## Abstract

In this work, we present single-step aeroacoustic calculations using the Lattice Boltzmann method (LBM). Our application case consists of the prediction of an acoustic field radiating from the outlet of a porous media silencer. It has been proved that the LBM is able to simulate acoustic wave generation and propagation. Our particular aim is to validate the LBM for aeroacoustics in porous media. As a validation case, we consider a spinning vortex pair emitting sound waves as the vortices rotate around a common centre. Non-reflective boundary conditions based on characteristics have been adopted from Navier–Stokes methods and are validated using the time evolution of a Gaussian pulse. We show preliminary results of the flow through the porous medium.

## 1. Introduction

Computational aeroacoustics (CAA) has emerged from classical computational fluid dynamics as a separate discipline with its own methods. The goal is to calculate acoustic pressure fluctuations that result from the flow. The traditional CAA method follows the analogy of Lighthill [1], where the isentropic Euler equations are rewritten into a wave equation for a medium at rest. The sound generation is modelled by fluctuating forces applied on the right-hand side of the equation that results from internal stresses of the fluid. This is a hybrid approach in which the aerodynamic flow is solved separately from the acoustic field and then used to evaluate the acoustic sources. Opposed to that, the direct computation of the acoustic field solves the time-dependent and compressible Navier–Stokes equations and resolves the acoustic and flow variables without the use of models. Problems arise with the different scales on which the aerodynamic and acoustic fluctuations occur. Acoustic oscillations are several orders of magnitude smaller than the aerodynamic counterparts, which makes it difficult to resolve them accurately. For the different time and length scales, many time steps have to be calculated in order to resolve the acoustic field on a wide range of frequencies. Additionally, the boundary conditions have a great influence on the quality of such simulations. It must be ensured that the sound waves are radiated into open space without reflections on the boundaries, as this would severely disturb the calculated acoustic field. The specific goal of our work is to gain insight into the creation mechanisms of sound that is induced by vortical flow resulting from the flow through a porous medium.

The application we focus on is a pneumatic valve terminal that controls the air flow to pneumatic devices such as vacuum lifters. A porous flat plate silencer is placed in front of the air outlet to decrease the noise generation by reducing pressure and velocity levels. As the decay of vortices is a direct acoustic source, it is desirable to understand and predict the characteristics of the outflow. The porous medium flow and the fact that aeroacoustics is a weakly compressible phenomenon makes the Lattice Boltzmann method (LBM) an ideal candidate for solving this problem. First, we need to validate the non-reflective boundary conditions, the sound wave propagation and generation capabilities of the LBM and the correct flow through the porous medium.

We use a spinning vortex pair as a test case to validate the LBM for the process of aerodynamic sound creation. This numerical method was successfully applied to simulate acoustic wave propagation and aeroacoustic sound creation by Wilde [2] and Crouse *et al.* [3]. It was found to be a low-dispersion and dissipation scheme that has comparable accuracy to a higher order finite-difference scheme in space and time by Wilde [2], Crouse *et al.* [3], Marié *et al.* [4], and Li *et al.* [5]. In this paper, the sound generation process is studied and validated for the spinning vortex case because it resembles an acoustic quadrupole source. This is the kind of acoustic source that is typically found in turbulent shear flows. An analytical solution also exists. This validation case has been studied extensively both theoretically, e.g. by Müller & Obermeier [6] and Möhring [7], and numerically by, e.g. Lee & Koo [8].

The remainder of this paper is structured as followed. Section 2 serves as an overview of the numerical method, §3 gives a brief introduction to the acoustical properties of the underlying equations and the acoustic wave propagation properties of the LBM, and §4 presents the validation of the sound generation reproduction capabilities of the LBM with the spinning vortex pair test case. An outlook on future work is given in §5.

## 2. Numerical method

The LBM is employed to calculate the time evolution of a particle distribution function (pdf) where *f*_{i}(** x**,

*ξ*_{i},

*t*) defines the probability to find a particle at a given time

*t*at a certain position in space

**with a given velocity**

*x*

*ξ*_{i}. The simplified Lattice Boltzmann equation without a forcing term reads 2.1A discrete set,

*i*=1,…,

*n*, of microscopic velocities is used for the pdf

*f*

_{i}(

**,**

*x**t*). The collision operator

*Ω*is approximated with the Bhatnagar–Gross–Krook (BGK) model, where the pdfs are relaxed towards the thermodynamic equilibrium distributions

*f*

^{eq}with the collision frequency

*ω*. The Navier–Stokes equations are recovered in the small Knudsen number limit by the application of the Chapman–Enskog procedure. The viscosity of the fluid is expressed via the collision frequency

*ω*as

*ν*=

*c*

^{2}

_{s}(1/

*ω*−1/2)(Δ

*x*

^{2}/Δ

*t*). The model is valid for isothermal and weakly compressible assumptions. The adiabatic acoustic waves will therefore be modelled using an isothermal assumption relating pressure and density with the speed of sound as .

*Non-reflective boundary conditions* are used to model the propagation of waves into open space in the limited computational domain. Otherwise, the reflections disturb the acoustic field, which is already hard to capture using the numerical method because of the small pressure amplitudes. We use characteristic boundary conditions, which are commonly used within Euler-equation-based methods [9] and which were adopted to the LBM by Izquierdo & Fueyo [10]. A local one-dimensional inviscid system is solved, which is obtained from the Euler equations without source terms. The system employs characteristic variables *L*_{n} (acoustic, entropic and vorticity waves) that are directly related to the primitive hydrodynamic variables. As the LBM uses mesoscopic quantities instead of macroscopic ones, the primitive variables only lead to the equilibrium part of the pdfs. We omit the non-equilibrium part and set the distribution functions to their equilibrium values on the border nodes.

A one-dimensional Gaussian pulse in a three-dimensional domain is used to validate the characteristic boundaries in one direction, using periodic boundaries for the remaining directions. Outflow boundary conditions are employed, where the incoming wave is set to zero. A linear relaxation model serves as a pressure preservation with the relaxation factor *k*_{1}, which is defined by the Mach number and a characteristic length *L* as *k*_{1}=*σ*_{1}(1−Ma^{2})(*c*_{s}/*L*).

Setting *σ*_{1}=0 results in no information coming into the domain, which would be ideally absorbing, but can lead to a drift from the reference pressure. According to Selle *et al.* [11], a suitable range is 0.58<*σ*_{1}<*π*. In figure 1, the results for *σ*_{1}=10^{−4},…,2 are shown in terms of the pressure plotted versus the *x*-direction of the domain. The right border is the non-reflective one, and the plot shows the reflected wave after being absorbed at the non-reflective border. In opposition to the proposed values, we found *σ*_{1}=0.1 to be suitable in terms of reflection suppression and reference pressure preservation. Higher *σ*_{1} values resulted in a slow transition to the reference pressure level. The corner nodes proved to be a severe cause for instabilities. We dealt with the corners by using a sufficiently large computational domain in order to have only small perturbations on the border nodes from using the neighbours in the *x*-direction instead of diagonal ones.

## 3. Validation of sound propagation

Acoustic waves are characterized by small pressure fluctuations that are several orders of magnitude smaller than the pressure fluctuations of the flow field. We consider the fluid to be isothermal and at rest, ** u**=0, with the pressure perturbations

*P*=

*P*

_{0}+

*P*′ with

*P*′≪

*P*

_{0}. Wave propagation is governed by the lossy wave equation that can be derived from the linearized Navier–Stokes equations as 3.1where

*ν*=

*μ*/

*ρ*is the kinematic shear viscosity and

*ν*

_{bulk}=

*μ*

_{bulk}/

*ρ*is the bulk viscosity. For the BGK model, the bulk viscosity is

*ν*

_{bulk}=2/3

*ν*, resulting in an effective viscosity of

*ν*

_{eff}=2

*ν*[12]. Inserting the plane-wave solution into the lossy wave equation gives the dispersion relation 3.2The wave number and the angular frequency are complex values, and the solutions can be expressed by temporal or spatial analysis. We will only investigate temporal decay of sound waves, as this is the mechanism by which sound waves are generated by turbulent vorticity decay. The propagation of sound waves is exposed to dissipation, i.e. the amplitude error and dispersion (the phase error). Marié

*et al.*[4] examined dissipation and dispersion properties of acoustic waves with the LBM compared with other Navier–Stokes-based methods. The performance of the LBM was shown to be between a second-order and an optimized third-order finite-difference scheme in space, where the low dissipation in the LBM was emphasized.

Brès *et al.* [13] used a one-dimensional standing wave travelling through a periodic domain to validate the LBM for wave propagation accuracy that we used for validating our code. The initial conditions are
3.3The analytical solution of the temporal pressure fluctuation at a fixed point in space caused by a one-dimensional travelling wave is
3.4The numerical errors of dispersion and dissipation are quantified by and , and these are evaluated for resolutions of *N*_{ppw}=2,…,24 grid points per wavelength. The results in figure 2 of our code LBC show good agreement with results obtained by Brès *et al.* [13]. For a wave resolution of *N*_{ppw}=6, the error of dispersion *ε*_{c} drops below 3 per cent and dissipation *ε*_{α} below 1 per cent. Both values converge towards zero with an increasing number of points per wavelength. In further validation simulations, the waves are resolved with a resolution of *N*_{ppw}>100, where *ε*_{c},*ε*_{α}<0.1 per cent.

## 4. Validation of sound generation

We use the spinning vortices to validate the sound generation process. As the vortices spin around a common centre, they emit acoustic waves, equivalent to a spinning acoustic quadrupole that is the typical sound production mechanism of turbulent shear flow. The vortices have equal circulation *Γ* and spin on a circle of radius *r*_{0} with a velocity *u*_{(r=r0)}=*Γ*/(4*πr*) at angular speed and a rotating Mach number Ma_{rot}=*u*/*c*_{s}=*Γ*/(4*πr*_{0}*c*_{s}). Müller & Obermeier [6] derived the analytical solution for the acoustic far field from the co-rotating vortex pair, where the vortices are considered as point sources in an inviscid fluid. One solution for the incompressible near flow field and one for the acoustic far field are matched asymptotically in an intermediate domain to give an asymptotically valid solution expressed by a flow potential *Φ* with ∇*Φ*=** u**. The inner flow solution is expressed by the potential
4.1with

*z*=

*r*e

^{iθ}and

*b*=

*r*

_{0}e

^{iωt}. Assuming

*z*/

*b*≫1, the equation above can be approximated by splitting it into a steady flow potential

*Φ*

_{0}and a fluctuation

*Φ*

_{1}with the frequency of the vortex rotation 4.2In the outer domain, the flow is compressible and obeys the homogeneous wave equation 4.3Applying correct matching conditions, the solution of the complex potential can be written as 4.4with

*k*=2

*ω*/

*c*

_{s}and being the second-kind Hankel function of order two. The pressure fluctuations in the far-field solution can be extracted from the equation above in the outer limits 4.5The amplitude is the real part of the complex potential

*Φ*4.6The initial velocity field in Cartesian coordinates

**={**

*u**u*

_{1},

*u*

_{2}} can be obtained by differentiating the potential equation with respect to

*z*4.7The initial pressure field is chosen to be constant throughout the domain, as the analytical solution is prone to a pressure singularity in the vortex cores. This causes perturbations in the flow field that dissolve after some hundred time steps.

### (a) Simulation set-up and numerical results

All variables below are given in lattice units. The simulation is performed on an equidistant grid and the rotational radius *r*_{0} is resolved with 10 grid points. The total domain is 400 *r*_{0} in each direction. The maximum velocity located at the core radius *r*_{c} of both vortices is , resulting in an initial rotational Mach number Ma_{Rot,ini}=5.196×10^{−2}. The viscosity is chosen to be *ν*=4.168×10^{−5}, which corresponds to a collision frequency of *ω*=1.9995. The simulation was run with 32 cores on an Intel Nehalem Cluster.

A temporal and a spatial distribution of the acoustic pressure from the simulation results is compared with the analytical solution. As a single-step simulation is performed, the total pressure is obtained, i.e. the incompressible and acoustic pressures are superimposed and they have to be separated before the analysis. The incompressible pressure, obtained in the analytical solution, is subtracted from the numerical results in order to match the numerical results with the analytical acoustic pressure. The evaluation uses the dimensionless pressure *p*′/(*c*^{2}_{s}*ρ*_{0}), the dimensionless radius *r*/*r*_{0} and the dimensionless time *tc*_{s}/*r*_{0}.

### (b) Spatial pressure behaviour

Figure 3 shows a snapshot at time step *t*=750 of the acoustic pressure, ranging from the centre of rotation through one vortex centre to the domain limit. Near the vortex centre, the amplitude error *ε*_{α} is around 30 per cent, but decreases when advancing outside from the centre of spinning motion. The analytical solution works with point sources that are infinitely small in size and hence results in an infinitely large intensity in their centres. The amplitude is slightly underpredicted in the acoustic near field, which stretches until it is around three wavelengths away from the vortex centres. The phase shifts slightly and the waves propagate faster than predicted in the analytical solution.

This creates an amplitude error near the vortex centres.

### (c) Temporal pressure behaviour

Figure 4 shows the temporal behaviour of the acoustic pressure at a distance *r*=150 *r*_{0} from the centre of rotation. The angle of the line through the two vortex centres regarding the *x*-axis is *θ*=25^{°}. The plot starts after 11 vortex rotations with the phase matched to the analytical solution. This is done because of the perturbation by the constant pressure field in the initialization process. The amplitude is slightly lower than predicted in the analytical solution. The phase matches well at *t*=800, but then the wave periods increase. A slightly longer phase in the numerical solution can be observed after around six revolutions. The increasing oscillation period is considered to be due to the neglected viscosity in the analytical solution. In the simulation, however, the vortex revolution speed is decreased over time by the inclusion of viscosity. This results in an oscillation frequency that decreases over time in the numerical experiments. The decreased amplitude can also be attributed to the viscosity effects as the dissipation is added to the spatial energy distribution of the spreading waves.

## 5. Preliminary results for the flow through porous medium

The ongoing work consists of the simulation of a diverging channel with a porous medium. The porous geometry is obtained by a micro computer tomography (μCT) scan of the real sample. A large eddy turbulence model with Smagorinsky sub-grid scale modelling as proposed by Hou *et al.* [14] is applied owing to the high Reynolds numbers. We use 155 grid points for resolving the inlet diameter and a Smagorinsky constant of *C*_{S}=0.17. In figure 5, the vorticity is plotted, and the strongly vortical flow at the outflow after the porous medium is clearly visible. As the flow is strongly wall bounded, a dynamic turbulence model such as presented by Lévêque *et al.* [15] will be applied.

## 6. Conclusions

In this paper, a method for the calculation of aeroacoustic sound creation using flow through a porous medium has been described. The propagation of sound waves has been validated with a travelling planar wave, whereas the generation has been tested with a spinning vortex pair. We use non-reflective boundary conditions based on characteristics. For these test cases, quantitative comparisons were performed. We showed that the LBM is a suitable tool for aeroacoustic simulations. Future work will investigate the flow through the porous medium in detail, especially focusing on the sound generation by turbulent vortex decay.

## Acknowledgements

We thank the DLR for the μCT scans, Festo AG & Co. KG for providing geometric data, and S. Izquierdo and C. Janssen for prolific discussions.

## Footnotes

One contribution of 25 to a Theme Issue ‘Discrete simulation of fluid dynamics: applications’.

- This journal is © 2011 The Royal Society