Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples：turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder （GL） model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.

Phase Diagrams of the Semi-Infinite Blume-Capel Model with Mixed Spins (SA = 1 and SB = 3/2) by Migdal Kadanoff Renormalization Group

We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R1 and R2 of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.

Memory,Time and Technique Aspects of Density Matrix Renormalization Group Method

We present the memory size,computational time,and technique aspects of density matrix renormalization group (DMRG) algorithm.We show how to estimate the memory size and computational time before starting a large scale DMRG calculation.We propose an implementation of the Hamiltonian wavefunction multiplication and a wavefunction initialization in DMRG with block matrix data structure.One-dimensional Heisenberg model is used to illustrate our study.

Improved hybrid parallel strategy for density matrix renormalization group method

We propose a new heterogeneous parallel strategy for the density matrix renormalization group(DMRG)method in the hybrid architecture with both central processing unit(CPU)and graphics processing unit(GPU).Focusing on the two most time-consuming sections in the finite DMRG sweeps,i.e.,the diagonalization of superblock and the truncation of subblock,we optimize our previous hybrid algorithm to achieve better performance.For the former,we adopt OpenMP application programming interface on CPU and use our own subroutines with higher bandwidth on GPU.For the later,we use GPU to accelerate matrix and vector operations involving the reduced density matrix.Applying the parallel scheme to the Hubbard model with next-nearest hopping on the 4-leg ladder,we compute the ground state of the system and obtain the charge stripe pattern which is usually observed in high temperature superconductors.Based on simulations with different numbers of DMRG kept states,we show significant performance improvement and computational time reduction with the optimized parallel algorithm.Our hybrid parallel strategy with superiority in solving the ground state of quasi-two dimensional lattices is also expected to be useful for other DMRG applications with large numbers of kept states,e.g.,the time dependent DMRG algorithms.

Mechanism of local scour around submarine pipelines based on numerical simulation of turbulence model

The mechanism of local scour around submarine pipelines is studied numerically based on a renormalized group （RNG） turbulence model. To validate the numerical model, the equilibrium profiles of local scour for two cases are simulated and compared with the experimental data. It shows that the RNG turbulence model can give an appropriate prediction for the configuration of equilibrium scour hole, and it is applicable to this situation. The local scour mechanism around submarine pipelines including the flow structure, shear stress distribution and pressure field is then analyzed and compared with experiments. For further comparison and validation, especially for the flow structure, a numerical calculation employing the large eddy simulation （LES） is also conducted. The numerical results of RNG demonstrate that the critical factor governing the equilibrium profile is the seabed shear stress distribution in the case of bed load sediment transport, and the two-equation RNG turbulence model coupled with the law of wall is capable of giving a satisfying estimation for the bed shear stress. Moreover, the piping phenomena due to the great difference of pressure between the upstream and downstream parts of pipelines and the vortex structure around submarine pipelines are also simulated successfully, which are believed to be the important factor that lead to the onset of local scour.

Renormalization-group theory of first-order phase transition dynamics in field-driven scalar model

Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ^3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar Ca model, below its critical temperature and near the instability points. Finite- time scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang-Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.

Performances of Different Turbulence Models for Simulating Shallow Water Sloshing in Rectangular Tank

Liquid sloshing is a common phenomenon in the transportation of liquid-cargo tanks.Liquid waves lead to fluctuating forces on the tank walls.If these fluctuations are not predicted or controlled,for example,by using baffles,they can lead to large forces and momentums.The volume of fluid(VOF)two-phase numerical model in Open FOAM open-source software has been widely used to model the liquid sloshing.However,a big challenge for modeling the sloshing phenomenon is selecting a suitable turbulence model.Therefore,in the present study,different turbulence models were studied to determine their sloshing phenomenon prediction accuracies.The predictions of these models were validated using experimental data.The turbulence models were ranked by their mean error in predicting the free surface behaviors.The renormalization group(RNG)k-ε and the standard k–ω models were found to be the best and worst turbulence models for modeling the sloshing phenomena,respectively;moreover,the SST k-ω model and v2-f k-ε results were very close to the RNG k-εmodel result.

Entanglement and quantum phase transition in the Heisenberg-Ising model

We use the quantum renormalization-group（QRG） method to study the entanglement and quantum phase transition（QPT） in the one-dimensional spin-1/2 Heisenberg-Ising model [Lieb E,Schultz T and Mattis D 1961 Ann.Phys.（N.Y.） 16 407].We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations.We also investigate the scaling behavior of the system close to the quantum critical point,which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size.Also,the first derivative of concurrence between two blocks diverges at the quantum critical point,which is directly associated with the divergence of the correlation length.

A MONTE CARLO RENORMALIZATION GROUP STUDY OF ISING MODEL ON 2-DIMENSIONAL RANDOM TRIANGLE LATTICE

Ⅰ. INTRODUCTIONMonte Carlo renormalization group method (MCRG) has become a powerful technical tool for the study of critical behaviour and continuum limit of discrete systems since 1976. The method is therefore widely applied to the study of fixed point and critical exponents in statistical models, and to the study of phase transitions, β function and scaling behaviour for some gauge fields in lattice gauge theories.

Phase transitions of the five-state clock model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative tem- peratures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as To2 = 0.9565（7）. Investigating the fixed-point tensor, we precisely locate the transition temperatures at Tcl = 0.9029（1） and Tc2 = 0.9520（1）, consistent well with the Monte Carlo and the density matrix renormalization group results.

Equilibrium dynamics of the sub-Ohmic spin-boson model under bias

Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C（ω） of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior C（ω） ∝ ω～s is found to be universal and independent of the bias ε and the coupling strength α（except at the quantum critical point α = αc and ε = 0）. Our NRG data also show C（ω） ∝ χ～2ω～s for a wide range of parameters, including the biased strong coupling regime（ε = 0 and α 〉 αc）, supporting the general validity of the Shiba relation. Close to the quantum critical point αc,the dependence of C（ω） on α and ε is understood in terms of the competition between ε and the crossover energy scale ω0^＊of the unbiased case. C（ω） is stable with respect to ε for ε《ε^＊. For ε 》ε^＊, it is suppressed by ε in the low frequency regime. We establish that ε^＊∝（ω0^＊）^1/θ holds for all sub-Ohmic regime 0≤s 〈 1, with θ = 2/（3s） for 0 〈 s≤1/2 and θ = 2/（1 ＋ s） for 1/2 〈 s 〈 1. The variation of C（ω） with α and ε is summarized into a crossover phase diagram on the α–ε plane.

Dynamical correlation functions of the quadratic coupling spin-Boson model

The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO（ω）, with the operator taken as σx, σz, and X, respectively. In the weak-coupling regime α 〈 αc, these functions show power law ω-dependence in the small frequency limit, with the powers 1 ＋ 2s, 1 ＋ 2s, and s, respectively. At the critical point α = αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx= yσz= 1-2s and yX=-s, respectively. Here s is the bath index and X is the boson displacement operator. Close to the spin flip point, the high frequency peak of Cσx（ω） is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.

Renormalization group theory for temperature-driven first-order phase transitions in scalar models

We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of φ3 theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found： a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from φ3 theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two- loop results for the static and dynamic exponents for the Yang-Lee edge singularity, respectively, which falls into the same universality class as φ3 theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.

No-core Monte Carlo shell model calculations with unitary correlation operator method and similarity renormalization group

The unitary correlation operator method （UCOM） and the similarity renormalization group theory （SRG） are compared and discussed in the framework of the no-core Monte Carlo shell model （MCSM） calculations for ^3H and ^4He. The treatment of spurious center-of-mass motion by Lawson＇s prescription is performed in the MCSM calculations. These results with both transformed interactions show good suppression of spurious center-of-mass motion with proper Lawson＇s prescription parameter βc.m. values. The UCOM potentials obtain faster convergence of total energy for the ground state than that of SRG potentials in the MCSM calculations, which differs from the cases in the no-core shell model calculations （NCSM）. These differences are discussed and analyzed in terms of the truncation scheme in the MCSM and NCSM, as well as the properties of the potentials of SRG and UCOM.

Critical Behavior of Gaussian Model on X Fractal Lattices in External Magnetic Fields

Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and -dimensional Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality (or the fractal dimensionality ). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.

Critical Behavior of the Gaussian Model with Periodic Interactions on Diamond—Type Hierarchical Lattices in External Magnetic Fields

The Gaussian spin model with periodic interactions on the diamond-type hierarchical lattices is constructed by generalizing that with uniform interactions on translationally invariant lattices according to a class of substitution sequences. The Gaussian distribution constants and imposed external magnetic fields are also periodic depending on the periodic characteristic of the interaction bonds. The critical behaviors of this generalized Gaussian model in external magnetic fields are studied by the exact renormalization-group approach and spin rescaling method. The critical points and all the critical exponents are obtained. The critical behaviors are found to be determined by the Gaussian distribution constants and the fractal dimensions of the lattices. When all the Gaussian distribution constants are the same, the dependence of the critical exponents on the dimensions of the lattices is the same as that of the Gaussian model with uniform interactions on translationally invariant lattices.

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).

Assessment of Turbulence Models on Bridge-Pier Scour Using Flow-3D

The main purpose of this paper is to analyze the influence of different turbulence flow models on scouring pit of bridge-pier. Flow-3D software is applied in line with the purpose. The key motivation for this study is to contribute to the Flow-3D software by means of some modification and adjustment in the sediment scour model and shallow water model. An assessment of turbulence model adopted with the parameters of the Melville experiment to estimate the maximum scour-depth was performed. In the simulation results, the alternate eddy formation and shedding were repeated while the Karman vortex street formed behind the pier for the large eddy simulation LES turbulence model is more realistic in the flow phenomenon. The results of the scour development of large eddy simulation (LES) turbulence model were found to be more satisfied than the Renormalized group (RNG) turbulence model and close to the prior experiment results. The simulated scour results were significantly different with the observed data collected from previous literature in the reason of some unsuitability of meshing method in Flow-3D software.

Critical Finite Size Scaling Relation of the Order-Parameter Probability Distribution for the Three-Dimensional Ising Model on the Creutz Cellular Automaton

We study the order parameter probability distribution at the critical point for the three-dimensional spin-1/2 and spin-1 Ising models on the simple cubic lattice under periodic boundary conditions. The finite size scaling relation for the order parameter probability distribution is tested and verified numerically by microcanonical Creutz cellular automata simulations. The state critical exponent δ, which characterizes the far tail regime of the scaling order parameter probability distribution, is estimated for three-dimensional Ising models using the cellular automaton simulations at the critical temperature. The results are in good agreement with the Monte Carlo calculations.

胶凝原油临界损伤屈服应力计算模型研究

含蜡原油管道停输再启动压力与原油结构强度密切相关。为了理论上描述胶凝原油屈服应力,以胶凝原油微元强度服从Weibull分布为基础,结合统计力学理论及重正化群模型,建立了胶凝原油临界损伤屈服应力计算模型,并利用胜利及苏丹原油在胶凝点附近不同温度条件下的屈服应力对所建立模型进行可靠性验证。结果表明:在胶凝点附近温度条件下,随着测试温度的降低原油屈服应变呈下降的趋势;在测试范围内随着油温的升高,胶凝原油屈服应力越来越小,油温与屈服应力整体呈现幂律关系;相同温度条件下,随着剪切速率的增加,胶凝原油屈服应力逐渐增大;利用所建立的胶凝原油临界损伤屈服应力计算模型对不同温度条件下胜利原油屈服应力进行拟合,该模型能较好地描述原油屈服应力,计算结果可为含蜡原油管道停输再启动提供参考。