Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method

By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.

Transition to chaos in lid-driven square cavity flow

To date,there are very few studies on the transition beyond second Hopf bifurcation in a lid-driven square cavity,due to the difficulties in theoretical analysis and numerical simulations.In this paper,we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us.We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of(13944.7021,13946.5333)by the method of bisection.Through Fourier analysis,it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval.Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram,Kolmogorov entropy and maximal Lyapunov exponent.The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.

SUPG finite element method based on penalty function for lid-driven cavity flow up to Re = 27500

A streamline upwind/Petrov-Galerkin （SUPG） finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for- mulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pres- sure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.

The second Hopf bifurcation in lid-driven square cavity

To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of(11093.75, 11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs.

Mixed Convection Heat Transfer for Nanofluids in a Lid-Driven Shallow Rectangular Cavity Uniformly Heated and Cooled from the Vertical Sides:The Opposing Case

An investigation on flow and heat transfer due to mixed convection, in a lid-driven rectangular cavity filled with Cu- water nanofluids and submitted to uniform heat flux along with its vertical short sides, has been conducted numerically by solving the full governing equations with the finite volume method and the SIMPLER algorithm. In the case of a slender enclosure, these equations are considerably reduced by using the parallel flow concept. Solutions, for the flow and temperature fields, and the heat transfer rate, have been obtained depending on the governing parameters, which are the Reynolds, the Richardson numbers and the solid volume fraction of nanoparticles. A perfect agreement has been found between the results of the two approaches for a wide range of the abovementioned parameters. It has been shown that at low and high Richardson numbers, the convection is ensured by lid and buoyancy-driven effects, respectively, whereas between these extremes, both mechanisms compete. Moreover, the addition of Cu-nanoparticles, into the pure water, has been seen enhancing and degrading heat transfer by lid and buoyancy-driven effects, respectively.

Coupling of discrete-element method and smoothed particle hydrodynamics for liquid-solid flows

Particle based methods can be used for both the simulations of solid and fluid phases in multiphase medium, such as the discrete-element method for solid phase and the smoothed particle hydrodynamics for fluid phase. This paper presents a computational method combining these two methods for solid-liquid medium. The two phases are coupled by using an improved model from a reported Lagrangian-Eulerian method. The technique is verified by simulating liquid-solid flows in a two-dimensional lid-driven cavity.

Numerical research on lid-driven cavity flows using a three-dimensional lattice Boltzmann model on non-uniform meshes

A lattice Boltzmann model combined with curvilinear coordinate is proposed for lid-driven cavity three-dimensional (3D) flows. For particle velocity distribution, the particle collision process is performed in physical domain, and the particle streaming process is carried out in the corresponding computational domain, which is transferred from the physical domain using interpolation method. For the interpolation calculation, a second-order upwind interpolation method is adopted on internal lattice nodes in flow fields while a second-order central interpolation algorithm is employed at neighbor-boundary lattice nodes. Then the above-mentioned model and algorithms are used to numerically simulate the 3D flows in the lid-driven cavity at Reynolds numbers of 100, 400 and 1000 on non-uniform meshes. Various vortices on the x-y, y-z and x-z symmetrical planes are successfully predicted, and their changes in position with the Reynolds number increasing are obtained. The velocity profiles of u component along the vertical centerline and w component along the horizontal centerline are both in good agreement with the data in literature and the calculated results on uniform meshes. Besides, the velocity vector distributions on various cross sections in lid-driven cavity predicted on non-uniform meshes are compared with those simulated on uniform meshes and those in the literature. All the comparisons and validations show that the 3D lattice Boltzmann model and all the numerical algorithms on non-uniform meshes are accurate and reliable to predict effectively flow fields.

Chaotic Lid-Driven Square Cavity Flows at Extreme Reynolds Numbers

This paper investigates the chaotic lid-driven square cavity flows at extreme Reynolds numbers.Several observations have been made from this study.Firstly,at extreme Reynolds numbers two principles add at the genesis of tiny,loose counterclockwise-or clockwise-rotating eddies.One concerns the arousing of them owing to the influence of the clockwise-or counterclockwise currents nearby;the other,the arousing of counterclockwise-rotating eddies near attached to the moving(lid)top wall which moves from left to right.Secondly,unexpectedly,the kinetic energy soon reaches the qualitative temporal limit’s pace,fluctuating briskly,randomly inside the total kinetic energy range,fluctuations which concentrate on two distinct fragments:one on its upper side,the upper fragment,the other on its lower side,the lower fragment,switching briskly,randomly from each other;and further on many small fragments arousing randomly within both,switching briskly,randomly from one another.As the Reynolds number Re→∞,both distance and then close,and the kinetic energy fluctuates shorter and shorter at the upper fragment and longer and longer at the lower fragment,displaying tall high spikes which enlarge and then disappear.As the time t→∞(at the Reynolds number Re fixed)they recur from time to time with roughly the same amplitude.For the most part,at the upper fragment the leading eddy rotates clockwise,and at the lower fragment,in stark contrast,it rotates counterclockwise.At Re=109 the leading eddy-at its qualitative temporal limit’s pace—appears to rotate solely counterclockwise.

掺气减蚀设施后二维空腔流动计算

本文发展了文献 [4 ]中提出的求解自由面重力流的h2 类边值问题理论 .运用解析函数的Riemann Hilbert混合边值问题的h3 类边值问题理论 ,导出了计算有重力作用情况下双自由表面二维空腔势流流动的边界程分方程 ,提出了确定空腔区水舌下自由表面的计算格式 ,并在物理平面上进行了数值求解 .表明 ,该数值方法具有收敛速度快、边界适应能力强、计算量小、计算精度高和对初值要求不高等特点 .计算结果与实测资料对比分析表明 ,来流佛氏数Fr <7情况下 ,无需考虑阻力影响 ,采用势流模型即可获得与实测值相符合的结果 ;而在来流佛氏数Fr >7情况下 ,由于射流掺气加剧 。

非稳定不可压缩流动模拟的改进有限元数值方法(英文)

泰勒-伽辽金有限元法在对流扩散问题的数值模拟中存在数值耗散和伪振荡等问题.本文提出改进的二阶和三阶欧拉-泰勒-伽辽金有限元法,求解了粘性不可压缩流动的Navier-Stokes方程.为克服由不可压缩条件引起的压力场振荡问题,引入压力修正法和泰勒-胡德单元.对方腔拖曳流动进行了数值模拟,以验证改进后算法的性能.最后,分析了改进后算法的精度和计算效率.

A semi-implicit three-step method based on SUPG finite element formulation for flow in lid driven cavities with different geometries

A numerical algorithm using a bilinear or linear finite element and semi-implicit three-step method is presented for the analysis of incompressible viscous fluid problems. The streamline upwind/Petrov-Galerkin (SUPG) stabilization scheme is used for the formulation of the Navier-Stokes equations. For the spatial discretization, the convection term is treated explicitly, while the viscous term is treated implicitly, and for the temporal discretization, a three-step method is employed. The present method is applied to simulate the lid driven cavity problems with different geometries at low and high Reynolds numbers. The results compared with other numerical experiments are found to be feasible and satisfactory.

自由流线问题有限元分析

在文 ［７］ ［８］ ［９］ 的基础上，研究自由流线问题有限元解的通用计算方法。将各结点的坐标，代替流函数值，定义为有限元方程的广义坐标，使自由流线理论中对应于正、逆问题以及空腔流动问题的有限元方程有相同形式。用 Ｇａｌｅｒｋｉｎ 方法构成关于流速的变分方程，解决了弱解求导的困难。给出二种类型的空腔流动及机翼逆问题三个典型算例，表明方法有较好的适应能力。

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.

A Simplified Lattice Boltzmann Method for Turbulent Flow Simulation

To simulate the incompressible turbulent flows,two models,known as the simplified and highly stable lattice Boltzmann method(SHSLBM)and large eddy simulation(LES)model,are employed in this paper.The SHSLBM was developed for simulating incompressible viscous flows and showed great performance in numerical stability at high Reynolds numbers,which means that this model is capable of dealing with turbulent flows by adding the turbulence model.Therefore,the LES model is combined with SHSLBM.Inspired by the less amount of grids required for SHSLBM,a local grid refinement method is used at relatively high Reynolds numbers to improve computational efficiency.Several benchmark cases are simulated and the obtained numerical results are compared with the available results in literature,which show excellent agreement together with greater computational performance than other algorithms.

Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks(PINNs)

This paper explores the difficulties in solving partial differential equations(PDEs)using physics-informed neural networks(PINNs).PINNs use physics as a regularization term in the objective function.However,a drawback of this approach is the requirement for manual hyperparameter tuning,making it impractical in the absence of validation data or prior knowledge of the solution.Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate.Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence.To address these challenges,we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients.Consequently,we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible.Our method also provides a mechanism to focus on complex regions of the domain.Besides,we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model’s prediction,with adaptive and independent learning rates inspired by adaptive subgradient methods.We apply our approach to solve various linear and non-linear PDEs.

A Third-Order Upwind Compact Scheme on Curvilinear Meshes for the Incompressible Navier-Stokes Equations

This paper presents a new version of the upwind compact finite difference scheme for solving the incompressible Navier-Stokes equations in generalized curvilinear coordinates.The artificial compressibility approach is used,which transforms the elliptic-parabolic equations into the hyperbolic-parabolic ones so that flux difference splitting can be applied.The convective terms are approximated by a third-order upwind compact scheme implemented with flux difference splitting,and the viscous terms are approximated by a fourth-order central compact scheme.The solution algorithm used is the Beam-Warming approximate factorization scheme.Numerical solutions to benchmark problems of the steady plane Couette-Poiseuille flow,the liddriven cavity flow,and the constricting channel flow with varying geometry are presented.The computed results are found in good agreement with established analytical and numerical results.The third-order accuracy of the scheme is verified on uniform rectangular meshes.

Improving the Stability of theMultiple-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-relaxationtime 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 threedimensional(3D)rectangular jet.Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBMhas better accuracy and stability than that for the SRT-LBM.