In our study, the dual time-stepping strategy of the gas-kinetic scheme is constructed and used for the simulation of unsteady flows. In comparison to the previous implicit gas-kinetic scheme, both the inviscid and viscous flux Jacobian are considered in our work, and the linear system of the pseudo-steady-state is solved by applying generalized minimal residual algorithm. The accuracy is validated by several numerical cases, the incompressible flow around blunt bodies (stationary circular cylinder and square cylinder), and the transonic buffet on the NACA0012 airfoil under hybrid mesh. The numerical cases also demonstrate that the present method is applicable to approach the fluid flows from laminar to turbulent and from incompressible to compressible. Finally, the case of acoustic pressure pulse is carried out to evaluate the effects of enlarged time step, and the side effect of enlarged time step is explained. Compared with the explicit gas-kinetic scheme, the proposed scheme can greatly accelerate the computation and reduce the computational costs for unsteady flow simulations.
In this paper, a variant of the acoustic multipole source (AMS) method is proposed within the framework of the lattice Boltzmann method. A quadrupole term is directly included in the stress system (equilibrium momentum flux), and the dependency of the quadrupole source in the inviscid limit upon the fortuitous discretization error reported in the works of E. M. Viggen [Phys. Rev. E 87, 023306 (2013)PLEEE81539-375510.1103/PhysRevE.87.023306] is removed. The regularized lattice Boltzmann (RLB) method with this variant AMS method is presented for the 2D and 3D acoustic problems in the inviscid limit, and without loss of generality, the D3Q19 model is considered in this work. To assess the accuracy and the advantage of the RLB scheme with this AMS for acoustic point sources, the numerical investigations and comparisons with the multiple-relaxation-time (MRT) models and the analytical solutions are performed on the 2D and 3D acoustic multipole point sources in the inviscid limit, including monopoles, x dipoles, and xx quadrupoles. From the present results, the good precision of this AMS method is validated, and the RLB scheme exhibits some superconvergence properties for the monopole sources compared with the MRT models, and both the RLB and MRT models have the same accuracy for the simulations of acoustic dipole and quadrupole sources. To further validate the capability of the RLB scheme with AMS, another basic acoustic problem, the acoustic scattering from a solid cylinder presented at the Second Computational Aeroacoustics Workshop on Benchmark Problems, is numerically considered. The directivity pattern of the acoustic field is computed at r=7.5; the present results agree well with the exact solutions. Also, the effects of slip and no-slip wall treatments within the regularized boundary condition on this pure acoustic scattering problem are tested, and compared with the exact solution, the slip wall treatment can present a better result. All simulations demonstrate that the RLB scheme with the AMS method is capable of accurately simulating 2D and 3D acoustic generation, propagation, and scattering at zero viscosity.
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