Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming

We present a 2D- and 3D-lattice Boltzmann model for the treatment of free surface flows including gas diffusion. Interface advection and related boundary conditions are based on the idea of the lattice Boltzmann equation. The fluid dynamic boundary conditions are approximated by using the mass and momentum fluxes across the interface, which do not require explicit calculation of gradients. A similar procedure is applied to fulfill the diffusion boundary condition. Simple verification tests demonstrate the correctness of the algorithms. 2D- and 3D-foam evolution examples demonstrate the potential of the method.     

Surface Tension of a Metal-Electrolyte Boundary: Exactly Solvable Model

An ideal conductor electrode in contact with a semi-infinite two-dimensional two-component plasma in an external potential is considered. The model is mapped onto an integrable sine-Gordon theory with Dirichlet boundary conditions. The information gained from the mapping provides an explicit form of the surface tension in the plasma-stability regime.     

典型自由面流动问题的SPH-ALE数值模拟

应用基于黎曼解的SPH-ALE方法对两种典型自由面流动问题进行数值模拟,并提出一种一阶核函数修正压力计算方法,通过对临近边界的水粒子压力进行核积分,近似估算固壁边界压力.给出不同时刻的流场压力分布及自由液面演化过程,将计算结果与相关的试验值及数值解进行对比,分析结果表明：SPH-ALE方法较传统SPH方法在流场压力计算精度上有较大的改进,在处理强非线性自由面流动问题时能够达到较高的精度.     

CP violation from five-dimensional QED

It is shown that QED in (1 + 4)-dimensional space-time, with the fifth dimension compactified on a circle, is, in general, a CP violating theory. Depending on the fermionic boundary conditions, CP violation may be either explicit (through the Scherk-Schwarz mechanism) or spontaneous (via the Hosotani mechanism). The fifth component of the gauge field acquires (at the one-loop level) a nonzero vacuum expectation value which, in the presence of two fermionic fields, leads to spontaneous CP violation when the boundary conditions are CP symmetric. Phenomenological consequences are illustrated by a calculation of the electric dipole moment for the fermionic zero modes.     

An immersed boundary method based on discrete stream function formulation for two- and three-dimensional incompressible flows

An immersed boundary method is proposed in the framework of discrete stream function formulation for incompressible flows. In order to impose the non-slip boundary condition, the forcing term is determined implicitly by solving a linear system. The number of unknowns of the linear system is the same as that of the Lagrangian points representing the body surface. Thus the extra cost in force calculation is negligible if compared with that in the basic flow solver. In order to handle three-dimensional flows at moderate Reynolds numbers, a parallelized flow solver based on the present method is developed using the domain decomposition strategy. To verify the accuracy of the immersed-boundary method proposed in this work, flow problems of different complexity (decaying vortices, flows over stationary and oscillating cylinders and a stationary sphere, and flow over low-aspect-ratio flat-plate) are simulated and the results are in good agreement with the experimental or computational data in previously published literatures.     

Thickness of a boundary layer attributed to the wave motion on the charged free surface of a viscous liquid

It is shown that the analytical estimator for the boundary layer thickness that contains the wave frequency in the denominator and is proposed for approximate calculation of the wave motion on the free surface of a viscous liquid cannot be formally applied to the wave motion on the uniformly charged liquid surface. The fact is that, when the surface charge density attains a value critical in terms for the Tonks-Frenkel instability, the wave frequency tends to zero. From the analysis of liquid motions near the electric charge critical density, a technique is proposed for calculating the thickness of a boundary layer attributed to flows of various kinds. It is found that the thickness of the boundary layer due to aperiodic flows with amplitudes exponentially growing with time (such flows take place at the stage of instability against the surface charge) does not exceed a few tenths of the wavelength, whereas the thickness of the boundary layer due to exponentially decaying liquid flows is roughly equal to the wavelength.     

Roughness-induced critical phenomena in a turbulent flow

I present empirical evidence that turbulent flows are closely analogous to critical phenomena, from a reanalysis of friction factor measurements in rough pipes. The data collapse found here corresponds to Widom scaling near critical points, and implies that a full understanding of turbulence requires explicit accounting for boundary roughness.     

Simulation of viscous flows with undulatory boundaries. Part I: Basic solver

A numerical method for the simulation of viscous flows with undulatory walls and free surfaces is presented. The simulation domain is discretized by a boundary-fitted and time-dependent grid. The Navier–Stokes equations, subject to fully nonlinear kinematic and dynamic boundary conditions at the free surface and no-slip boundary condition at the wall, are simulated by a hybrid pseudo-spectral and finite difference method in space and a semi-implicit fractional-step method in time. The performance of the method is demonstrated by a series of test cases including flows over wavy boundaries, various surface waves, and interaction between vortices and free surfaces. Validation by convergence test and extensive comparisons with previous theoretical, experimental, and numerical studies indicate the accuracy and efficiency of the method. Finally, a simulation example of turbulence and free surface interaction is presented. Results show that the rich features of the free surface such as surface waves, splats, anti-splats, dimples, and scars are captured accurately. Characteristic vortical structures and variation of turbulence statistics in the near-surface region are also elucidated.     

椭圆外导体-矩形内导体同轴线的特性阻抗

本文提出用图形逼近和取尺寸上、下限值的几何平均值的方法,计算椭圆外导体-矩形内导体同轴线特性阻抗。矩形截面内导体的内切共焦椭圆和外接共焦椭圆,通过保角变换,变成同心圆。矩形变成近于矩形的闭合曲线,二同心圆分别为它的最小值和最大值,取二者的几何平均值,与同轴椭圆截面外导体同时变成的同心圆,构成一个标准同轴线,得到了特性阻抗公式。并由此推导出椭圆外导体-微带内导体、圆形外导体-方形内导体、圆形外导体-矩形内导体及圆形外导体-微带内导体等各种同轴线的特性阻抗的初等函数表示式。给出了上述几种同轴线的特性阻抗准确数值。 关键词：     

Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

Entanglement entropy for a Dirac fermion in three dimensions: Vertex contribution

In three dimensions there is a logarithmically divergent contribution to the entanglement entropy which is due to the vertices located at the boundary of the region considered. In this work we find the corresponding universal coefficient for a free Dirac field, and extend a previous work in which the scalar case was treated. The problem is equivalent to find the conformal anomaly in three-dimensional space where multiplicative boundary conditions for the field are imposed on a plane angular sector. As an intermediate step of the calculation we compute the trace of the Green function of a massive Dirac field in a two-dimensional sphere with boundary conditions imposed on a segment of a great circle.     

The local geometry of chaotic billiards

For a billiard of a general shape a transformation is introduced which projects the boundary on the unit circle. This introduces a non-Euclidean metric on the plane which contains all relevant information of the shape of the boundary. Classically the straight lines of the free motion correspond to geodesics and quantum mechanically the energy spectrum is that of Laplace–Beltrami operator with Dirichlet boundary conditions on the unit circle. The geodesic equations are highly non-linear. Nevertheless for the interval between two consecutive scatterings we have two integrals of motion, the kinetic energy and the angular momentum. This fact helps to solve explicitly the geodesic equations. These solutions can be used to derive interesting properties for the classical scattering. Quantum mechanically the spectrum of the above billiards is obtained for certain parameter values both perturbatively for small values of the parameter and also using a diagonalization procedure. This method is applicable to any particular form of a billiard for which the transformation is invertible and can be used on one hand as a quick method of approximate spectral determination and as a theoretical tool to analyse specific properties of integrability and chaos through the associated connection form and the Laplace–Beltrami operator. Finally as a first indication of the potentiality of this method we present a graphical test where for very small deviations from the circular billiard an integrable and two non-integrable billiards can be distinguished by the distribution of the differences of the first order corrections while this distinction is not evident by the usual test for the nearest neighbor level spacings.     

Renormalization of singlet NN-scattering with one pion exchange and boundary conditions

We present a simple and physically compelling boundary condition regularization scheme in the framework of effective field theory as applied to nucleon–nucleon interaction. It is free of off-shell ambiguities and ultraviolet divergences and provides finite results at any step of the calculation. Low-energy constants and their non-perturbative evolution can directly be obtained from experimental threshold parameters in a completely unique, one-valued and model independent way when the long range explicit pion effects are removed. This allows to compute scattering phase shifts which are, by construction consistent with effective range expansion to a given order in the CM momentum and are free from finite cut-off artifacts. We illustrate how the method works in the ¹S₀ channel for the one pion exchange potential.     

On the Distribution and Gap Structure of Lee–Yang Zeros for the Ising Model: Periodic and Aperiodic Couplings

In this work, we present some results on the distribution of Lee–Yang zeros for the ferromagnetic Ising model on the rooted Cayley Tree (Bethe Lattice), assuming free boundary conditions, and in the one-dimensional lattice with periodic boundary conditions. In the case of the Cayley Tree, we derive the conditions that the interactions between spins must obey in order to ensure existence or absence of phase transition at finite temperature (T0). The results are first obtained for periodic interactions along the generations of the lattice. Then, using periodic approximants, we are also able to obtain results for aperiodic sequences generated by substitution rules acting on a finite alphabet. The particular examples of the Fibonacci and the Thue-Morse sequences are discussed. Most of the results are obtained for a Cayley Tree with arbitrary order d. We will be concerned in showing whether or not the zeros become dense in the whole unit circle of the fugacity variable. Regarding the one-dimensional Ising model, we derive a general treatment for the structure of gaps (regions free of Lee–Yang zeros) around the unit circle.     

Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications

A version of immersed boundary-lattice Boltzmann method (IB-LBM) is proposed in this work. It is based on the lattice Boltzmann equation with external forcing term proposed by Guo et al. [Z. Guo, C. Zheng, B. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E 65 (2002) 046308], which can well consider the effect of external force to the momentum and momentum flux as well as the discrete lattice effect. In this model, the velocity is contributed by two parts. One is from the density distribution function and can be termed as intermediate velocity, and the other is from the external force and can be considered as velocity correction. In the conventional IB-LBM, the force density (external force) is explicitly computed in advance. As a result, we cannot manipulate the velocity correction to enforce the non-slip boundary condition at the boundary point. In the present work, the velocity corrections (force density) at all boundary points are considered as unknowns which are computed in such a way that the non-slip boundary condition at the boundary points is enforced. The solution procedure of present IB-LBM is exactly the same as the conventional IB-LBM except that the non-slip boundary condition can be satisfied in the present model while it is only approximately satisfied in the conventional model. Numerical experiments for the flows around a circular cylinder and an airfoil show that there is no any penetration of streamlines to the solid body in the present results. This is not the case for the results obtained by the conventional IB-LBM. Another advantage of the present method is its simple calculation of force on the boundary. The force can be directly calculated from the relationship between the velocity correction and the force density.     

An adaptive implicit–explicit scheme for the DNS and LES of compressible flows on unstructured grids

An adaptive implicit–explicit scheme for Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) of compressible turbulent flows on unstructured grids is developed. The method uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme. The solution is marched in time using an Implicit–Explicit Runge–Kutta (IMEX-RK) time-advancement scheme. A novel adaptive algorithm for splitting the system into implicit and explicit sets is developed. The method is validated using several canonical laminar and turbulent flows. Load balance for the new scheme is achieved by a dual-constraint, domain decomposition algorithm. The scalability and computational efficiency of the method is investigated, and memory savings compared with a fully implicit method is demonstrated. A notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed.     

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

The localized artificial diffusivity method is investigated in the context of large-eddy simulation of compressible turbulent flows. The performance of different artificial bulk viscosity models are evaluated through detailed results from the evolution of decaying compressible isotropic turbulence with eddy shocklets and supersonic turbulent boundary layer. Effects of subgrid-scale (SGS) models and implicit time-integration scheme/time-step size are also investigated within the framework of the numerical scheme used. The use of a shock sensor along with artificial bulk viscosity significantly improves the scheme for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The proposed combination of Ducros-type sensor with a negative dilatation sensor removes unnecessary bulk viscosity within expansion and weakly compressible turbulence regions without shocks and allows it to localize near the shocks. It also eliminates the need for a wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. For the numerical schemes used, better results are obtained without adding an explicit SGS model than with SGS model at moderate Reynolds number. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity results in additional damping of the resolved turbulence. However, investigations at higher Reynolds numbers suggest the need for an explicit SGS model. The flow statistics obtained using the second-order implicit time-integration scheme with three sub-iterations closely agrees with the explicit scheme if the maximum Courant–Friedrichs–Lewy is kept near unity.     

Shape deformation and circle instability in two-dimensional lipid domains by dipolar force: a shape- and size-dependent line tension model

The dipolar energy of a solid monolayer domain surrounded by a fluid phase at an air-water interface is derived approximately as a sum of an additionally negative line tension and a curvature-elastic energy at the boundary. Variation of the domain energy yields an equilibrium domain shape equation. The obvious solutions of the domain shape equation clearly predict a circle, torus, D-form, S-form, and serpentine manner shape found experimentally, depending on the difference in the Gibbs free energy between the solid and fluid phases and the total line tension. Analysis of linear instability for a circle with a fixed area shows that, above a threshold size, the circle can be deformed into an m-sided quasipolygon. The good agreement with the observation and numerical calculation reported by Lee and McConnell [J. Phys. Chem. 91, 9532 (1993)]] shows the quantitative validity of the present theory.     

Solving 1D plasmas and 2D boundary problems using Jack polynomials and functional relations

The general one-dimensional log-sine gas is defined by restricting the positive and negative charges of a two-dimensional Coulomb gas to live on a circle. Depending on charge constrannts, this problem is equivalent to different boundary field theories.We study the electrically neutral case, which is equivalent to a two-dimensional free boson with an impurity cosine potential. We use two different methods: a perturbative one based on Jack symmetric functions, and a nonperturbative one based on the thermodynamic Bethe ansatz and functional relations. The first method allows us to find an explicit series expression for all coefficients in the virial expansion of the free energy and the experimentally measurable conductance. Some results for correlation functions are also presented. The second method gives an expression for the full free energy, which yields a surprising fluctuation-dissipation relation between the conductance and the free energy.