Direct Aeroacoustical Simulation in Weakly Compressible Flow

Human speech is generated in fluid-structure interaction of a stream of air with the vocal folds generating a pulsating pressure  which is modulated in the vocal tract into a sound wave from the mouth to reach an ear at some distance, studied e.g. in the EUNISON project. Lighthill derived a model from the compressible Navier-Stokes equations of the form

• $\square\rho =\sum_{i,j}\frac{\partial^2}{\partial x_i\partial_j}(\rho u_iu_j)$,

where $\rho$ is air density, $u=(u_1,u_2,u_3)$ flow velocity depending on space coordinates $x=(x_1,x_2,x_3)$ and time $t$, 

• $\square =\frac{\partial^2}{\partial t^2} -c^2\Delta$,

is the wave operator and $c$ the speed of sound. Here the underlying turbulent flow as source enters the wave equation for a density fluctuation, with simulation proceeding in two steps solving first Navier-Stokes equations for the flow velocity and then a wave equation for the density fluctuation.

Since in speech the flow velocity is much smaller than the speed of sound and thus compressibility effects are small, it is natural so consider the Navier-Stokes equations for a weakly compressible fluid (omitting viscosity effects):

1. $\frac{\partial u}{\partial t} +u\cdot\nabla u +\nabla p = 0$,
2. $\alpha^2\frac{\partial p}{\partial t} - \delta\Delta p = - \nabla\cdot u$,  

which with $\alpha =0$ is the model used in Computational Turbulent Incompressible Flow with $\delta\sim h$ with $h$ a computational mesh parameter. We thus consider a pressure equation augmented by a time derivative as a relaxation of Poisson's equation into a heat equation.

Splitting now $u=\bar u +U$ and $P=\bar p + P$ into components with slow and fast time variation suggests that the model 1-2 contains the following equations for the fast components:

• $\frac{\partial U}{\partial t} +\nabla P = 0$,
• $\alpha^2\frac{\partial P}{\partial t}= - \nabla\cdot U$,  

which corresponds to a homogeneous wave equation for the pressure variation $P$:

• $\square P =0$

with $c=\frac{1}{\alpha}$. The model 1-2 thus emerges as a one-step  alternative to the two-step Lighthill model for direct simulation of aeroacoustics of turbulent flow by solving Navier-Stokes equations for weakly compressible flow.