Author: fumiya

This is the second in a series of posts about the computational aeroacoustics and I will try to introduce the very basics of the acoustic wave with some pictures.

We consider an oscillatory motion with small amplitude in a compressible fluid as shown in the following picture. It shows the distribution of the sound pressure (acoustic pressure) that is defined as the local pressure deviation from the equilibrium pressure \(p_0\) caused by the sound wave propagating from left to right direction.

Since we now consider small oscillations, we can write the local pressure \(p\) and density \(\rho\) in the form

\begin{align}
p &= p_0 + p^{´}, \tag{1a} \label{eq:pressure} \\
\rho &= \rho_0 + \rho^{´}, \tag{1b} \label{eq:density}
\end{align}

where \(p_0\) and \(\rho_0\) are the constant equilibrium pressure and density and \(p^{´}\) and \(\rho^{´}\) are their variations in the sound wave (\(p^{´} \ll p_0, \rho^{´} \ll \rho_0\)), so the above figure is the contour of \(p^{´}\).

We hereafter ignore the fluid viscosity so that only the effect of compressibility is taken into account. Then, the governing equations of the fluid flow is the continuity equation

\begin{align}
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{u}) = 0 \tag{2} \label{eq:continuity}
\end{align}

and the Euler’s equation

\begin{align}
\frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla)\boldsymbol{u} + \frac{1}{\rho}\nabla p = 0 \tag{3} \label{eq:euler}
\end{align}

where \(\boldsymbol{u}\) is the velocity field. Substituting the eqns. \eqref{eq:pressure} and \eqref{eq:density} into the governing equations and neglecting small quantities of the second order, we get

\begin{align}
\frac{\partial \rho^{´}}{\partial t} + \rho_0 \nabla \cdot \boldsymbol{u} = 0, \tag{4} \label{eq:continuity2}
\end{align}

and

\begin{align}
\frac{\partial \boldsymbol{u}}{\partial t} + \frac{1}{\rho_0} \nabla p^{´} = 0. \tag{5} \label{eq:euler2}
\end{align}

We notice that a sound wave in an ideal fluid is adiabatic and the following relationship holds between the small changes in the pressure and density

\begin{align}
p^{´} = \left(\frac{\partial p}{\partial \rho} \right)_s \rho^{´} \tag{6} \label{eq:adiabatic}
\end{align}

where the subscript s denotes that the partial derivative is taken at constant entropy. Substituting it into \eqref{eq:continuity2}, we get

\begin{align}
\frac{\partial p^{´}}{\partial t} + \rho_0 \left(\frac{\partial p}{\partial \rho} \right)_s \nabla \cdot \boldsymbol{u} = 0. \tag{7} \label{eq:continuity3}
\end{align}

If we introduce the velocity potential \(\boldsymbol{u} = \nabla \phi\), we can derive the relationship between \(p^{´}\) and the potential \(\phi\) from \eqref{eq:euler2}

\begin{align}
p^{´} = -\rho_0 \frac{\partial \phi}{\partial t}. \tag{8} \label{eq:pandphi}
\end{align}

We then obtain the following wave equation from \eqref{eq:continuity3}

\begin{align}
\frac{\partial^2 \phi}{\partial t^2} – c^2 \Delta \phi = 0 \tag{9} \label{eq:waveEqn}
\end{align}

where \(c\) is the speed of sound in an ideal fluid

\begin{align}
c = \sqrt{\left(\frac{\partial p}{\partial \rho} \right)_s}. \tag{10} \label{eq:soundSpeed}
\end{align}

Applying the gradient operator to \eqref{eq:waveEqn}, we find that the each of the three components of the velocity \(\boldsymbol{u}\) satisfies an equation having the same form, and on differentiating \eqref{eq:waveEqn} with respect to time we see that the pressure \(p^{´}\) (and therefore \(\rho^{´}\)) also satisfies the wave equation.

– Landau and Lifshitz, Fluid Mechanics

In a travelling plane wave,

we find
\begin{align}
u_x = \frac{p^{´}}{\rho_0 c}. \tag{11} \label{eq:uxandp}
\end{align}

Substituting here from \eqref{eq:adiabatic} \(p^{´} = c^2 \rho^{´}\), we find the relation between the velocity and the density variation:

\begin{align}
u_x = \frac{c \rho^{´}}{\rho_0}. \tag{12} \label{eq:uxandrho}
\end{align}

– Landau and Lifshitz, Fluid Mechanics

This book is freely accessible from the link.

The following picture shows the velocity distribution \(u_x\) calculated using a compressible solver in OpenFOAM at the time corresponding to the pressure variation shown in the above picture. We can calculate the velocity using \eqref{eq:uxandp}

\begin{align}
u_x = \frac{10}{1.2 \times 340} = 0.0245 [m/s]
\end{align}

and it is in good agreement with the result obtained using OpenFOAM.

This is the first in a series of posts about the computational aeroacoustics that is abbreviated as CAA. Aeroacoustics is mainly concerned with the generation of sound or noise through a fluid flow. Familiar examples are

• Aeolian tone
• Edge tone
• Cavity tone.

They can be studied by both experimental and computational methods. In an experimental approach, the aerodynamically generated sound is measured in the anechoic chamber that is designed to prevent reflections of sound on the wall, making the room echo-free. The following video created by Microsoft will help us to understand the structure of it.

The computational techniques for simulating flow-generated noise can be classified into two broad categories:

The governing equations of the compressible fluid flow is the compressible Navier-Stokes equations and they also describe the generation and propagation of acoustic noise, so we can solve the computational aeroacoustics problems by solving the transient compressible Navier-Stokes equations both in the source region where the flow disturbances generate noise and propagation region where the generated acoustic waves propagate. Acoustic waves have to be resolved in both regions so that the noise can be accurately simulated at observation locations. However, the solution of the Navier–Stokes equations with fine mesh over large domains to determine farfield noise is computationally very expensive.

As is the case in the experimental approaches, it is essential to prevent the reflection of the acoustic waves on the artificially truncated boundary (i.e. it does not stretch to infinity) of the computational domain in order to obtain an accurate result. A variety of numerical techniques have been developed for this purpose:

1. Navier-Stokes Characteristic Boundary Conditions (NSCBC)
2. Artificial dissipation and damping in an absorbing zone
3. Grid stretching and numerical filtering in a “sponge layer” or “exit zone”
4. Perfectly matched layer (PML)

In OpenFOAM, acousticDampingSource is categorized into Ⅱ and the waveTransmissive boundary condition is an approximation of NSCBC.

David P. Lockard and Jay H. Casper [1] states that:

The physics-based, airframe noise prediction methodology under investigation is a hybrid of aeroacoustic theory and
computational fluid dynamics (CFD). The near-field aerodynamics associated with an airframe component are simulated to obtain the source input to an acoustic analogy that propagates sound to the far field. The acoustic analogy employed within this current framework is that of Ffowcs Williams and Hawkings, who extended the analogies of Lighthill and Curle to the formulation of aerodynamic sound generated by a surface in arbitrary motion.

• Lighthill’s analogy

Lighthill derived the following wave equation \eqref{eq:Lighthill} from the compressible Navier-Stokes equations:
\begin{align}
\frac{\partial^2 \rho}{\partial t^2} – c_0^2 \nabla^2 \rho = \frac{\partial^2 T_{ij}}{\partial x_i \partial x_j}, \tag{1} \label{eq:Lighthill}
\end{align}
where the so-called Lighthill (turbulence) stress tensor is expressed as

\begin{align}
T_{ij} = \rho u_i u_j + \left( p-c_0^2 \rho \right)\delta_{ij} -\mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} – \frac{2}{3}\delta_{ij}\frac{u_k}{x_k} \right), \tag{2} \label{eq:Tij}
\end{align}
\(\delta_{ij}\) is the Kronecker delta and \(c_0\) is the speed of sound in the medium in its equilibrium state.

• Curle’s analogy

The existence of the objects (solid walls) is not considered in the Lighthill’s theory and Curle developed the theory so that it can be dealt with. The density variation at the observer location \(\boldsymbol{x}\) is calculated from the following equation \eqref{eq:Curle}
\begin{align}
\acute{\rho}(\boldsymbol{x}, t) &= \rho(\boldsymbol{x}, t)\;- \rho_0 \\
&=\frac{1}{4 \pi c_0^2} \frac{\partial^2}{\partial x_i \partial x_j} \int_{V}\frac{[T_{ij}]}{r}dV – \frac{1}{4 \pi c_0^2} \frac{\partial}{\partial x_i}\int_{S}\frac{[P_i]}{r}dS, \tag{3} \label{eq:Curle}
\end{align}
where
\begin{align}
P_i = -n_j \{ \delta_{ij}p -\mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} – \frac{2}{3}\delta_{ij}\frac{u_k}{x_k} \right) \}, \tag{4} \label{eq:Pi}
\end{align}
\(r\) is the distance between the receiver \(\boldsymbol{x}\) and the source position \(\boldsymbol{y}\) in \(V\) (or on \(S\)) and the operator [] denotes evaluation at the retarded time \(t-r/c_0\).

• Ffowcs-Williams and Hawkings (FW-H) analogy

In the following video, we can watch the lecture by James Lighthill and John Ffowcs Williams !

• [1] Permeable Surface Corrections for Ffowcs Williams and Hawkings Integrals
• [2] CFD Online – Acoustic Solver with openfoam
• [3] WIKIBOOKS – Engineering Acoustics/Analogies in aeroacoustics

• [4] 加藤千幸，大規模数値流体解析による流体音の予測
• [5] 飯田明由，送風機（ファン）の騒音発生メカニズムと低減・静粛化対策とその事例
• [6] 大嶋拓也，流れ中の柱状物体列から発生する空力音の数値予測に関する研究
• [7] Newsletters Nagare Dec. issue 2010