R. C. K. Leung

One-Step Aeroacoustics Simulation Using Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is a numerical simplification of the Boltzmann equation of the kinetic theory of gases that describes fluid motions by tracking the evolution of the particle velocity distribution function based on linear streaming with nonlinear collision. If the Bhatnagar‐Gross‐Krook (BGK) collision model is invoked, the velocity distribution function in this mesoscopic description of nonlinear fluid motions is essentially linear. This intrinsic feature of LBM can be exploited for convenient parallel programming, which makes itself particularly attractive for one-step aeroacoustics simulations. It is shown that the compressible Navier‐Stokes equations and the ideal gas equation of state can be correctly recovered by considering the translational and rotational degrees of freedom of diatomic gases in the internal energy and using a multiscale Chapman‐Enskog expansion. Assuming two relaxation times in the BGK model allows the temperature dependence of the first coefficient of viscosity of diatomic gases to be replicated. The modified LBM model is solved using a two-dimensional 9-discretized and a twodimensional 13-discretized velocity lattices. Three cases are selected to validate the one-step LBM aeroacoustics simulation. They are the one-dimensional acoustic pulse propagation, the circular acoustic pulse propagation, and the propagation of acoustic, vorticity, and entropy pulses in a uniform stream. The accuracy of the LBM is established by comparing with direct numerical simulation (DNS) results obtained by solving the governing equations using a finite difference scheme. The tests show that the proposed LBM and the DNS give identical results, thus suggesting that the LBM can be used to simulate aeroacoustics problems correctly.

Lattice Boltzmann Method Simulation of Aeroacoustics and Nonreflecting Boundary Conditions

A lattice Boltzmann method that can recover the first coefficient of viscosity and the specific heat ratio correctly has been adopted for one-step aeroacoustic simulations because it can recover the speed of sound correctly. Instead of solving the Navier-Stokes equations as in the case of direct numerical simulation, the lattice Boltzmann method only needs to solve one transport equation for the collision function. Other flow properties are obtained by integrating this collision function over the particle velocity space. The lattice Boltzmann method is effective only if appropriate nonreflecting boundary conditions for open computational boundaries are available, just like the direct numerical simulation. Four different nonreflecting boundary conditions are commonly used with direct numerical simulation for one-step aeroacoustic simulations. Among these are the characteristics-based method, the perfectly matched layer method, the C 1 continuous method, and the absorbing layer method. Not all nonreflecting boundary conditions are applicable when used with the lattice Boltzmann method; some might not be appropriate, whereas others could be rather effective. This paper examines some existing nonreflecting boundary conditions plus other new proposals, their appropriateness, and their suitability for the lattice Boltzmann method. The assessment is made against two classic aeroacoustic problems: propagation of a plane pressure pulse and propagation of acoustic, entropy, and vortex pulses in a uniform stream. A reference solution is obtained using direct numerical simulation assuming a relatively large computational domain without any specified nonreflecting boundary conditions. The results, obtained with a computational domain half the size of that used for the direct numerical simulation calculations, show that the absorbing layer method and the extrapolation method with assumed filter perform the best.

Propagation speed, internal energy, and direct aeroacoustics simulation using lattice Boltzmann method

ThevalidityofthelatticeBoltzmannmethodfordirectaeroacousticssimulationsdependsonitsabilitytocorrectly recovertheequationofstateofthegasanditsdynamicviscosity.ThispaperpresentsalatticeBoltzmannmethodwith tworelaxationtimestocarryoutthedirectaeroacousticssimulationsofatwo-dimensionalGaussiansoundpulseina uniform flowoverarangeofMachnumbers(M)varyingfrom0.01to0.9.Itisassumedthatthereisnoshockpresent intherangeofMachnumberstested.Asixth-order finite-differenceschemeisusedtoevaluatetheconvectivetermin the modeled Boltzmann equation, and a second-order Runge–Kutta scheme is used to forward march in time. Thus solved, the calculations show that the wave propagation speed (c) over the range 0:01 � M � 0:9, determined from the deduced equation of state and from the propagation of the pulse, are in good agreement with theoretical analysis and direct numerical simulation results obtained by solving the unsteady compressible Navier–Stokes equations using a low-dispersive and low-dissipative finite-difference scheme. The specific heat ratio (� ) for a diatomic gas is recovered correctly and so is the dependence of the internal energy on � . Thus, the proposed lattice Boltzmann method is valid for direct aeroacoustics simulations at very low to near transonic M.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Hartens first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

Acoustic radiation by vortex induced flexible wall vibration

Sound radiation due to unsteady interaction between an inviscid vortex (which models a turbulent eddy) and a finite length flexible boundary in a two-dimensional space is studied using potential theory and the matched asymptotic expansion technique. The Mach number of the vortex propagation is kept below 0.15. Results suggest that the monopole field created by the volumetric flow induced by the vibrating flexible boundary dominates the overall acoustic power radiation. The longitudinal dipole directly due to the transverse vortex acceleration is only important when the vortex is moving over the flexible boundary. The longitudinal dipole resulting from the boundary vibration gains slightly in importance in the strong vortex case, but the corresponding transverse dipole remains negligible for the cases considered in the present study. The two longitudinal dipoles give rise to biased radiation directivities on both sides of the flexible boundary.

Aeroacoustics of T-junction merging flow

This paper reports a numerical study of the aeroacoustics of merging flow at T-junction. The primary focus is to elucidate the acoustic generation by the flow unsteadiness. The study is conducted by performing direct aeroacoustic simulation approach, which solves the unsteady compressible Navier-Stokes equations and the perfect gas equation of state simultaneously using the conservation element and solution element method. For practical flows, the Reynolds number based on duct width is usually quite high (>10(5)). In order to properly account for the effects of flow turbulence, a large eddy simulation methodology together with a wall modeling derived from the classical logarithm wall law is adopted. The numerical simulations are performed in two dimensions and the acoustic generation physics at different ratios of side-branch to main duct flow velocities VR (=0.5,0.67,1.0,2.0) are studied. Both the levels of unsteady interactions of merging flow structures and the efficiency of acoustic generation are observed to increase with VR. Based on Curles analogy, the major acoustic source is found to be the fluctuating wall pressure induced by the flow unsteadiness occurred in the downstream branch. A scaling between the wall fluctuating force and the efficiency of the acoustic generation is also derived.

Finite Difference Lattice Boltzmann Method for Compressible Thermal Fluids

a splitting method to solve the modeled lattice Boltzmann equation. The splitting technique permits the boundary conditions for the lattice Boltzmann equation to be set as conveniently as those required for the finite difference solution of the Navier–Stokes equations. It is shown that the compressible Navier–Stokes equation can be recovered fully from this approach; however, the formulation requires the solution of a Poisson equation governing a secondorder tensor. Thus constructed, the method has no arbitrary constants. The proposed method is used to simulate thermalCouette flow,aeroacoustics,andshockstructureswithanextendedthermodynamicsmodel.Thesimulations are carried out using a high-order finite difference scheme with a two-dimensional, nine-velocity lattice. All simulations are performed using a single relaxation time and a set of constants deduced from the derivation. It is found that the finite difference lattice Boltzmann method is able to correctly replicate viscous effects in thermal Couette flows,aeroacoustics,andshockstructures.Thesolutionsobtainedareidenticaleithertoanalyticalresults,or obtained by solving the compressible Navier–Stokes equations using a direct numerical simulation technique.

Comparative study of nonreflecting boundary condition for one-step duct aeroacoustics simulation

Introduction O NE-STEP numerical simulations of aeroacoustics problems have been studied for quite some time. The proposed methods usually solve the fully unsteady compressible Navier–Stokes equations, thus allowing the far-field sound and the near-field aerodynamics to be determined without modeling the source terms in the wave equation. Because of the very small energy of the acoustics field, a low dispersive and low dissipative scheme is always required if wave propagation were to be resolved accurately in an aeroacoustics computation. Besides, precise boundary conditions also play a key role in aeroacoustics computations. At the inflow and outflow boundaries, the assumed computational boundaries should allow the aerodynamic field to pass freely with minimal reflection, while at the same time they should be nonreflecting for the incident acoustic waves. Otherwise the spurious erroneous waves reflecting from the boundaries would contaminate the numerical simulation, decrease the computational accuracy, and might even drive the solution toward a wrong time-stationary state. Therefore, it is necessary to formulate truly nonreflecting conditions at these computational boundaries. The most widely used Navier–Stokes characteristics-based boundary conditions (NSCBC) (see Refs. 1–4) for unsteady flows has been proven accurate only if the wave incidence is normal to the computational boundary. Numerical instabilities at other incident angles are intolerable; the instabilities are further amplified in the presence of strong mean shear. Consequently, damping techniques or filtering with a buffer region between the physical domain and outflow boundaries is commonly invoked in the use of NSCBC.5 In addition to suppressing instabilities, the damping or filtering is so constructed that it renders the flow at the end of the buffer region more one dimensional; thus, the residual wave will be relatively more normal to the computational boundary prescribed for NSCBC. Two kinds of buffer regions are commonly adopted for one-step aeroacoustics simulation: the absorbing boundary condition (ABC)

Modeled Boltzmann Equation and Its Application to Direct Aeroacoustics Simulation

The Bhatnagar, Gross, and Krook modeled Boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success. However, its application to direct aeroacoustic computation is less successful. One possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound. The present study reports on the development of an improved modeled Boltzmann equation for aeroacoustics simulation. The approach is to modify the Maxwellian distribution normally assumed for the equilibrium particle distribution function. Constraints imposed are the exact recovery of the state equation for a diatomic gas and the Euler equations without invoking the small Mach number assumption. Thus formulated, a distribution function consisting of the Maxwellian distribution plus three other terms that attempt to account for particle-particle collisions is obtained. A velocity lattice method is used to solve the improved modeled Boltzmann equation using an equivalent lattice equilibrium distribution function. The simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the Euler equations. The results of the improved modeled Boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain, thus verifying the viability and correctness of the modified equilibrium distribution function.

Validation of CE/SE Scheme in Low Mach Number Direct Aeroacoustic Simulation

Abstract The space-time conservation element and solution element (CE/SE) scheme has caught many attention in aeroacoustic research community as an alternative numerical strategy for direct aeroacoustic simulation (DAS). As a result of its strict conversation of flow flux in both space and time, the low-order CE/SE scheme possesses excellent non-dissipative characteristics, expedient in calculating low Mach number DAS which requires uniform numerical accuracy to resolve the widely disparate flow and acoustic scales of the problem. In this paper, an attempt of validating a simplified Courant Number Insensitive CE/SE scheme using carefully selected aeroacoustic benchmark problems is reported. Excellent agreement with the benchmark results obtained firmly establishes that CE/SE scheme is a viable scheme for resolving the nonlinear physics of low Mach number aeroacoustic problems.