Computation of the effective potential for the two-dimensionalO(N) nonlinear σ-models

A Monte Carlo method for the computation of effective Hamiltonians for theO(N) nonlinear lattice σ-models is described. The procedure is based on simulations of auxiliary statistical mechanical systems with fixed block spins and thus avoids the simulation of systems with large correlation length. The method is applied to study the renormalization group flow of the effective potential for the Ising model, the XY model, and the Heisenberg ferromagnet in two dimensions. Some of the results are compared with second order high temperature expansions. For small block size a very good agreement is found in a large range of the inverse temperature β.     

Three-dimensional simulations of instabilities in a Marangoni-driven, low Prandtl number liquid bridge with magnetic stabilization to verify linear stability theory

The Full-Zone model of a liquid bridge encountered in crystal growth is analyzed via linear stability analysis and three-dimensional spectral element simulations, neglecting gravitational forces, for Prandtl number 0.02. The base state is axisymmetric and steady state. Linear stability predicts the character of flow transitions and the value of Re _FZ , the thermocapillary Reynolds number, at which instabilities occur. Previous linear stability findings show that application of a steady, axial magnetic field stabilizes the base state. Previous three-dimensional simulations with no magnetic field predict a first transition that agrees well with linear stability theory. However, these simulations also demonstrated that continued time integration at just slightly higher Re _FZ leads to what appears to be periodic flow. Closer inspection and comparison with linear stability theory revealed that this apparent periodicity was actually competition between two steady modes with different axial symmetries. Here an axial magnetic field is applied in three-dimensional simulations and it is verified that the magnetic field does have the intended effect of stabilizing the flow and removing modal competition. The azimuthal flow shows excellent agreement with eigenvectors predicted by linear stability theory.     

偏晶合金液-液相分离的格子玻尔兹曼方法模拟

本文建立了一个模拟在弥散相液滴的扩散长大、碰撞凝并和Ostwald熟化等因素的作用下偏晶合金液-液相分离过程的二维格子玻尔兹曼方法 (lattice Boltzmann method, LBM) 模型.该模型结合了Shan-Chen的两相流模型和Qin的介观粒子相互作用势模型的优点,并在LB演化方程中引入了反映相变的源项.应用该模型模拟研究了偏晶合金液-液相分离过程中单液滴的生长、两液滴的合并和多液滴的生长规律.结果表明在两液相区中第二相单个液滴的生长是一个通过扩散从非平衡态到平衡态过渡的过程.两液滴合并 关键词： 偏晶合金 液-液相分离 格子玻尔兹曼方法     

基于大密度差多相流SPH方法的二维液滴碰撞数值模拟

该文结合了Ott提出的修正连续性方程和Adami改进的动量方程, 对空气中的液滴碰撞问题进行了二维数值模拟. 为有效提高计算精度, 推导了适用于大密度差多相流的人工黏性和人工应力方程. 通过表面张力作用下方形液滴自然变化和空气中两液滴互溶的算例, 验证了算法的有效性; 对不同韦伯数 (8.8, 19.8)、不同碰撞参数 (0, 0.5)下的液滴碰撞过程进行了数值模拟, 并与VOF方法对比,取得了较为一致的结果; 进一步计算多个韦伯数、多个碰撞参数下的液滴碰撞, 得到了空气中二维液滴碰撞结果分布图,与实验结果相符合. 结果表明, 该算法对于求解涉及大密度差多相流的液滴碰撞破碎问题十分有效,而且该方法容易拓展到三维, 从而为进一步模拟火箭发动机的二次雾化过程奠定了基础. 关键词： 光滑粒子流体动力学 大密度差 多相流 液滴碰撞     

Dimension reduction in heterogeneous neural networks: Generalized Polynomial Chaos (gPC) and ANalysis-Of-VAriance (ANOVA)

We propose, and illustrate via a neural network example, two different approaches to coarse-graining large heterogeneous networks. Both approaches are inspired from, and use tools developed in, methods for uncertainty quantification (UQ) in systems with multiple uncertain parameters – in our case, the parameters are heterogeneously distributed on the network nodes. The approach shows promise in accelerating large scale network simulations as well as coarse-grained fixed point, periodic solution computation and stability analysis. We also demonstrate that the approach can successfully deal with structural as well as intrinsic heterogeneities.     

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

We present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in three-dimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R³. We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R³. The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable ‘material surface’ in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics — a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C² interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady three-dimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.     

Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry

The gas-kinetic numerical algorithm solving the Boltzmann model equation is extended and developed to study the three-dimensional hypersonic flows of spacecraft re-entry into the atmosphere in perfect gas. In this study, the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic Boltzmann–Shakhov model. The discrete velocity ordinate technique and numerical quadrature methods, such as the Gauss quadrature formulas with the weight function 2/π^1/2exp(?V²) and the Gauss–Legendre numerical quadrature rule, are studied to resolve the barrier in simulating complex flows from low Mach numbers to hypersonic problems. Specially, the gas-kinetic finite-difference scheme is constructed for the computation of three-dimensional flow problems, which directly captures the time evolution of the molecular velocity distribution function. The gas-kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function. The HPF (high performance fortran) parallel implementation technique for the gas-kinetic numerical method is developed and applied to study the hypersonic flows around three-dimensional complex bodies. The main purpose of the current research is to provide a way to extend the gas-kinetic numerical algorithm to the flow computation of three-dimensional complex hypersonic problems with high Mach numbers. To verify the current method and simulate gas transport phenomena covering various flow regimes, the three-dimensional hypersonic flows around sphere and spacecraft shape with different Knudsen numbers and Mach numbers are studied by HPF parallel computing. Excellent results have been obtained for all examples computed.     

Droplet-Laden Mist Film Cooling Effectiveness of Cylindrical Holes Embedded in a Transverse Trench

A three-dimensional numerical model of calculating in the Euler approach is developed to calculate a two-phase turbulent near-wall flow; simulation of thermal efficiency of a gas-droplet shielding injected into a transverse trench through inclined cylindrical holes is fulfilled. The influence of the main thermo-gas-dynamic characteristics of the two-phase flow on thermal efficiency is analyzed. Significant increase in thermal efficiency was obtained by adding droplets in the nearwall coolant flow (up to 2 times in comparison with a single-phase flow). A particular advantage of this method of coolant injection is achieved at high injection parameters. It is shown that the use of two-phase gas-droplet near-wall shielding is promising for protection of surfaces against thermal influence of the heated gas flows.     

Reference motion in deformable bodies under rigid body motion and vibration. Part II: evaluation of the coefficient of restitution for impacts

The dynamic analysis of rigid body impacts is usually performed by using the coefficient of restitution as a measure of the mechanical energy lost in the process. The coefficient of restitution provides an algebraic equation that allows impulse-momentum balance equations to be solved. This paper reports a method for calculating the kinematic coefficient of restitution for the impact of deformable bodies by using a numerical simulation procedure. Calculations were done within the framework of the floating reference frame approach. As shown here for the first time, discriminating between reference velocities and rigid body equivalent velocities is essential with a view to accurately calculating the kinematic coefficient of restitution. Thus, the velocities at which the contact points approach and depart must be calculated as rigid body equivalent velocities. If references velocities are used instead, the resulting coefficient of restitution lacks physical significance. The proposed method is illustrated with two applications, namely: an axial impact of a rigid body on a deformable rod and a transverse impact of a beam on a fixed stop.     

An examination of the flow induced by the motion of many buoyant bubbles

The results of direct numerical simulations of the motion of many three-dimensional buoyant bubbles in periodic domains are examined. The bubble motion is computed by solving the full Navier-Stokes equations by a parallelized finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is about 25. Two cases are examined. In one, the bubbles are nearly spherical; in the other, the bubbles rise with an ellipsoidal shape. The ellipsoidal bubbles show a much larger fluctuation velocity and by visualizing the flow field it is possible to show that the difference is due to larger vorticity generation and stronger interactions of the deformable bubbles. The focus here is on the early stage of the flow, when both the spherical and the deformable bubbles are nearly uniformly distributed.     

Unsteady analysis of six-DOF motion of a 6:1 prolate spheroid in viscous fluid

Free-moving simulations of airplanes, submarines and other automobiles under extreme and emergency conditions are becoming increasingly important from operational and tactical perspectives. Such simulations are fairly challenging due to the extreme unsteady motions and high Re(Reynolds) numbers. The aim of this study is to perform a six-DOF motion simulation of a 6:1prolate spheroid that is falling in a fluid field. Prior to conducting the six-DOF simulation, some verification simulations were performed. First, a laminar flow past an inclined prolate spheroid at a Re number of 1000 and incidence angle of 45. with a tetrahedral mesh was simulated to verify the relevant targeted discrete method for an unstructured mesh. Second, to verify the LES(large eddy simulation) models and dependent parameters for the DDES(delayed detached eddy simulation), a turbulent flow past a sphere was performed at a subcritical Re number of 10000. Third, a steady maneuvering problem about a prolate spheroid pitching up from 0. to 30. incidence at a uniform angular velocity was established based on a dynamic tetrahedral mesh with changing topology and the ALE(arbitrary Lagrangian-Eulerian) method of fluid-structure coupling at a Re number of 4.2 × 10~6.Finally, two six-DOF motions of an inclined 6:1 prolate spheroid at an initial incidence of 45. were simulated at different Re numbers of 10000 and 4.2 × 10~6.     

An interface capturing method for the simulation of multi-phase compressible flows

A novel finite-volume interface (contact) capturing method is presented for simulation of multi-component compressible flows with high density ratios and strong shocks. In addition, the materials on the two sides of interfaces can have significantly different equations of state. Material boundaries are identified through an interface function, which is solved in concert with the governing equations on the same mesh. For long simulations, the method relies on an interface compression technique that constrains the thickness of the diffused interface to a few grid cells throughout the simulation. This is done in the spirit of shock-capturing schemes, for which numerical dissipation effectively preserves a sharp but mesh-representable shock profile. For contact capturing, the formulation is modified so that interface representations remain sharp like captured shocks, countering their tendency to diffuse via the same numerical diffusion needed for shock-capturing. Special techniques for accurate and robust computation of interface normals and derivatives of the interface function are developed. The interface compression method is coupled to a shock-capturing compressible flow solver in a way that avoids the spurious oscillations that typically develop at material boundaries. Convergence to weak solutions of the governing equations is proved for the new contact capturing approach. Comparisons with exact Riemann problems for model one-dimensional multi-material flows show that the interface compression technique is accurate. The method employs Cartesian product stencils and, therefore, there is no inherent obstacles in multiple dimensions. Examples of two- and three-dimensional flows are also presented, including a demonstration with significantly disparate equations of state: a shock induced collapse of three-dimensional van der Waal’s bubbles (air) in a stiffened equation of state liquid (water) adjacent to a Mie-Grüneisen equation of state wall (copper).     

Accelerated boundary integral method for multiphase flow in non-periodic geometries

An accelerated boundary integral method for Stokes flow of a suspension of deformable particles is presented for an arbitrary domain and implemented for the important case of a planar slit geometry. The computational complexity of the algorithm scales as $O(N)$ or $O(N\log N$), where N is proportional to the product of number of particles and the number of elements employed to discretize the particle. This technique is enabled by the use of an alternative boundary integral formulation in which the velocity field is expressed in terms of a single layer integral alone, even in problems with non-matched viscosities. The density of the single layer integral is obtained from a Fredholm integral equation of the second kind involving the double layer integral. Acceleration in this implementation is provided by the use of General Geometry Ewald-like method (GGEM) for computing the velocity and stress fields driven by a set of point forces in the geometry of interest. For the particular case of the slit geometry, a Fourier-Chebyshev spectral discretization of GGEM is developed. Efficient implementations employing the GGEM methodology are presented for the resulting single and the double layer integrals. The implementation is validated with test problems on the velocity of rigid particles and drops between parallel walls in pressure driven flow, the Taylor deformation parameter of capsules in simple shear flow and the particle trajectories in pair collisions of capsules in simple shear flow. The computational complexity of the algorithm is verified with results from several large scale multiparticle simulations.     

Experimental evidence for non-linear growth in compressible mixing layer

An experimental study of compressible mixing layers(CMLs)was conducted using planar laser Mie scattering(PLMS)visualizations from condensed ethanol droplets in the flow.Large ensembles of digital images were collected for two flow conditions at convective Mach numbers Mc=0.11 and 0.47.The coherent vortices,braids and eruptions in the mixing zone were observed,interpreted as evidence of multi-scale,three-dimensional structures at a high Reynolds number.The mixing layers with a large visualized range present two stages along the streamwise direction,corresponding to the initial mixing and the well-developed stage.A new method,the gray level ensemble average method(GLEAM),by virtue of the similarity of the mixing layer,was applied to measure the growth rate of the CML thickness.New evidence for a nonlinear growth of CML is reported,providing an interpretation of previous observations of the scattering of the growth rate.     

A new domain decomposition method with overlapping patches for ultrascale simulations: Application to biological flows

We address the failure in scalability of large-scale parallel simulations that are based on (semi-)implicit time-stepping and hence on the solution of linear systems on thousands of processors. We develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources. It is a non-intrusive approach as it does not require modification of existing codes. Specifically, we present here a two-stage domain decomposition method for the Navier–Stokes equations that combines features of discontinuous and continuous Galerkin formulations. At the first stage the domain is subdivided into overlapping patches and within each patch a C⁰ spectral element discretization (second stage) is employed. Solution within each patch is obtained separately by applying an efficient parallel solver. Proper inter-patch boundary conditions are developed to provide solution continuity, while a Multilevel Communicating Interface (MCI) is developed to provide efficient communication between the non-overlapping groups of processors of each patch. The overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch. This dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency. The accuracy of the method has been evaluated in solutions of steady and unsteady 3D flow problems including blood flow in the human intracranial arterial tree. Benchmarks on BlueGene/P, CRAY XT5 and Sun Constellation Linux Cluster have demonstrated good performance on up to 96,000 cores, solving up to 8.21B degrees of freedom in unsteady flow problem. The proposed method is general and can be potentially used with other discretization methods or in other applications.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Towards an entropy-based detached-eddy simulation

A concept of entropy increment ratio()is introduced for compressible turbulence simulation through a series of direct numerical simulations(DNS). represents the dissipation rate per unit mechanical energy with the benefit of independence of freestream Mach numbers.Based on this feature,we construct the shielding function fs to describe the boundary layer region and propose an entropy-based detached-eddy simulation method(SDES).This approach follows the spirit of delayed detached-eddy simulation(DDES)proposed by Spalart et al.in 2005,but it exhibits much better behavior after their performances are compared in the following flows,namely,pure attached flow with thick boundary layer(a supersonic flat-plate flow with high Reynolds number),fully separated flow(the supersonic base flow),and separated-reattached flow(the supersonic cavity-ramp flow).The Reynolds-averaged Navier-Stokes(RANS)resolved region is reliably preserved and the modeled stress depletion(MSD)phenomenon which is inherent in DES and DDES is partly alleviated.Moreover,this new hybrid strategy is simple and general,making it applicable to other models related to the boundary layer predictions.