On acoustic radiation by a rigid object in a fluid flow

A problem of sound radiation by an absolutely rigid object, moving with respect to the surrounding fluid, is considered on the basis of the Lighthill's equation for aerodynamic sound. An integral representation of the radiated acoustic field is utilized, where the field is characterized as the sum of three fields, generated by a volume distribution of monopoles and by distributions of monopoles and dipoles on the surface of the rigid object. It is shown that, due to a discontinuity of Lighthill's stress tensor on the rigid boundary, a layer of surface divergence of hydrodynamic stresses on the boundary must be taken into account when evaluating the volume integral over Lighthill's quadrupole sources. When the contribution of the surface divergence is included in the solution of Lighthill's equation, amplitudes of the monopole and dipole sound radiated by the rigid object are shown to depend on the potential components of the normal velocity and the pressure on the rigid surface. The obtained solution is compared with Curle's solution for this problem, which establishes that the sound radiation by a rigid object is determined by the force exerted by the object upon the fluid. Both solutions are applied to two known problems of sound scattering and radiation by a rigid sphere in variable pressure and velocity fields. It is shown that predictions based on the obtained solution are equivalent to the results known from literature, whereas Curle's solution gives predictions contradicting the known results. It is also shown that the Ffowcs Williams and Hawkings equation, which coincides with Curle's equation for an immoveable rigid object, does not lead to the correct predictions as well.     

Calculating the pressure in simulations using periodic boundary conditions

Because it is not immediately clear how to write down a proper Hamiltonian for a system in periodic boundary conditions, particularly with Coulombic interactions, we consider a large, finite array of copies of a basic simulation cell containingN particles with some interaction between them. We also putN independent copy particles in each of the copy cells of the array and write down a constrained Lagrangian for the whole system. Constraints on the velocities of the particles of the whole array together with an appropriate initial condition implement the periodic structure in the cells of the array of copies. We derive a Hamiltonian for the whole system with constraints and then derive the equations of motion and a virial expression for the pressure tensor in terms of the forces on the system. In the limit as the array of cell copies becomes large, the equations of motion become the standard ones used in periodic-boundaryconditions simulations. The method also provides an unequivocal algorithm for the pressure in this limit in terms of a virial expression. Particular attention is paid to the case of Coulombic interactions.     

Influence of wall motion on particle sedimentation using hybrid LB-IBM scheme

We integrate the lattice Boltzmann method (LBM) and immersed boundary method (IBM) to capture the coupling between a rigid boundary surface and the hydrodynamic response of an enclosed particle laden fluid. We focus on a rigid box filled with a Newtonian fluid where the drag force based on the slip velocity at the wall and settling particles induces the interaction. We impose an external harmonic oscillation on the system boundary and found interesting results in the sedimentation behavior. Our results reveal that the sedimentation and particle locations are sensitive to the boundary walls oscillation amplitude and the subsequent changes on the enclosed flow field. Two different particle distribution analyses were performed and showed the presence of an agglomerate structure of particles. Despite the increase in the amplitude of wall motion, the turbulence level of the flow field and distribution of particles are found to be less in quantity compared to the stationary walls. The integrated LBM-IBM methodology promised the prospect of an efficient and accurate dynamic coupling between a non-compliant bounding surface and flow field in a wide-range of systems. Understanding the dynamics of the fluid-filled box can be particularly important in a simulation of particle deposition within biological systems and other engineering applications.     

Driving force of acoustic streaming caused by aperiodic sound beam in unbounded volumes

Instantaneous driving force of acoustic streaming in the thermoviscous medium is the subject of investigation. Dynamic equation of the Eulerian streaming velocity is a result of splitting the hydrodynamic equations into acoustic and non-acoustic parts. The acoustic force represents a sum of three parts, one is the classic one, which being averaged over the sound period coincides with the well-known expression. The second one is connected to the periodicity of the sound, it becomes exact zero after averaging for the strictly periodic sound but is not zero for other acoustic wave. The last term originates from the sound divergence. All terms are nonlinear and proportional to the overall attenuation. The consistent comparative analysis of both formula for quasi-periodic and modulated sound is proceeded.     

Direct assessment of lattice Boltzmann hydrodynamics and boundary conditions for recirculating flows

A hydrodynamic boundary condition is developed for lattice Boltzmann hydrodynamics using a square, orthogonal grid. A constraint based on energy considerations is developed to provide closure for the equations which govern the particle distribution at the boundaries. This boundary condition is applied to the two-dimensional, steady flow of an incompressible fluid behind a grid, known as Kovasznay flow. The results are compared to those using alternate boundary conditions using the known exact solution. The hydrodynamic boundary condition produces quadratic spatial convergence, while alternate techniques fail to maintain this second-order accuracy.     

Majorana zero modes induced by skyrmion lattice

One-dimensional s-wave superconductor with spin-orbit coupling is a platform for the realization of Majorana zero modes. The spin-exchange with the magnetic skyrmion lattice can induce spin-orbit coupling in a s-wave superconductor system and the effects are different from the constant spin-orbit coupling. The strength of the effective spin-orbit coupling as well as the rich topoloigcal phase diagram are directly connected to the radius of the skyrmion lattice R. We obtain the rich topological phase diagram of this system with different skyrmion lattice radii by numerically evaluating the spectrum of the system under the periodic boundary condition, and we also find the Majorana zero modes under the open boundary condition to verify the bulk-edge correspondence.     

Pressure boundary conditions for computing incompressible flows with SPH

In Smoothed Particle Hydrodynamics (SPH) methods for fluid flow, incompressibility may be imposed by a projection method with an artificial homogeneous Neumann boundary condition for the pressure Poisson equation. This is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries. We resolve this problem by utilizing a “rotational pressure-correction scheme” with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity. We show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects, and extends the scope of SPH simulation beyond the usual periodic boundary conditions.     

高精度有限差分法模拟Kelvin-Helmholtz不稳定性

 WENO有限差分格式有较高的分辨精度,适合复杂流场的计算,在国际上被广泛采用。本文利用WENO有限差分格式求解2维守恒型欧拉方程,实现了对无粘流体中Kelvin-Helmholtz不稳定性的数值模拟。速度剪切方向采用周期边界条件;扰动增长方向采用嵌边出流边界条件,一个不稳定波长分布64个网格。数值模拟给出的扰动幅值线性增长率与线性稳定性分析给出的结果很好符合,显示了该格式的有效性和精度。数值模拟给出了清晰的密度等值线,表明该方法还具有较好的界面变形捕捉能力。     

A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

We present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing immersed boundary methods: the proposed algorithms calculate and apply the force density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use the MAC (marker and cell) algorithm to solve the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of the force density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of the force density. We validate the proposed algorithms via the classical benchmark problem of flow past a cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of a cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement.     

A Kato Type Theorem for the Inviscid Limit of the Navier-Stokes Equations with a Moving Rigid Body

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Kato (Math Sci Res Inst Publ 2:85?C98, 1984) says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body??s dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, which seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.     

Tiling a plane in a dynamical process and its applications to arrays of quantum dots, drums, and heat transfer

We present a reaction-diffusion system consisting of N components. The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large N, the stationary state becomes a periodic array of hexagonal cells. We present a functional of the densities of the components, which decreases monotonically during the evolution and attains its minimal value in the stationary state. This value is equal to the sum of the first Laplacian eigenvalues for all cells. Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem.     

A New Method for Analysis of the Fluid Interaction with a Deformable Membrane

The lattice Boltzmann cellular automaton method has been successfully extended for analysis of fluid interactions with a deformable membrane or web. The hydrodynamic forces on the solid web are obtained through computation of the fluid flow stress at the moving boundary using the lattice Boltzmann method. Analysis of solid boundary deformation or vibration due to hydrodynamic force is based on Newtonian dynamics and a molecular dynamic type approach.     

ELECTRODYNAMICS OF A QUASI-(1+1)DIMENSIONAL ELECTRON GAS

A hydrodynamic model. is used to study the electrodynamics of a quasi-(1+1)-dimensional electron gas arranged in a periodic array and neutralized by rigid inert positive background The theory is applied to discuss a new model of quasi-(1+1)-dimensional electron gas with a complex unit cell and an analytical expression of the dispersion relation for this system is derived. The results are the same as. those der-ived caith the semi-classical theory dealing with collective excitations.     

Simulation method of colloidal suspensions with hydrodynamic interactions: fluid particle dynamics

We develop a new simulation method of colloidal suspensions, which we call a "fluid particle dynamics" (FPD) method. This FPD method, which treats a colloid as a fluid particle, removes the difficulties stemming from a solid-fluid boundary condition in the treatment of hydrodynamic interactions between the particles. The importance of interparticle hydrodynamic interactions in the aggregation process of colloidal particles is demonstrated as an example. This method can be applied to a wide range of problems in colloidal science.     

不同边界条件下一维双原子链的晶格振动

在自由边界条件和驻波边界条件下，讨论分析了一维双原子链晶格振动的振动模和色散关系，并与周期性边界条件下的结果进行了比较；明确了在不同边界条件下讨论实际原子链晶格振动的振动模和色散关系，其结果是相同的。     

Zero Surface Tension Limit of Viscous Surface Waves

We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the “semi-small” initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.     

Flow pattern in a polymer coil placed into a stationary longitudinal flow

The flow pattern of solvent in a polymer coil placed into a stationary flow is examined. In contrast to the previous works, the flow of solvent at large distances from the macromolecule has a constant longitudinal gradient. The calculations are based on a simple model of macromolecule dynamics in flowing solutions proposed earlier. An analysis of the results shows that, in the first-order approximation in the longitudinal velocity at a certain threshold value of the parameter of hydrodynamic interactions P, the coil acquires a hydrodynamic boundary at which the radial component of the flow velocity is zero. The threshold value of P coincides with that for a stationary shear flow, determined earlier. At large P, i.e., large molecular mass, the hydrodynamic boundary of the coil encompasses a major part of the macromolecule, while the longitudinal intrinsic viscosity takes a form analogous to that characteristic of a suspension of solid balls with a radius equal to the radius of inertia of the polymer coil. In the second-order approximation in the flow velocity, the radial component of the flow velocity is nonzero. As a result, the mass transfer of solvent between the regions separated by the hydrodynamic boundary only slows down, without hindering the speedup of reactions by mixing the reagents and macromolecules.     

SOME PROGRESS IN THE LATTICE BOLTZMANN MODEL

A lattice Boltzmann equation model has been developed by using the equilibrium distribution function of the Maxwell－Boltzmann-like form, which is third order in fluid velocity u_α. The criteria of energy conservation between the macroscopic physical quantities and the microscopic particles are introduced into the model, thus the thermal hydrodynamic equations containing the effect of buoyancy force can be recovered in terms of the Taylor and Chapman－Enskog asymptotic expansion methods. The two-dimensional thermal convection phenomena in a square cavity and between two concentric cylinders have been calculated by implementing a heat flux boundary condition. Both numerical results are in good agreement with the conventional numerical results.     

多层网格法在分离流计算中的应用

文中应用多层网格法计算圆柱绕流,获得了流体的分离现象。文中采用MAC类型的一个分裂格式,推导和给出了固壁处的压力边界条件,用插值法给出了对称面处速度的边界条件,对这些条件都进行了试算和比较。为了更好的计算分离流,我们令z=1nr,使网格步长在r方向以指数形式增长,这就大量地节省了总网格数。文中也用多层网格法来选取微分方程的初值,节省了总的计算时间。最后在文中给出了计算结果和计算机打印的流线图。