# Computational Aeroacoustics (CAA) part2

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.

## Author: fumiya

CFD engineer in Japan