots shown above, together with 2 \pi times the value (known as the viscous/thermal wavelength). The two characteristic lengths are related by the dimensionless Prandtl number Pr:

\textrm{Pr} = \frac{C_p \mu}{\textrm{k}} \qquad \delta_\textrm{visc} = \sqrt{\textrm{Pr}} \: \delta_\textrm{therm}

which gives a measure of the ratio of the viscous to thermal losses in a system. For air, this number is 0.7, while it is around 7.1 for water. In air, the thermal and viscous effects are roughly equal in importance, while for water (and most other fluids), the thermal losses only play a more minor role. The viscous and thermal boundary layer thicknesses exist as pre-defined variables for use in postprocessing in the Acoustics Module, and they are denoted by ta.d_visc and ta.d_therm. The Prandtl number is denoted by ta.Pr.

The plane wave problem can be solved analytically and expressions for the viscous (d_\textrm{visc}) and thermal (d_\textrm{therm}) boundary layer thickness subsequently derived. They are given by:

\delta_\textrm{visc} = \sqrt{\frac{2\mu}{\omega\rho_0}} \qquad \delta_\textrm{therm} = \sqrt{\frac{2 \textrm{k}}{\omega\rho_0 C_p}}

The value of d_\textrm{visc} is 0.22 mm for air and 0.057 mm for water at 100 Hz, 20°C and 1 atm. Over a range of frequencies, the viscous and thermal boundary layer thickness can be plotted, such as the figures below:

Value of the viscous and thermal boundary layer thickness for air
Value of the viscous and thermal boundary layer thickness for water

The value of the viscous (d_\textrm{visc}) and thermal (d_\textrm{therm}) boundary layer thickness as functions of frequency for (left) air and (right) water.

This shows the diminishing effect of viscous and thermal losses at increasing acoustic wave propagation frequencies. Finally, another important effect that is captured when modeling with the Thermoacoustic interface is the transition from adiabatic to isothermal acoustics at low frequencies in small devices. This effect occurs when the thermal boundary layer stretches over the full device and is important in, for example, condenser microphones, such as the B&K 4133 condenser microphone. At isothermal conditions the speed of sound changes to the isothermal speed of sound.

Bulk Losses and Attenuation

It is important to note that viscous and thermal losses also exist in the bulk of the fluid. These are losses that typically occur when acoustic signals propagate over long distances and are attenuated. One example of this is sonar signals. These types of losses are, in air, only dominating at very high frequencies (they can be neglected at audio frequencies). The bulk losses are, of course, also described by the governing equations for thermoacoustics as they include all the physics. However, modeling large domains with the thermoacoustics equations is very computationally expensive. In the Acoustics Module, you should instead use the Pressure Acoustics interface and select one of the available fluid models: Viscous, Thermally conducting, or Thermally conducting and viscous.

Material Parameters

Solving a full thermoacoustic model involves defining several material parameters:

  • Dynamic viscosity \mu:
    • The dynamic viscosity measures the fluid’s resistance to shearing in the fluid. It is the constant that relates stress to velocity gradients. The dynamic viscosity is related to the kinematic viscosity \nu by the relation \mu = \rho_0 \: \nu. The symbol for the dynamic viscosity \eta is also sometimes used.
  • Bulk viscosity \mu_\textrm{B}:
    • The bulk viscosity is also known as the volume viscosity, the second viscosity, or the expansive viscosity. It is related to losses that appear due to the compression and expansion of the fluid. \mu_\textrm{B} appears in the stress tensor term (right side of equation 3), which has to do with the compressibility (\nabla\cdot\mathbf{u}) of the bulk fluid. This factor is difficult to measure and is often seen to depend on the frequency.
  • Heat capacity at constant pressure (specific) C_p:
    • This material parameter gives a measure of how much energy is required to change the temperature of the fluid (at constant pressure).
  • Coefficient of thermal conduction \textrm{k}:
    • The coefficient of proportionality between the temperature gradient and the heat flux in Fourier’s heat conduction law.
  • Coefficient of thermal expansion (isobaric) \alpha_0:
    • This is the volumetric thermal expansion of the fluid and expresses the ability of the fluid to expand when its temperature rises.
  • Isothermal compressibility \beta_T:
    • Important parameter in the equation of state of the fluid. It relates changes in pressure to changes in volume in the fluid. The isothermal compressibility is related to the usual (isentropic) compressibility through the ratio of specific heats by \beta_T = \gamma \beta_s.

Conclusion and Next Steps

Now that you know the theory behind thermoacoustics and the associated equations, we can move on to tips and tricks for setting up a thermoacoustic model using COMSOL Multiphysics and the Acoustics Module. We will discuss that as well as examples and applications in the next blog post of this series.

Further Reading and References

  • COMSOL Documentation: Acoustics Module User’s Guide
  • COMSOL Documentation: Acoustics Module User’s Guide > The Thermoacoustics Branch
  • D. T. Blackstock, “Fundamentals of Physical Acoustics”, John Wiley and Sons, 2000
  • S. Temkin, “Elements of Acoustics”, Acoustical Society of America, 2001
  • B. Lautrup, “Physics of Continuous Matter”, Second Edition, CRC Press, 2011
  • P. M. Morse and K. U. Ingard, “Theoretical Acoustics” Princeton University Press
  • A. D. Pierce, “Acoustics; An Introduction to Its Physical Principles and Applications”, Acoustical Society of America, 1989
  • A. S. Dukhin and P. J. Goetz, “Bulk viscosity and compressibility measurements using acoustic spectroscopy”, J. Chem. Phys. 130, 124519 (2009)

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