Perturbed Convective Wave Model

The Perturbed Convective Wave model is a hybrid acoustics model that uses a wave equation to calculate the sound generation and propagation in incompressible fluid flows. This wave equation solves for an acoustic potential from which acoustic quantities such as acoustic pressure, acoustic velocity, or acoustic density are derived.

The Perturbed Convective Wave model is based on the Linearized Perturbed Compressible Equation (LPCE) as proposed by Seo and Moon [914] and reformulated by Piepiorka and Von Estorff [908]. Assuming incompressible fluid flow and a constant speed of sound, the wave equation for the acoustic potential $ϕ^{a}$ is defined as:

Figure 1. EQUATION_DISPLAY

\frac{D^{2} ϕ^{a}}{D t^{2}} - c_{0}^{2} Δ (ϕ^{a} + τ \frac{\partial ϕ^{a}}{\partial t}) - \frac{1}{ρ} \nabla ϕ^{a} \cdot \nabla p_{i n c}^{f} = - \frac{1}{ρ} \frac{D p_{i n c}^{f}}{D t} + S_{ϕ^{a}}

(4735)

where:

$c_{0}$ is the speed of sound.
$t$ is time.
$ρ$ is the density.
$p_{i n c}^{f}$ is the incompressible pressure fluctuation.
$S_{ϕ^{a}}$ is the user-defined noise source.

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

Figure 2. EQUATION_DISPLAY

τ = b_{d} \frac{Δ x}{π c_{0}}

(4736)

where:

$b_{d}$ is the acoustic damping coefficient.
$Δ x = \sqrt[3]{V}$ where $V$ is the cell volume.

The definition of the substantial (total) derivative $\frac{D}{D t}$ is based on the flow velocity $v$ as:

Figure 3. EQUATION_DISPLAY

\frac{D}{D t} = \frac{\partial}{\partial t} + v \cdot \nabla

(4737)

The acoustic pressure $p^{a}$ , the acoustic density $ρ^{a}$ , and the acoustic velocity $v^{a}$ are derived from the acoustic potential as:

Figure 4. EQUATION_DISPLAY

p^{a} = ρ \frac{D ϕ^{a}}{D t}

(4738)

Figure 5. EQUATION_DISPLAY

ρ^{a} = \frac{p^{a}}{c_{0}^{2}} + ρ

(4739)

Figure 6. EQUATION_DISPLAY

v^{a} = - \nabla ϕ^{a}

(4740)

Boundary Conditions

Depending on the acoustic specification at the boundary, the following boundary conditions apply to

ϕ^{a}

Non-reflective: The non-reflective boundary condition allows the acoustic potential to leave the domain through the boundary without any spurious reflections.; The gradient of the acoustic potential at the boundary is given as:; Figure 7. EQUATION_DISPLAY

$\nabla ϕ^{a} \cdot a_{f} = \frac{c_{0} - v \cdot a_{f}}{c_{0}^{2} - {‖ a_{f} ‖}^{2}} \frac{\partial ϕ^{a}}{\partial t}$

(4741); where $a_{f}$ is the face normal area and $‖ a_{f} ‖$ is its magnitude.

Reflective: For reflective boundaries, a Neumann boundary condition is imposed. For reflective walls that are moving, the acoustic potential is additionally constrained as:; Figure 8. EQUATION_DISPLAY

$\nabla ϕ^{a} = - v_{w a l l} \cdot a_{f}$

(4742); where $v_{w a l l}$ is the velocity of the wall.
Specified $ϕ^{a}$: A Dirichlet boundary condition where the specified value for $ϕ^{a}$ is imposed.

Newmark Alpha Method

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

Figure 9. EQUATION_DISPLAY

\frac{1}{c_{0}^{2}} \int_{V} {(\frac{\partial^{2} ϕ^{a}}{\partial t^{2}})}^{n + 1} d V - (1 + α) \int_{s} \nabla [{(ϕ^{a})}^{n + 1} - τ {(\frac{\partial ϕ^{a}}{\partial t})}^{n + 1}] \cdot a d s = \int_{V} [(1 + α) {(\nabla\cdot \nabla p_{i n c}^{f})}^{n + 1} - α {(\nabla\cdot \nabla p_{i n c}^{f})}^{n}] d V - α \int_{s} \nabla [{(ϕ^{a})}^{n} - τ {(\frac{\partial ϕ^{a}}{\partial t})}^{n}] \cdot a d s

(4743)

where $α$ is the Newmark alpha parameter.