19. Acoustics

Assuming negligible viscosity and conductivity, Navier-Stokes equations for compressible fluids become Euler equations1

The conservative form of Navier-Stokes equations

\[\begin{split}\begin{cases} \partial_t \rho + \nabla \cdot \left( \rho \mathbf{u} \right) = 0 \\ \partial_t \left( \rho \mathbf{u} \right) + \nabla \cdot \left( \rho \mathbf{u} \mathbf{u} \right) = \rho \mathbf{g} + \nabla \cdot \mathbb{T} \\ \partial_t \left( \rho e^t \right) + \nabla \cdot \left( \rho e^t \mathbf{u} \right) = \rho \mathbf{g} \cdot \mathbf{u} + \nabla \cdot \left( \mathbb{T} \cdot \mathbf{u} \right) - \nabla \cdot \mathbf{q} \end{cases}\end{split}\]

need

• boundary conditions

• constitutive laws for relating stress tensor \(\mathbb{T}\) and conductive heat flux \(\mathbf{q}\) to dynamic variables \(\rho\), \(\mathbf{m} := \rho \mathbf{u}\), \(E^t := \rho e^t\). As a common example, stress tensor for Newtonian fluids is the sum of pressure and viscosity stress, and this latter contribution is linear and isotropic w.r.t. first order spatial derivatives of the velocity field2,

  \[\mathbb{T} = - p \mathbb{I} + \mathbb{S} = - p \mathbb{I} + 2 \mu \mathbb{D} + \lambda \left( \nabla \cdot \mathbf{u} \right) \mathbb{I} \ ,\]

  being \(p\) the pressure field, \(\mathbb{D} = \frac{1}{2} \left[ \nabla \mathbf{u} + \nabla^T \mathbf{u} \right]\) the deformation velocity tensor and \(\mu\), \(\lambda\) the viscosity coefficients. Fourier’s law provides a linear isotropic relation between the conductive heat flux and the temperature gradient

  \[\mathbf{q} = - k \nabla T \ .\]

• state equations, for relating thermodynamic variables appearing in constitutive laws with dynamic variables. Using constitutive laws for Newtonian fluids and Fourier’s law, there’s a need for state equations relating pressure, temperature and - if not possible to model as constant - viscosity coefficients and heat conductivity,

  \[\begin{split}\begin{aligned} p & = p(\rho, \mathbf{m}, E_t) \\ T & = T(\rho, \mathbf{m}, E_t) \\ & ... \end{aligned}\end{split}\]

  Usually internal energy is used in state equations. Mass density of total energy is defined as the sum of internal energy and kinetic energy

  \[e^t = e + \frac{|\mathbf{u}|^2}{2} \ .\]

Introducing the constitutive laws and the state equations in Navier-Stokes equations and collecting terms with 1-st order and 2-nd order space derivatives,

\[\begin{split}\begin{cases} \partial_t \rho + \nabla \cdot \left( \rho \mathbf{u} \right) = 0 \\ \partial_t \left( \rho \mathbf{u} \right) + \nabla \cdot \left( \rho \mathbf{u} \mathbf{u} + p \mathbb{I} \right) = \rho \mathbf{g} + \nabla \cdot \mathbb{S} \\ \partial_t \left( \rho e^t \right) + \nabla \cdot \left( \rho e^t \mathbf{u} + p \mathbf{u} \right) = \rho \mathbf{g} \cdot \mathbf{u} + \nabla \cdot \left( \mathbb{S} \cdot \mathbf{u} \right) - \nabla \cdot \mathbf{q} \end{cases}\end{split}\]

Euler equations immediately follow by setting equal to zero the contribution of viscosity and heat conduction, i.e. the terms collecting the 2-nd order space derivatives, those ones that have diffusive nature,

\[\begin{split}\begin{cases} \partial_t \rho + \nabla \cdot \left( \rho \mathbf{u} \right) = 0 \\ \partial_t \left( \rho \mathbf{u} \right) + \nabla \cdot \left( \rho \mathbf{u} \mathbf{u} + p \mathbb{I} \right) = \rho \mathbf{g} \\ \partial_t \left( \rho e^t \right) + \nabla \cdot \left( \rho h^t \mathbf{u} \right) = \rho \mathbf{g} \cdot \mathbf{u} \end{cases}\end{split}\]

todo discuss the hyperbolic mathematical nature of Euler equations, and link to Math:PDEs:Hyperbolic equations

• Convective form…

\[\begin{split}\begin{cases} D_t \rho + \rho \nabla \cdot \mathbf{u} = 0 \\ \rho D_t \mathbf{u} + \nabla p = \rho \mathbf{g} \\ \rho D_t e^t + \nabla \cdot ( p \mathbf{u} ) = \rho \mathbf{g} \cdot \mathbf{u} \end{cases}\end{split}\]

• Alternative sets of equations, if differential equations hold (no discontinuities: no shocks,…): internal energy instead of total energy, entropy,…

  • internal energy, differente of total energy and kinetic energy (momentum equation \(\cdot \mathbf{u}\)):

    \[\rho D_t e + p \nabla \cdot \mathbf{u} = 0\]

  • entropy, from the 1-st principle of theromdynamics \(de = T ds + \frac{p}{\rho^2} d \rho\)

    \[\begin{split}\begin{aligned} \rho D_t s & = \frac{\rho}{T} \left[ D_t e - \frac{p}{\rho^2} D_t \rho \right] = \\ & = \frac{\rho}{T} \left[ - \frac{p}{\rho} \nabla \cdot \mathbf{u} - \frac{p}{\rho^2} \left( - \rho \nabla \cdot \mathbf{u} \right) \right] = 0 \ , \end{aligned}\end{split}\]

    i.e.

    \[D_t s = 0\]

    todo \(1)\) entropy and shocks, \(2)\) entropy and vorticity,…

19.1. Linearized Euler equations

Linearization around a reference state

…

19.1.1. Uniform and steady reference state

\(\overline{\rho}, \overline{\mathbf{u}}, \overline{s}\). Time and space derivatives of the reference state are identically zero, thus Euler equations in convective form

\[\begin{split}\begin{aligned} & \partial_t \rho + \mathbf{u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{u} = 0 \\ & \rho \partial_t \mathbf{u} + \rho \mathbf{u} \cdot \nabla \mathbf{u} + \nabla p = \rho \mathbf{g} \\ & \partial_t s + \mathbf{u} \cdot \nabla s = 0 \end{aligned}\end{split}\]

are linearized as

\[\begin{split}\begin{aligned} & \partial_t \rho' + \mathbf{u} \cdot \nabla \rho' + \rho \nabla \cdot \mathbf{u}' = 0 \\ & \rho \partial_t \mathbf{u}' + \rho \mathbf{u} \cdot \nabla \mathbf{u}' + \nabla p' = \rho' \mathbf{g} \\ & \partial_t s' + \mathbf{u} \cdot \nabla s' = 0 \end{aligned}\end{split}\]

If the entropy is uniform everywhere (also at the boundary, so that no entropy enters from it), its perturbation is identically zero \(s' = 0\), \(s(\mathbf{r},t) = \overline{s}\). Thus, for any thermodynamic variable written as function of density and entropy as the pair of independent variable, e.g. pressure \(p(\rho, s)\), its differential is a function of the differential of the density only, being \(ds = 0\)

\[\begin{split}\begin{aligned} dp & = \left( \frac{\partial p}{\partial \rho} \right)_s d \rho + \left( \frac{\partial p}{\partial s} \right)_\rho \underbrace{d s}_{=0} = \\ & = \left( \frac{\partial p}{\partial \rho} \right)_s d \rho = \\ & = c^2(\rho, s) \, d\rho \ . \end{aligned}\end{split}\]

Now, for negligible volume forces \(\mathbf{g} = \mathbf{0}\), and writing \(\nabla p = c^2 \nabla \rho\), mass and momentum linearized equations read

\[\begin{split}\begin{aligned} & \partial_t \rho' + \overline{\mathbf{u}} \cdot \nabla \rho' + \overline{\rho} \nabla \cdot \mathbf{u}' = 0 \\ & \overline{\rho} \partial_t \mathbf{u}' + \overline{\rho} \overline{\mathbf{u}} \cdot \nabla \mathbf{u}' + c^2(\overline{\rho}, \overline{s}) \nabla \rho' = \mathbf{0} \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} & D^{\overline{\mathbf{u}}}_t \rho' + \overline{\rho} \nabla \cdot \mathbf{u}' = 0 \\ & \overline{\rho} D^{\overline{\mathbf{u}}}_t \mathbf{u}' + \overline{c}^2 \nabla \rho' = \mathbf{0} \ . \end{aligned}\end{split}\]

being \(D^{\overline{\mathbf{u}}}_t\) the material time derivative with the reference velocity field,

\[D^{\overline{\mathbf{u}}}_t = \partial_t + \overline{\mathbf{u}} \cdot \nabla \ .\]

19.1.2. Wave equations

A wave equation for density perturbation \(\rho'(\mathbf{r},t)\) is obtained taking the difference of time derivative of the mass equation and the divergence of momentum equation,

\[\begin{split}\begin{aligned} & D^{\overline{\mathbf{u}}}_{tt} \rho' + \overline{\rho} D^{\overline{\mathbf{u}}}_{t} \nabla \cdot \mathbf{u}' = 0 \\ & \overline{\rho} \nabla \cdot D^{\overline{\mathbf{u}}}_t \mathbf{u}' + \overline{c}^2 \nabla^2 \rho' = 0 \ , \end{aligned}\end{split}\]

it follows

\[\frac{1}{\overline{c}^2} D^{\overline{\mathbf{u}}}_{tt} \rho'- \nabla^2 \rho' = 0 \ .\]

A wave equation for velocity perturbation \(\mathbf{u}'(\mathbf{r},t)\) is obtained taking the difference of time gradient of the mass equation and the time derivative of momentum equation,

\[\begin{split}\begin{aligned} & \nabla D^{\overline{\mathbf{u}}}_{t} \rho' + \overline{\rho} \nabla \left( \nabla \cdot \mathbf{u}' \right) = 0 \\ & \overline{\rho} D^{\overline{\mathbf{u}}}_{tt} \mathbf{u}' + \overline{c}^2 D^{\overline{\mathbf{u}}}_{t} \nabla \rho' = \mathbf{0} \ , \end{aligned}\end{split}\]

and using vector identity \(\nabla^2 \mathbf{v} = \nabla \left( \nabla \cdot \mathbf{v} \right) - \nabla \times \nabla \times \mathbf{v}\), it follows

\[\frac{1}{\overline{c}^2} D^{\overline{\mathbf{u}}}_{tt} \mathbf{u}' - \nabla^2 \mathbf{u}' - \nabla \times \boldsymbol\omega' = \mathbf{0} \ ,\]

being \(\boldsymbol\omega' = \nabla \times \mathbf{u}'\) the vorticity of the velocity perturbation.

For isentropic flows, differential of density perturbation \(d \rho'\) can be related to differential of pressure perturbation, \(d \rho' = \frac{1}{\overline{c}^2} dp'\), so that pressure perturbation field is governed by a wave equation as well.

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1

NS equations are second-order in space due to the diffusive contributions of viscous stress \(\nabla \cdot \mathbb{T} = \nabla \cdot \left[ 2 \mu \nabla^s \mathbf{u} + \lambda \left( \nabla \cdot \mathbf{u} \right) \mathbb{T} \right]\) for Newtonian fluids in momentum equation and heat conduction flux \(-\nabla \cdot \mathbf{q} = \nabla \cdot \left( k \nabla T \right)\) with Fourier law in total energy equation. Boundary conditions needed for the mathematical problem change as well: while NS equations require no-slip boundary conditions apply on solid bodies, \(\mathbf{u} = \mathbf{u}_b\), Euler equations need only no-penetration boundary conditions \(\mathbf{u} \cdot \hat{\mathbf{n}} = \mathbf{u}_b \cdot \hat{\mathbf{n}}\).

2

todo discuss isotropic tensors, or link to Math:Vector and Tensor Algebra and Calculus:Isotropic Tensors