Application of the Lattice-Boltzmann method in flow acoustics

Transcription

1 Application of the Lattice-Boltzmann method in flow acoustics Andreas Wilde Fraunhofer Institut für Integrierte Schaltungen, Außenstelle EAS Zeunerstr. 38, Dresden Abstract The Lattice-Boltzmann method for simulation of low Mach number flows is evaluated for the application in flow acoustics. By Linearization and von-neumann analysis quantitative measures for the accuracy of phase speed and attenuation of linearly propagating sound waves in presence of a mean flow are derived. It is shown that phase errors are the only relevant problem when simulating sound waves in the audible range in air. The deviations in phase speed are below 1% as long as the wave is resolved with at least 12 points per wavelength. The LBM is applied to the problem of a Helmholtz resonator under a grazing flow and to the trailing edge noise generation problem. The results, though preliminary, clearly demonstrate the ability to reproduce relevant flow acoustic effects. 1 Introduction In recent years the Lattice-Boltzmann method (LBM) for numerical simulation of low Mach number flows attracted a lot of attention. While many applications in different disciplines were studied extensively [2] the use of LBM in flow acoustics was explored by a small number of authors. Skordos simulated 3d sound generation in a flue organ pipe using a simple LBM on a cluster of workstations [10]. Buick et al. studied the propagation of high amplitude sound waves and compared the results with theoretical predictions based on Burgers equation [1]. While both authors found good agreement between the results from LBM and theoretical or experimental data, little information on theoretical properties of sound wave generation and propagation capabilities of the Lattice-Boltzmann scheme is given. Lallemand and Luo studied the wave propagation properties of several Lattice-Boltzmann schemes focused on the so-called Multiple-Relaxation-Time model, which provide better stability than the standard single relaxation time LBM [6, 7]. The present work follows the ansatz described there to linearize the equilibrium distribution function in order to obtain information on properties of the standard LBM by von-neumann analysis. This paper is organized as follows: Section 2 gives a theoretical derivation of acoustic properties of the standard D2Q9 single relaxation time Lattice-Boltzmann model based on the linearized equilibrium distribution function. Section 3 describes two applications of the Lattice Boltzmann method: The flow over a partially covered cavity and the flow past the trailing edge of a flat plate. Section 4 concludes the paper. 2 Acoustic properties of the D2Q9 Lattice-Boltzmann model The basic theory of the Lattice-Boltzmann method has been described in numerous publications (see e.g. [11, 12, 9]), therefore only a brief outline shall be given here. An ideal gas may be described by the Boltzmann equation, which essentially enables to compute the particle distribution f ( x, v, t). This function gives the probability P = f ( x, v,t) x 3 v 3 to find a particle in a small volume x 3 around the location x with a velocity within v 3 around v at time t. For computational purposes this equation may be discretized and approximated numerically. One particularly effective way to do this is to introduce a lattice. At discrete times all particles are assumed to be located at lattice sites, where they collide and exchange energy and momentum. Given a finite number of lattice sites this step limits the number of positions and velocities available for the particles to a finite number. The velocities are further limited to such values, that particles may be at rest or travel to the nearest neighbor site during one time step. For a two dimensional rectangular lattice this results in 9 different velocity states: one rest state, four states for traveling along the axes and four states for traveling along the diagonals to the next neighbor. This defines a so-called D2Q9 model. The particle density is assumed to be large, such that the discretized particle distribution function is approximated by a vector of nine real numbers at any lattice site. The Boltzmann equation in its discretized form may be written as where f i ( x + x i,t + t) = f i ( x,t) + Ω i ( f i ( x,t)) (1) Ω i 1 τ ( f i f eq i ) is an approximation to the collision operator and f eq i (ρ, u) = w i ρ(1 + 3( x i u) ( x i u) u2 ) is the equilibrium distribution function [11, 12]. The index i refers to the i-th direction x i (see fig. 1). The

2 ccs k Figure 1: The Boltzmann equation is discretized on a lattice. The particles may travel from on lattice site to the i-th neighbor within one time step. relaxation time τ determines the viscosity of the fluid. The equilibrium distribution function is only dependent on the macroscopic values of the local density and velocity, which in turn are given by the first moments of the distribution function: ρ = f i i x i ρ u = i t f i Eq. (1) is nonlinear in the f i only through the equilibrium distribution function f eq. It is therefor sufficient to linearize the equilibrium distribution function to enable the standard von-neumann analysis of the resulting linear scheme. Consider the distribution divided into a stationary and a fluctuating part f i = fi 0 + f i, then the linearized version of eq. (1) reads with f i ( x + x i,t + t) = L i j f j( x,t) (2) L i j = 1 τ ( f eq i f j ) + δ i j (τ 1) f j = f 0 j Now define the advection operator A through its action on f : A f i ( x,t) = f i ( x + x i,t). Taking the spatial Fourier transform of eq. (2) yields [6] f ( k,t + t) = A 1 L f ( k,t) (3) Looking for plane wave solutions of eq. (3) in an infinite space is an eigenvalue problem, where the eigenvalues determine the dispersion relation. Eq. (3) cannot be solved analytically, because the advection operator contains transcendent functions of the wave vector k. Here the eigenvalues of A 1 L are computed numerically using Mathematica. Fig. 2 shows the speed of sound waves traveling in several directions relative to the lattice axes as a function of the wave number k. The spatial step was set to x = 1 m, the kinematic viscosity was set to kg/(m Figure 2: Phase speed of sound waves in the D2Q9 Lattice- Boltzmann scheme as a function of wave number measured in lattice units. Solid line: propagation direction 0 relative to lattice, dotted line: 22.5, dashed: 45 ccs k Figure 3: Phase speed of sound waves in the D2Q9 Lattice- Boltzmann scheme compared to the FDTD scheme as a function of the wave number measured in lattice units. The propagation direction was 0 relative to lattice. Solid line: D2Q9-LBM, dotted line: FDTD s). From the plot one can see that the numerical error in phase speed exceeds 1% at wave numbers above 0.5, which corresponds to one wave length being resolved with less than approximately 12 points. Up to this wave number the wave propagation shows virtually no anisotropy, the corresponding curves deviate from each other significantly only for higher values of k. The numerical phase error was compared to that of the Finite Difference Time Domain scheme by Yee [13], which was applied to solve the linearized Navier-Stokesequations with viscous effects neglected and mean velocity set to zero. This scheme provides second order accuracy in time and space. At the stability limit given by x = c 2 t with c being the speed of sound this scheme is free of numerical dissipation and also, it is numerically equivalent to the so-called Transmission-Line- Matrix method (TLM), which in turn has conceptual similarities to the Lattice-Boltzmann method when applied to sound wave propagation [5, 8]. Fig. 3 shows the results for waves traveling parallel to the lattice axes. One characteristic of the FDTD scheme is constant phase speed of waves traveling at 45 to the lattice, while the largest error occurs for the 0 direction. Thus the FDTD scheme features a significant anisotropy at high wave numbers. On the other hand, even in the "worst case" the phase error of the FDTD is below that of the LBM scheme. Unlike the simple FDTD scheme used here the LBM is capa-

3 ccs k Figure 4: Phase speed of sound waves in the D2Q9 Lattice- Boltzmann scheme traveling upstream and downstream with a uniform flow at Mach number 0.1 along a lattice axis. ble of simulating advection (Doppler) effects on sound waves. Fig. 4 shows the absolute value of the phase speed of sound waves traveling along a lattice axis with likewise oriented mean flow at Mach number 0.1. The phase speeds are shifted by the amount of the Mach number. Also, a small additional anisotropy is introduced with respect to the phase error at higher wave numbers: The deviation from the constant value at infinite wave length is smaller for waves traveling upstream than for wave traveling downstream. The attenuation of sound waves in ideal monoatomic gases can described by the attenuation exponent µ = 2 νω 2 3 ρc 3 3d Flow simulations presently can cover a range of a few 10 wavelengths, which means that in the audible range of frequencies the attenuation of sound due to viscous or heat conduction effects is negligible. Inspection of the damping in LBM reveals that for the range of applications considered here (spatial resolution m, air) the attenuation of sound waves is of the same order of magnitude as the theoretical predictions, thus is negligible. It also turns out that the absorption is highly anisotropic, being twice as strong for waves traveling 45 to the lattice compared to those traveling parallel to the lattice. Because of the smallness of the absorption effect this doesn t pose a problem for simulations. 3 Applications of LBM in flow acoustics The Lattice Boltzmann Method was applied to two test cases. The first was a flow over a partially covered cavity and the second was a trailing edge of a flat plate. The cavity case was a benchmark problem posed for the 3rd Computational Aeroacoustic Workshop on benchmark problems in Ohio It consists of a partially covered cavity as shown in fig. 5. The problem definition requires a boundary layer thickness of 1.6 cm at the mouth of the cavity for a mean flow velocity of 26.8 m/s and 2.2 cm at 50.9 m/s. This case was setup for calcu- Figure 5: Sketch of the partially covered cavity used as first test case length of upstream plate boundary layer thickness inflow velocity medium spatial resolution dimensions of the cavity: stream wise depth span wise opening width opening depth 53 cm 12 mm 50 m/s air mm 15.9 mm 24.7 mm 24.7 mm 8.8 mm 3 mm Table 1: Parameters of the simulation of the cavity. lation with PowerFLOW, which is a commercially available Lattice-Boltzmann code. PowerFLOW employs a standard turbulence model to account for unresolved fluctuations and a wall model to achieve realistic results in boundary layers. In the calculations shown here a 3 dimensional model was used. To get a correct boundary layer thickness the flat plate in which the cavity was located reached upstream. The parameters of the simulation are given in table 1. Fig. 6 shows the spectral density of the sound pressure measured in the middle of the left wall of the cavity. The solid line gives the result of the simulation while the dashed line indicates the results of an experiment conducted by Henderson [3]. The pressure fluctuation recorded in the cavity consists of a small bandwidth tone (1824 Hz, 144 db 1 ) which stands out of the broadband background by 30 db. The first harmonic of his tone with a sound pressure 30 db less than the main peak is also visible. The simulated result deviates from the experimental value by 10 db in magnitude and 50 Hz in frequency. The broadband noise background is more 20 db lower than in the simulation. This is probably due to the turbulence 1 All sound pressure levels are referenced to Pa, intensity levels to Pa 2.

4 sound pressure level in db frequency in Hz thickness of the plate length of the plate thickness of the boundary layer distance ribbon and trailing edge height of ribbon above the plate ribbon vibration frequency ribbon velocity amplitude flow velocity medium spatial resolution time step calculation volume 0.4 mm 63.5 mm 5 mm 43.5 mm 1 mm Ma 3 m/s m/s air 0.1 mm s points Figure 6: Spectral density of the probe inside the cavity. Table 2: Parameters of the trailing edge noise simulation. model applied here which damps out small scale fluctuations in the boundary layer, which is poorly resolved with about 20 cells. The other test case was a 2d simulation of a flat plate trailing edge flow. Here the standard D2Q9-LBMscheme as described above was employed in conjunction with a 4-th order low pass filter to damp high wave number fluctuations [10]. Again the boundary layer was not resolved well enough to simulate turbulent fluctuations, therefor a vibrating ribbon was modeled on the upper side of the plate upstream of the trailing edge. The ribbon vibrated normal to the wall with a fixed velocity amplitude of 3 m/s, and a frequency proportional to the inflow Mach number. The simulation was repeated for inflow velocities of m/s in 5 m/s increments to calculate the scaling of the sound intensity with respect to Mach number. At the left and right side of the control volume the velocity and the density were prescribed, respectively. To allow fluctuations and especially sound waves to leave the control volume an ad-hoc relax to mean condition was employed, which means that the actual value of the velocity or the pressure were calculated as v BC = 0.9 v v mean At the top and bottom boundaries of the control volume equilibrium conditions were used, which simply means that the distribution functions at top and bottom lines of cells were not updated. This conditions result in reflections of sound waves of about 10% of the pressure amplitude. It was observed that the transition of vortices across the outflow boundary causes strong spurious acoustic disturbances. Therefore the vortices were damped out by increasing the viscosity in the rightmost quarter of the control volume. Fig. 7 shows a snapshot of the high pass filtered pressure field at time step The inflow velocity was 40 m/s. On the upper side of the plate a regular pattern of vortices has developed which is convected across the trailing edge. The passing of a vortex triggers an acoustic wave, which propagates upstream. pressure fluctuation in Pa pressure fluctuation in Pa Figure 8: Directivity of sound radiation from the trailing edge. intensity fluctuations in db Pa velocity in m/s Figure 9: Pressure fluctuation levels 20 mm above the trailing edge as a function of mean inflow velocity.

5 Figure 7: Snapshot of the high pass filtered pressure field at time step Blue indicates 2.31 Pa below mean pressure, red 2.31 Pa above mean pressure. The directivity of the acoustic wave is depicted in fig.8. It was recorded at 72 points located on a circle with a radius of 200 spatial steps centered on the trailing edge. Though the sampling positions are in the near field of the source it clearly shows the pattern of a dipole normal to the plate at the trailing edge. The deviations from the theoretical curve towards higher values are due pressure fluctuations caused by convected vortices near the plate and in the wake of the plate, respectively. The minimum for the small upstream angles on the other hand can be explained by a destructive interference of the pressure field of the convecting vortices with the sound wave. Because of the monochromatic forcing by the ribbon the pressure fluctuations at the sampling location mentioned above show the same period, with the fundamental frequency carrying most of the energy. Fig. 9 shows the level of intensity fluctuations at the sampling point at the stream wise position of the trailing edge, i.e. 90 with respect to mean flow, as a function of Mach number. In spite of being forced with a constant velocity amplitude the turbulent velocity fluctuations near the trailing edge were dependent on the mean flow velocity. On the other hand, the fluctuations in the boundary layer are due to artificial forcing and therefor do not scale with the power laws appropriate for real turbulent boundary layers. From theoretical arguments one expects a power law of p 2 v 2 Ma 2 where p is the far field sound pressure, v are the turbulent velocity fluctuations and Ma is the mean flow Mach number [4]. Thus if the influence of turbulent velocity fluctuations is accounted for, a dependence of p 2 Ma 2 is expected. For this reason, the levels depicted in fig. 9 are corrected by 20 log 10 (v). The intensity fluctuation levels exhibit an ambiguous behavior: Between 10 m/s and 20 m/s mean flow velocity the levels scale as Ma 1.0, while between 25 m/s and 40 m/s the levels follow p 2 Ma Conclusions The Lattice-Boltzmann-Method (LBM) for the numerical simulation of flows was studied with respect to application on flow acoustics problems. The theoretical analysis of linearly propagating sound waves shows that with a D2Q9 LBM the propagation of sound can be simulated with an error in phase speed of less than 1% if the spatial resolution is better that 12 grid points per wave length. This result means that the LBM cannot compete with specialized Finite Difference DRP schemes for Euler equations in terms of efficiency of simulating propagation of sound waves. On the other hand, for simulation of flows a high spatial resolution is needed to reproduce flow structures, while the typical scale of the sound waves is larger by the order of Ma 1. The damping of sound waves in LBM simulations is of the same order of magnitude as theoretical values for monoatomic gases and is therefor negligible for all practical purposes of simulation. Doppler effects could be shown to be accounted for with high precision. As the LBM solves the full weakly compressible Navier-Stokes equations, it should be seen as a solver for the flow with excellent properties with respect to sound propagation. In a set of numerical experiments the ability of the LBM to reproduce weakly compressible flow effects, which are relevant for flow acoustic problems at low Mach numbers, were investigated. The preliminary results of the simulation of the trailing edge noise generation mechanism in 2d and of a Helmholtz cavity under a grazing flow in 3d clearly show, that the LBM is capable of delivering meaningful results for sound generation of low Mach number flows. For both numerical experiments a number of questions still needs to answered: Additional calculations for the Helmholtz resonator are necessary to explain the observed deviations of the resonance parameters in the simulation results compared to the experiment.

6 It is well known that the frequency and the magnitude of the tone strongly depend on the boundary layer, about which little information is given in the experimental report. For the trailing edge noise experiment the intensity doesn t scale with a uniform exponent to the Mach number. Probably the simple theoretical model employed here doesn t capture the physical effects when a boundary layer is forced by a vibration ribbon. Observation of the pressure fields suggests, that at low mean flow velocities the vibrating ribbon produces other types of disturbances than at higher mean velocities. [11] S. Succi. The Lattice Boltzmann Equation - For Fluid Dynamics and Beyond. Oxford University Press, [12] D. Wolf-Gladrow. Lattice-Gas Cellular Automata and Lattice-Boltzmann Models. Springer-Verlag Berlin Heidelberg, [13] K. Yee. Numerical solutions of initial boundary value problems involving Maxwell s equations in isotropic media. IEEE Transactions on Antennas and Propagation, AP-14: , References [1] J. M. Buick, C. A. Greated, and D. M. Campbell. Lattice BGK simulation of sound waves. Europhysics Letters, 43(3): , [2] S. Chen and G.D. Doolen. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30: , [3] B. Henderson. Automobile noise involving feedback sound generation by low speed cavity flows. In The 3rd Computational aeroacoustics (CAA) Workshop on Benchmark problems, NASA/CP , [4] M.S. Howe. A review of the theory of trailing edge noise. Journal of Sound and Vibration, 61(3): , [5] Y. Kagawa, T. Tsuchiya, B. Fujii, and K. Fuchioka. Discrete Huygens model approach to sound wave propagation. Journal of Sound and Vibration, 218(3): , [6] P. Lallemand and L.-S. Luo. Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, galilean invariance, and stability. Physical Review E, 61(6): , [7] P. Lallemand and L.-S. Luo. Theory of the Lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions. Physical Review E, 68(3), [8] P.O. Lüthi. Lattice Wave Automata. PhD thesis, Université de Genéve, [9] D. H. Rothman and S. Zaleski. Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics. Cambridge University Press, [10] P.A. Skordos. Modeling flue organ pipes: subsonic flow, Lattice Boltzmann, and parallel distributed computers. PhD thesis, Massachusetts Institute of Technology, Artificial Intelligence Laboratory, 1995.