Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT–LBM schemes

Lattice Boltzmann methods (LBMs) are very efficient for computational fluid dynamics, and for capturing the dynamics of weak acoustic fluctuations. It is known that multi-relaxation-time lattice Boltzmann method (MRT–LBM) appears as a very robust scheme with high precision. There exist several free relaxation parameters in the MRT–LBM. Although these parameters have been tuned via linear analysis, the sensitivity analysis of these parameters and other related parameters is still not sufficient for describing the behavior of the dispersion and dissipation relations of the MRT–LBM. Previous researches have shown that the bulk dissipation in the MRT–LBM induces a significant over-damping of acoustic disturbances. This indicates that the classical MRT–LBM is not best suited to recover the correct behavior of pressure fluctuations. In wave-number space, the first/second-order sensitivity analyses of matrix eigenvalues are used to address the sensitivity of the wavenumber magnitudes to the dispersion-dissipation relations. By the first-order sensitivity analysis, the numerical behaviors of the group velocity of the MRT–LBM are first obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues corresponding to the linearized form of the MRT–LBM are investigated in the complex plane. Based on the sensitivity analysis and an effective algorithm of recovering linearized Navier–Stokes equations (L-NSEs) from linearized MRT–LBM (L-MRT–LBM), we propose some simplified optimization strategies to determine the free relaxation parameters of the MRT–LBM. Meanwhile, the dispersion and dissipation relations of the optimal MRT–LBM are quantitatively compared with the exact dispersion and dissipation relations. At last, some numerical validations on classical acoustic benchmark problems are shown to assess the new optimal MRT–LBM.     

过渡区微尺度流动的有效黏性多松弛系数格子Boltzmann模拟

为提高采用二维九速离散速度模型的格子Boltzmann方法 (LBM)模拟微尺度流动中非线性现象的精度和效率,引入Dongari等提出的有效平均分子自由程对黏性进行修正(Dongari N,Zhang Y H,Reese J M2011 J.Fluids Eng.133 071101);并针对以往研究微尺度流动时采用边界处理格式含有离散误差的问题,采用多松弛系数格子Boltzmann方法结合二阶滑移边界条件,对微尺度Couette流动和周期性Poiseuille流动进行模拟,并将速度分布以及质量流量等模拟结果与直接模拟蒙特卡罗方法模拟数据、线性Boltzmann方程的数值解以及现有的LBM模型模拟结果进行对比.结果表明,相对于现有的LBM模型,引入新的修正函数所建立的有效黏性多松弛系数LBM模型有效提高了LBM模拟过渡区的微尺度流动中的非线性现象的能力.     

Improving the Stability of the Multiple-Relaxation-Time Lattice Boltzmann Method by a Viscosity Counteracting Approach

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method (LBM). The multiple-relaxation-time (MRT) model of the LBM can improve the accuracy and stability, but is still subject to numerical instability when simulating flows with large single-grid Reynolds number (Reynolds number/grid number). The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM. However, its effectiveness was only verified in the single-relaxation-time model of the LBM (SRT-LBM). This paper aims to propose the viscosity counteracting approach for the multiple-relaxation-time model (MRT-LBM) and analyze its numerical characteristics. The verification is conducted by simulating some benchmark cases: the two-dimensional (2D) lid-driven cavity flow, Poiseuille flow, Taylor-Green vortex flow and Couette flow, and three-dimensional (3D) rectangular jet. Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBM has better accuracy and stability than that for the SRT-LBM.     

微尺度振荡Couette流的格子Boltzmann模拟

采用有效多松弛时间-格子Boltzmann方法(Effective MRT-LBM)数值模拟了微尺度条件下的振荡Couette和Poiseuille流动. 在微流动LBM中引入Knudsen边界层模型,对松弛时间进行修正. 模拟时平板或外力以正弦周期振动,Couette流中考虑了单平板振动、上下板同相振动这两类情况. 研究结果表明,修正后的MRT-LBM模型能有效用于这类非平衡的微尺度流动模拟;对于Couette流,随着Kn数的增大,壁面滑移效应变得越明显. St越大,板间速度剖面的非线性特性越剧烈;两板同相振荡时,若Kn,St均较小,板间流体受到平板拖动剪切的影响很小,板间速度几乎重叠在一起;在振荡Poiseuille流动中,St数增大到一定值时,相位滞后现象减弱;相对于Kn数,St数对振荡Couette 和Poiseuille流中不同位置处速度相位差的产生有较大影响. 关键词： 格子Boltzmann方法 有效MRT模型 Knudsen层 振荡流     

Power-independent technique to extract the dispersion parameters of an optical fiber using four wave mixing

Conventional methods which extract dispersion parameters of an optical fiber using four wave mixing rely on the measurement of optical power in the generated wavelengths, and hence are prone to measurement errors. We propose and demonstrate a power-independent method to extract the dispersion curve of a fiber using four wave mixing (FWM). The analytical equations that have been traditionally used to estimate the phase mismatch in FWM are modified to cater to low-dispersion fibers, and are verified. Experiments to obtain the dispersion curve of a low-dispersion and highly nonlinear fiber are discussed and the results are analyzed. Limitations of any FWM-based method to extract the dispersion curve of a fiber are presented for the first time.     

Optimized prefactored compact schemes

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and dissipation errors. In this paper we describe a strategy for developing high-order accurate prefactored compact schemes, requiring very small stencil support. These schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes. The prefactorization strategy splits the central implicit schemes into forward and backward biased operators. Using Fourier analysis, we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components. This ensures that when the forward and backward stencils are added, the original central compact scheme is recovered. To extend the resolution characteristic of the schemes, an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh. The resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth- and eighth-order compact schemes making them more suitable for high-resolution numerical simulations in gas dynamics and computational aeroacoustics. The efficiency, accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems.     

Self-consistent tomography of temporally correlated errors

The error model of a quantum computer is essential for optimizing quantum algorithms to minimize the impact of errors using quantum error correction or error mitigation. Noise with temporal correlations, e.g. low-frequency noise and context-dependent noise, is common in quantum computation devices and sometimes even significant. However, conventional tomography methods have not been developed for obtaining an error model describing temporal correlations. In this paper,we propose self-consistent tomography protocols to obtain a model of temporally correlated errors, and we demonstrate that our protocols are efficient for low-frequency noise and context-dependent noise.     

A high-resolution,hybrid compact-WENO scheme with minimized dispersion and controllable dissipation

In this paper,a high-resolution,hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique.Firstly,a suffcient condition for a family of tri-diagonal compact schemes to have independent dispersion and dissipation is derived.Then,a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed.Finally,the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme.Moreover,the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme.Several test cases are carried out to verify the highresolution as well as the robust shock-capturing capabilities of the proposed scheme.     

Microscopic modes in a Fermi superfluid. II. Dispersion relations

This is the second of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this paper we obtain approximate solutions to the linearized kinetic equations for the bogolon spin density and total density for the case of long-wavelength disturbances after long times when a fluctuating superfluid velocity is present. In so doing, we obtain microscopic expressions for the amplitude and dispersion relations for the spin diffusion mode, the two shear modes, and the four longitudinal modes (two first-sound modes and two second-sound modes).     

Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier–Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier–Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier–Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier–Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge–Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.     

An optimized spectral difference scheme for CAA problems

In the implementation of spectral difference (SD) method, the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes. The errors incurred in using extrapolation and interpolation would result in instability. On the other hand, the difference between the left and right conserved variables at the edge interface will introduce dissipation to the SD method when applying a Riemann solver to compute the flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers. The optimized coefficients of the extrapolation and interpolation are presented. And the dispersion error of the original and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The stability of the optimized fourth order SD scheme is analyzed. It is found that the stability of the 4th order scheme with Chebyshev–Gauss–Lobatto flux points, which is originally weakly unstable, has been improved through the optimization. The weak instability is eliminated completely if an additional second order filter is applied on selected flux points. One and two dimensional linear wave propagation analyses are carried out for the optimized scheme. It is found that in the resolvable wavenumber range the new SD scheme is less dispersive and less dissipative than the original scheme, and the new scheme is less anisotropic for 2D wave propagation. The optimized SD solver is validated with four computational aeroacoustics (CAA) workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with the original schemes.     

Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method

In this paper, we discuss the incorporation of dynamic subgrid scale (SGS) models in the lattice-Boltzmann method (LBM) for large-eddy simulation (LES) of turbulent flows. The use of a dynamic procedure, which involves sampling or test-filtering of super-grid turbulence dynamics and subsequent use of scale-invariance for two levels, circumvents the need for empiricism in determining the magnitude of the model coefficient of the SGS models. We employ the multiple relaxation times (MRT) formulation of LBM with a forcing term, which has improved physical fidelity and numerical stability achieved by proper separation of relaxation time scales of hydrodynamic and non-hydrodynamic modes, for simulation of the grid-filtered dynamics of large-eddies. The dynamic procedure is illustrated for use with the common Smagorinsky eddy-viscosity SGS model, and incorporated in the LBM kinetic approach through effective relaxation time scales. The strain rate tensor in the SGS model is locally computed by means of non-equilibrium moments of the MRT-LBM. We also discuss proper sampling techniques or test-filters that facilitate implementation of dynamic models in the LBM. For accommodating variable resolutions, we employ conservative, locally refined grids in this framework. As examples, we consider the canonical anisotropic and inhomogeneous turbulent flow problem, i.e. fully-developed turbulent channel flow at two different shear Reynolds numbers Re_∗ of 180 and 395. The approach is able to automatically and self-consistently compute the values of the Smagorinsky coefficient, C_S. In particular, the computed value in the outer or bulk flow region, where turbulence is generally more isotropic, is about 0.155 (or the model coefficient ) which is in good agreement with prior data. It is also shown that the model coefficient becomes smaller and approaches towards zero near walls, reflecting the dampening of turbulent length scales near walls. The computed turbulence statistics at these Reynolds numbers are also in good agreement with prior data. The paper also discusses a procedure for incorporation of more general scale-similarity based SGS stress models.     

Lattice Boltzmann Study of a Vortex Ring Impacting Spheroidal Particles

Interaction of vortex rings with solid is an important research topic of hydrodynamic. In this study, a multiple-relaxation time (MRT) lattice Boltzmann method (LBM) is used to investigate the flow of a vortex ring impacting spheroidal particles. The MRT-LBM is validated through the cases of vortex ring impacting a flat wall. The vortex evolution due to particle size, the aspect ratio of a prolate particle, as well as Reynolds $(Re)$ number are discussed in detail. When the vortex ring impacting a stationary sphere, the primary and secondary vortex rings wrap around each other, which is different from the situation of the vortex ring impacting a plate. For the vortex ring impacting with a prolate spheroid, the secondary vortex ring stretches mainly along the long axis of the ellipsoid particle. However, it is found that after the vortex wrapping stage, the primary vortex recovers along the short axis of the particle faster than that in the long axis, i.e., the primary vortex ring stretches mainly along the short axis of the particle. That has never been addressed in the literature.     

耦合不可压流场输运方程的格子Boltzmann方法研究

基于格子Boltzmann方法,提出了求解耦合不可压缩流场输运方程的一种改进数值方法. 该方法使用格子Boltzmann方法求解流场方程,并根据流场格子模型的密度分布函数构建了输运方程的二阶离散格式. 通过二维平板通道流场输运系统验证了该方法的有效性.数值结果表明,该方法可以有效地减少计算过程中出现的非物理耗散, 并克服了传统模型所需巨大存储量的缺点.     

Link-wise artificial compressibility method

The artificial compressibility method (ACM) for the incompressible Navier–Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the lattice Boltzmann method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be among the few which are compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.     

Phonon or electron friction of a quantum heavy particle

Summary In this paper we analyse, with the path integral method, the diffusion of a quantum heavy particle moving in a strongly corrugated periodic potential both in the case when the particle is interacting with a thermal bath of phonons or of electrons. In the first case, the integration over the phonon degrees of freedom is performed exactly and in the large mass limit of the heavy particle it gives rise to an ohmic effective action which includes a nonlocal self-interacting term whose strength is the classical friction coefficient. In the second case, the integration over the electronic degrees of freedom is more difficult; we are able to derive an approximate effective action for the heavy particle in two different limiting cases: i) arbitrary large coupling between heavy particle and electrons and linear dissipation; ii) weak coupling and nonlinear dissipation. In i) we obtain an effective action for the particle equal to that found for the phonons but with a friction coefficient given by that of a classical heavy particle in a fermionic bath. In ii) we obtain a nonlinear, but still ohmic, dissipative term. Using an instanton approach we evaluate the mobility (and the diffusion coefficient) of the particle, whose temperature dependence shows a crossover from diffusive to localized behaviour at a critical value of the friction. Finally we discuss whether the electronic and phononic frictions can reach such a critical value. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Stability is one of the main concerns in the lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of the lattice Boltzmann equation with the Bhatnagar–Gross–Krook collision operator (LBGK) for the advection–diffusion equation (ADE), and to understand the relationship between the stability of the LBGK and non-negativity of the equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify the non-negative domains of the dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistency between the theoretical findings and LBM solutions.     

Fokker-Planck equation,distribution and correlation functions for laser noise

As an application of a preceding paper we set up a Fokker-Planck equation with quantum mechanically defined dissipation and fluctuation coefficients for a distribution function of the atomic variables (dipole moments and level occupation numbers) as well as of the lasing light amplitude in a laser with a homogeneously broadened line. Since the nonlinear coefficients can be linearized in appropriate coordinates well below and well above threshold, the equation can be solved with the Wang-Uhlenbeck method. Then it is easy to obtain correlation functions, spectral densities and expressions for linewidth.     

Two-pump fiber optical parametric amplifiers using optimized photonic crystal fiber by genetic algorithm

In this paper, we investigate the gain spectra of fiber optical parametric amplifiers (FOPAs) consisting of dispersion-flattened fibers (DFFs) with different dispersion curves comparatively by means of the dispersion curve model of the DFFs. It is demonstrated that the broader gain spectrum of FOPAs can be achieved if the dispersion curve is lower and more flattened. Based on the preceding analyses, we propose to constitute FOPAs by using index-guiding photonic crystal fiber (PCF) with an anticipated ultra-flattened dispersion and ultra-low dispersion slope. The as-proposed PCF had different air-holes in the cladding rings. In total, four parameters of the PCF are optimized by using genetic algorithm. PACS 42.81.-i; 42.65.Hw; 42.65.Yj