Application Of Lattice Boltzmann Method In Parity Time-Revesal Of Viscous Fluid Flow

In quantum mechanics, If a system is Hermitian Hamiltonian, then the Hamiltonian eigenvalues are real numbers, the system has real spectrum. Carl Bender and co-authors had already showed that all eigenvalues of a class of non-Hermitian Hamiltonians were purely real and bounded below if the system satisfied parity time-reversal for the first time in1998. In the past few years, people find that PT symmetry in quantum system reflect to the balanced sink and source in macroscopic system. By introducing the source balanced with the sink, the original dissipative system which tends to failure will come to a sustained functioning. They represent systems as diverse as coupled optical waveguides, and electrical or mechanical oscillators, where light is absorbed in one side and amplified in another.In this paper, we use Lattice Boltzmann Method (LBM) to study the PT symmetry of the fluid. LBM is a mesoscopic method which is based on mass and momentum conservation to stimulate the fluid field that under the condition of continuity equation of hydromechanics or Navier-Stokes equation. The method had developed rapidly in the past decades. Instead of traditional stimulation method of computational fluid dynamics (CFD), LBM takes its own advantage in simpler programming, less memory space of computer, and easily to deal with complex boundary condition, what’s more, it is suitable to do parallel operation. Therefore, LBM has a wide range of applications just as magnetofluid, chemical reaction flow, multiphase and multicomponent flow, porous flow etc. On the other hand, GPU-LBM takes the advantage of parallel operation of LBM, make the calculating speed faster, so that it improves the ability to deal with practical issues accurately and effectively.We introduce PT symmetry into fluid system by setting up balanced inflow and outflow in a2D channel. The flow is governed by Navier-Stokes equation and we use lattice Boltzmann method to solve them. It is founded that all asymmetries scale power laws with the Reynolds number, Ρn～Ren. Our configurations have asymmetries that are orders of magnitude smaller than the asymmetries that occur in traditional configurations at low Reynolds numbers. In this work, we impose three different velocity profiles to drive the flow. It is demonstrated that in three driven modes, the power-law schedule maintains. It is concluded that PT asymmetry of the viscous flow is due to the properties of the fluid itself not the driven modes, implying the universality of the power-law scaling in viscous flow with balanced inflow and outflow.Then, we study the PT symmetry with different configuration of velocity boundary condition, and find that all asymmetries scale power laws with he Reynolds number. Compared with the traditional fully developed boundary condition, the asymmetries of balanced sink and source boundary condition are several orders of magnitude lower. At the end of this paper, we compare the results in three-dimensional with that in2D situation, get the same results, which makes the conclusion more persuasive.