An internal penalty discontinuous Galerkin method for simulating conjugate heat transfer in a closed cavity

Using the discontinuous Galerkin (DG) method for conjugate heat transfer problems can provide improved accuracy close to the fluid‐solid interface, localizing the data exchange process, which may further enhance the convergence and stability of the entire computation. This paper presents a framework for the simulation of conjugate heat transfer problems using DG methods on unstructured grids. Based on an existing DG solver for the incompressible Navier‐Stokes equation, the fluid advection‐diffusion equation, Boussinesq term, and solid heat equation are introduced using an explicit DG formulation. A Dirichlet‐Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux of interface quadrature points in the fluid‐solid interface. Formal h and p convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved. The algorithm is then further validated against 3 existing benchmark cases, including a thermally driven cavity, conjugate thermally driven cavity, and a thermally driven cavity with conducting solid, at Rayleigh numbers from 1000 to 100 000. The computational effort is documented in detail demonstrating clearly that, for all cases, the highest‐order accurate algorithm has several magnitudes lower error than first‐ or second‐order schemes for a given computational effort.     

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

discontinuous galerkin method for discontinuous temperature field problems

针对不连续温度场问题建立了一种间断Galerkin有限元方法,该方法的主要特点是允 许插值函数在单元边界上存在跳变. 在建立有限元方程时,通过在单元边界上引入数值通量 项和稳定性项来处理间断效应,并且数值通量可以直接由接触热阻的定义式导出. 数值算例 表明该方法可以很方便且准确地捕捉到结构内部由于接触热阻而引起的温度跳变,同时在局 部高梯度温度场的模拟方面也比常规连续Galerkin有限元方法效率明显要高. 该方法也为研 究由接触热阻引起的温度场与应力场之间的耦合问题提供了一种新的数值模拟手段.     

基于特征分裂有限元准隐格式的共轭传热整体耦合数值模拟方法

共轭传热现象在科学和工程领域中大量存在.随着计算能力的发展,对共轭传热现象进行准确有效的数值模拟,成为科学研究和工程设计上的重要挑战.共轭传热数值模拟的方法可以分为两大类:分区耦合和整体耦合.本文采用有限元法对共轭传热问题进行整体耦合模拟.固体传热求解采用标准的伽辽金有限元方法.流动求解采用基于特征分裂的有限元方法.该...     

Parallel three‐dimensional simulation of the injection molding process

This paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 John Wiley & Sons, Ltd.     

Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin (SUPG) method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid are performed by the Galerkin method. The second-order semiimplicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are linearized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effiectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.     

A staggered grid‐based explicit jump immersed interface method for two‐dimensional Stokes flows

In this paper the explicit jump immersed interface method (EJIIM) is applied to stationary Stokes flows. The boundary value problem in a general, non‐grid aligned domain is reduced by the EJIIM to a sequence of problems in a rectangular domain, where staggered grid‐based finite differences for velocity and pressure variables are used. Each of these subproblems is solved by the fast Stokes solver, consisting of the pressure equation (known also as conjugate gradient Uzawa) method and a fast Fourier transform‐based Poisson solver. This results in an effective algorithm with second‐order convergence for the velocity and first order for the pressure. In contrast to the earlier versions of the EJIIM, the Dirichlét boundary value problem is solved very efficiently also in the case when the computational domain is not simply connected. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

Study of airflow in a cold-region tunnel using a standard k − ε turbulence model and air-rock heat transfer characteristics: validation of the CFD results

An efficient computational fluid dynamics (CFD) method for simulating the flow and convective heat transfer process of airflow in a tunnel is required to analyze the freezing and thawing of surrounding rock and to apply the results to the design of the insulation layer for a tunnel located in a cold region. Comparisons of experimental data and CFD results using a standard k ? ε turbulence model, a wall function, a thermal function and an adaptive finite element method are presented. Comparison of the results indicated that the proposed model and simulation method are efficient at determining the solid–air interface heat coefficient in a thin and infinitely wide horizontal plate and the hydrodynamic and thermal fields in a 3-D cavity. After demonstrating that the necessary validations are satisfied, this paper presents an analysis of the characteristics of airflow and air–rock heat transfer in a cold-region tunnel.     

Stabilized finite element solution to handle complex heat and fluid flows in industrial furnaces using the immersed volume method

We consider the numerical simulation of conjugate heat transfer, incompressible turbulent flows for multicomponents systems using a stabilized finite element method. We present an immersed volume approach for thermal coupling between fluids and solids for heating high‐alloy steel inside industrial furnaces. It consists in considering a single 3D grid of the furnace and solving one set of equations with different thermal properties. A distance function enables to define precisely the position and the interface of any objects inside the volume and to provide homogeneous physical and thermodynamic properties for each subdomain. An anisotropic mesh adaptation algorithm based on the variations of the distance function is then applied to ensure an accurate capture of the discontinuities that characterize the highly heterogeneous domain. The proposed method demonstrates the capability of the model to simulate an unsteady three‐dimensional heat transfers and turbulent flows in an industrial furnace with the presence of three conducting solid bodies. Copyright © 2010 John Wiley & Sons, Ltd.     

A quasi‐dynamic procedure for coupled thermal simulations

This paper outlines the development and adaptation of a coupling strategy for transient temperature analysis in a solid via a conjugate heat transfer method. This study proposes a quasi‐dynamic coupling procedure to bridge the temporal disparities between the fluid and the solid. In this approach, dynamic thermal modeling in the solid is coupled with a sequence of steady states in the fluid. This quasi‐dynamic algorithm has been applied to the problem of convective heat transfer over, and transient conduction heat transfer within, a flat plate using the severe thermal conditions of a solid propellant rocket. Two different coupled thermal computations have been performed. In the first one—referred to as the reference computation—the coupling period is equal to the smallest solid time constant. In the second one, a very large coupling period is used. The results show that the procedure can predict accurate transient temperature fields at a reasonable computational cost. The simulation CPU time is approximately reduced by up to 90%, while maintaining a very good accuracy. All the details of the numerical test case are given in the paper. This application illustrates the capabilities and the overall efficiency of this coupled approach in a solid transient problem using long term simulations of time dependent flows. Copyright © 2013 John Wiley & Sons, Ltd.     

MPS-FEM数值分析带自由面的流固耦合问题

随着计算科学的发展,研究人员为探索流固耦合问题的物理机理而提出了众多的数值方法。其中,耦合的移动粒子半隐式方法 MPS(Moving Particle Semi-Implicit method)和有限单元法FEM(Finite Element method)为流固耦合问题的数值仿真工作提供了新的途径。本文所有流场的数值模拟工作均采用课题组自主开发的无网格法求解器MLParticle-SJTU来完成。该求解器在原始的MPS法基础上,对核函数、压力梯度模型、压力泊松方程的求解和自由面判断方式等方面进行了改进。此外,在该求解器框架内,基于FEM法拓展了针对结构场进行求解的功能。首先,对MPS和FEM方法的理论模型及其耦合策略进行了介绍。然后,采用该自研MPS-FEM耦合求解器,数值模拟了溃坝流动对弹性结构的冲击及其相互作用的标准问题。通过将结构变形及自由面波型变化等结果与已发表结果进行对比,验证了该求解器在处理带自由面剧烈变化的粘性流体和柔性变形结构的耦合作用问题上的可行性。     

近似不可压软材料动力分析的完全拉格朗日物质点法

物质点法(MPM)在模拟非线性动力问题时具有很好的效果,其已被广泛应用于许多大变形动力问题的分析中.然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁,极大地影响模拟精度和收敛性.本文针对近似不可压软材料的大变形动力学行为,提出一种混合格式的显式完全拉格朗日物质点法(TLMPM).首先基于近似不可压软材料的体积部分应变能密度,引入关于静水压力的方程;之后将该方程与动量方程基于显式物质点法框架进行离散,并采用完全拉格朗日格式消除物质点跨网格产生的误差,提升大变形问题的模拟精度;对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法,提升算法的准确性.此外,算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解.最后,给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性,计算结果表明该方法可以有效处理体积自锁,准确地模拟近似不可压软材料的大变形动力学行为.     

Monte Carlo method of radiative transfer applied to a turbulent flame modeling with LES

Radiative transfer plays an important role in the numerical simulation of turbulent combustion. However, for the reason that combustion and radiation are characterized by different time scales and different spatial and chemical treatments, the radiation effect is often neglected or roughly modelled. The coupling of a large eddy simulation combustion solver and a radiation solver through a dedicated language, CORBA, is investigated. Two formulations of Monte Carlo method (Forward Method and Emission Reciprocity Method) employed to resolve RTE have been compared in a one-dimensional flame test case using three-dimensional calculation grids with absorbing and emitting media in order to validate the Monte Carlo radiative solver and to choose the most efficient model for coupling. Then the results obtained using two different RTE solvers (Reciprocity Monte Carlo method and Discrete Ordinate Method) applied on a three-dimensional flame holder set-up with a correlated-k distribution model describing the real gas medium spectral radiative properties are compared not only in terms of the physical behavior of the flame, but also in computational performance (storage requirement, CPU time and parallelization efficiency). To cite this article: J. Zhang et al., C. R. Mecanique 337 (2009).     

On the role of (weak) compressibility for fluid-structure interaction solvers

In the present study, a weakly compressible formulation of the Navier-Stokes equations is developed and examined for the solution of fluid-structure interaction (FSI) problems. Newtonian viscous fluids under isothermal conditions are considered, and the Murnaghan-Tait equation of state is employed for the evaluation of mass density changes with pressure. A pressure-based approach is adopted to handle the low Mach number regime, ie, the pressure is chosen as primary variable, and the divergence-free condition of the velocity field for incompressible flows is replaced by the continuity equation for compressible flows. The approach is then embedded into a partitioned FSI solver based on a Dirichlet-Neumann coupling scheme. It is analytically demonstrated how this formulation alleviates the constraints of the instability condition of the artificial added mass effect, due to the reduction of the maximal eigenvalue of the so-called added mass operator. The numerical performance is examined on a selection of benchmark problems. In comparison to a fully incompressible solver, a significant reduction of the coupling iterations and the computational time and a notable increase in the relaxation parameter evaluated according to Aitken's Δ² method are observed.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

We present an efficient algorithm for simulation of deformable bodies interacting with two-dimensional incompressible fluid flows. The temporal and spatial discretizations of the Navier–Stokes equations in vorticity stream-function formulation are based on classical fourth-order Runge–Kutta scheme and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero over an optimal grid. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body which is interacting with the surrounding fluid due to the hydrodynamic forces and the torque calculated on the Eulerian reference grid. An efficient law for controlling the curvature of an anguilliform fish, swimming toward a prescribed goal, is proposed which is based on the geometrically exact theory of nonlinear beams and quaternions. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems.     

管内充分发展流动与传热数值模拟的教学方法探讨

由于缺乏数值计算的基础理论知识,工科本科生在学习流体流动与传热过程的数值模拟方法并进行程序设计时,往往觉得难度较大。本文围绕圆管内不可压缩流体充分发展流动与传热,采用边界层积分法,推导出无量纲控制方程组;针对层流工况和湍流工况,开发出相应的数值方法。基于该方法的程序代码易于理解,且计算结果表明该方法具有预测精度高、收敛速度快的优点。本文的工作可以为计算流体力学、数值传热学及热工计算等系列课程的本科教学提供参考。

     

A coupled solution of the vorticity–velocity formulation of the incompressible Navier–Stokes equations

A relatively novel formulation of the Navier-Stokes equations is used for obtaining solutions of two dimensional incompressible fluid flow and convective heat transfer problems. A vorticity transport equation along with two Poisson equations for the velocity components and the energy equation are solved by a finite difference scheme. A coupled solution procedure is used for solving simultaneously the dependent variables along a line, using a block tridiagonal matrix algorithm. The formulation is found to be stable and has features that may be desirable for solving a wide variety of flow and heat transfer problems.