# Computational aeroacoustics using lattice boltzmann model

 Author: Li, Xuemin Title: Computational aeroacoustics using lattice boltzmann model Degree: Ph.D. Year: 2007 Subject: Hong Kong Polytechnic University -- Dissertations.Aerodynamic noise -- Mathematical models.Sound-waves -- Mathematical models.Lattice gas -- Mathematical models.Maxwell-Boltzmann distribution law. Department: Dept. of Mechanical Engineering Pages: xvi, 182 leaves : ill. ; 30 cm. Language: English InnoPac Record: http://library.polyu.edu.hk/record=b2116558 URI: http://theses.lib.polyu.edu.hk/handle/200/1699 Abstract: Computational Aeroacoustics (CAA) has been developed for noise predictions, sound-flow interactions and noise control methodologies for some years. There is a major difficulty in any CAA; that of resolving correctly the aerodynamic and acoustic scales which differ by three to four orders of magnitudes. A frequently used method is to separate the calculation of the aerodynamics and acoustics field and to carry out the simulation sequentially, thus giving rise to the well known two-step method. In this method, the unsteady flow field can be determined first using any conventional and established flow simulation schemes, such as direct numerical simulation (DNS) or large eddy simulation (LES). Once the aerodynamic field is known, the sound in the far field can be analyzed using the acoustic analogy or the vortex sound theory. In other words, the two-step method solves the Navier-Stokes equation, and a given wave equation with a specified sound speed. Therefore, if the noise source generation mechanism is not of primary interest, the two-step method is most appropriate. If the noise source generation mechanism and the far field noise are to be determined, the acoustic disturbances in the whole field need to be deduced simultaneously with the aerodynamic field. This means that both the aerodynamic and acoustic fields have to be calculated simultaneously. This approach for CAA is either called the direct noise calculation (DNC) or the one-step method for CAA and is able to yield the aerodynamic and acoustic field as well as the sound propagation speed directly. In view of scale disparity (the aerodynamic disturbances are 103 to 104 times larger than acoustic fluctuations); the one-step method has to be highly accurate so that both aerodynamic and acoustic disturbances can be resolved accurately. Present research on one-step method indicates that a low-dispersion and low-dissipation scheme with high-order filters and high-order non-reflecting boundary conditions is required. Such a code is very complicated because of the complex non-linear unsteady compressible Navier-Stokes equations that need to be solved. Parallel computation could not be fully made used of; therefore, computational time becomes an important issue in any one-step method for CAA. This thesis proposes an alternative to conventional one-step methods and attempts to solve an improved Boltzmann equation (BE) rather than the non-linear Navier-Stokes equations. This method offers the following advantage over DNS or LES methods. Since the improved BE is linear, the computational code has a very simple structure. An effective numerical simulation of the improved BE is the lattice Boltzmann method (LBM), where the continuous velocity space is discretized and the particles are allowed to move with specific speeds. Conventional LBM has been developed mostly for incompressible and very low Mach number flows and is limited to mono-atomic gases. If the LBM were to be applicable to aeroacoustics simulation, it should recover the specific heat ratio and gas properties correctly for diatomic gases. Recent LBM simulations of compressible flows still invoke the mono-atomic gas model; hence the calculated specific heat ratio differs from 1.4 for diatomic gas and the Sutherland law and Fourier law of heat conduction are not recovered correctly. In other words, the calculated Reynolds and Prandtl numbers would be different from the specified values. Therefore, these methods could not be used to simulate aeroacoustic problems for air. If the LBM is to be adopted for one-step aeroacoustic simulation of low Mach number incompressible flows, the first task is to recover the specific heat ratio and the Sutherland law correctly, thus giving rise to a correct Reynolds number. This is accomplished by modifying the BGK model for the collision term in the BE by deriving an effective collision time that takes into account the role played by translational and rotational energy of the atoms. The resultant equation is linear on the left hand side and can be solved using a 6th-order compact finite difference method to evaluate the streaming term and a second order Runge-Kutta time scheme to deal with the time dependent term on the left hand side of the improved BE. This method of solving the improved BE proves to be quite successful in resolving the aerodynamic and acoustic scales accurately. Furthermore, implementation of the non-reflecting boundary conditions on the computational boundary is relatively simple compared with the DNS scheme. A fourth- to sixth-order accurate scheme is required for the boundary in a DNS solution while a first-order accurate boundary scheme in the improved LBM is sufficient to give the same accuracy as the DNS solutions. The improved LBM is tested against many classical problems of aeroacoustic propagation in stationary and moving medium. These include developing acoustic, vortical and entropy pulses, speed of sound recovery and sound scattering by a vortex. All improved LBM solutions are validated against DNS results and available theoretical predictions. The comparisons show that the one-step LBM scheme can be used to accurately resolve the attempted aeroacoustic problems. The advantage of this one-step CAA method is that the code is only about 400 lines for all cases tested, where as if the Navier-Stokes equations are solved using DNS a code with 1400 lines are not unusual. Also, the simple structure of the LBM is most suitable for parallel computation, which means a significant reduction in computational time. The other advantage of this one-step CAA method is that a very simple form of boundary condition could be developed for CAA problems. The next step is to require the improved BE to recover the Prandtl number correctly. Once that is accomplished, the improved LBM could be extended to calculate compressible flows with shocks.

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