Lighthill Wave Model

The Lighthill Wave model provides an efficient method to model sound propagation for incompressible fluid flow ( $M a < 0.2$ ).

To calculate the sound propagation in either quiescent or turbulent flows, the Lighthill Wave model introduces a separate wave equation. This equation solves for the transport of the Lighthill pressure $p^{l}$ , which can be represented as the sum of acoustic pressure $p^{a}$ and hydrodynamic pressure $p^{h}$ :

Figure 1. EQUATION_DISPLAY

p^{l} = p^{a} + p^{h}

(4729)

The transport equation for the Lighthill pressure $p^{l}$ is given as:

Figure 2. EQUATION_DISPLAY

\frac{1}{c_{0}^{2}} \frac{\partial^{2} p^{l}}{\partial t^{2}} - \nabla^{2} (p^{l} + τ \frac{\partial p^{l}}{\partial t}) = \nabla\cdot h

(4730)

where $h$ is defined as the divergence of the fluctuating part of the Lighthill Stress tensor $v \otimes v$ as:

Figure 3. EQUATION_DISPLAY

h = \nabla\cdot [b_{s} ρ (v \otimes v - \bar{v \otimes v})]

(4731)

where:

$p^{l}$ is the Lighthill pressure, which corresponds to the sum of the fluctuating hydrodynamic pressure and the acoustic pressure.
$c_{0}$ is the far-field speed of sound.
$t$ is time.
$v$ is the velocity.
$b_{s}$ is the Noise Source Weighting Coefficient.
$ρ$ is the density.

$τ$ is the noise source damping term, defined as:

Figure 4. EQUATION_DISPLAY

τ = b_{d} \frac{Δ x}{π c_{0}}

(4732)

where:

$b_{d}$ is the Acoustic Damping Coefficient.
$Δ x = \sqrt[3]{V}$ , where $V$ is the cell volume.

Non-reflective Boundary Condition

The non-reflective boundary condition allows acoustic waves to leave the domain through the boundary without any spurious reflections.

The gradient of the Lighthill pressure at a non-reflective boundary is given as:

Figure 5. EQUATION_DISPLAY

\nabla p^{l} \cdot a_{f} = {\frac{\partial p^{l}}{\partial t}}_{c} ‖ a_{f} ‖

(4733)

where $a_{f}$ is the face normal area and $‖ a_{f} ‖$ is its magnitude.

Newmark Alpha Method

The Newmark- $α$ method uses the Hilber-Hughes-Taylor- $α$ (HHT- $α$ ) scheme to express the Lighthill pressure time derivative as:

Figure 6. EQUATION_DISPLAY

\frac{1}{c_{0}^{2}} \int_{V} {(\frac{\partial^{2} p^{l}}{\partial t^{2}})}^{n + 1} d V - (1 + α) \int_{s} \nabla [{(p^{l})}^{n + 1} - τ {(\frac{\partial p^{l}}{\partial t})}^{n + 1}] \cdot a d s = \int_{V} [(1 + α) {(\nabla\cdot h)}^{n + 1} - α {(\nabla\cdot h)}^{n}] d V - α \int_{s} \nabla [{(p^{l})}^{n} - τ {(\frac{\partial p^{l}}{\partial t})}^{n}] \cdot a d s

(4734)

where:

$n + 1$ is the time-step for which the solution is being computed.
$n$ is the previous time-step.
$α$ is a weighting parameter, $- 0.33 \leq α \leq 0$ .