Lower-upper Symmetric Gauss-Seidel Research Articles

A matrix-free, implicit finite volume lattice Boltzmann method for steady flows

Abstract In the present paper, a matrix-free, implicit finite volume lattice Boltzmann method for steady flow on unstructured mesh is proposed. The approximate linear system arising from the implicit finite volume discretization of lattice Boltzmann equation (LBE) is solved by a novel algorithm, which combines the lower-upper symmetric Gauss–Seidel (LU-SGS) and the Jacobi iteration schemes. A remarkable feature of the present implicit method is that the storage of the Jacobian matrixes of the convection and collision terms can be completely eliminated by approximating the Jacobian matrix-solution incremental vector product with appropriate numerical flux incremental vector and the numerical increment of collision term, resulting a matrix-free implicit scheme. The present method is validated by several two-dimensional testing cases.

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

This paper is concerned with the application of k-exact finite volume methods for compressible Reynolds-Averaged Navier-Stokes computations of flows around aeronautical configurations including the NACA0012, RAE2822, MDA30P30N, ONERA-M6, CRM and DLR-F11. High-order spatial discretisation is obtained with the Weighted Essentially Non-Oscillatory and the Monotone-Upstream Central Scheme for Conservation Laws methods on hybrid unstructured grids in two- and three- dimensions. Schemes of fifth, third and second order comprise the foundation of the analysis, with main findings suggesting that enhanced accuracy can be obtained with at least a third-order scheme. Steady state solutions are achieved with the implicit approximately factored Lower-Upper Symmetric Gauss-Seidel time advancing technique, convergence properties of each scheme are discussed. The Spalart-Allmaras turbulence model is employed where its discretisation with respect to the high-order framework is assessed. A low-Mach number treatment technique is studied, where recovery of accuracy in low speed regions is exemplified. Results are compared with referenced data and discussed in terms of accuracy, grid dependence and computational budget.

Validation and analysis of gas-kinetic unified algorithm for solving Boltzmann model equation with vibrational energy excitation

With the increase of temperature in flow field,gas molecules possess not only rotational degree of freedom,but also vibrational energy excitation.In order to simulate and study the influence of internal energy excitation on polyatomic gas flow with high temperature and high Mach number,according to the general Boltzmann equation,we consider the rotational and vibrational energy modes as the independent variables of gas molecular velocity distribution function.It is assumed that the rotational and vibrational energy modes are described by continuous distribution with degree of freedom and temperature.Based on the Borgnakke-Larsen collision model used in direct simulation Monte Carlo (DSMC) method, the collision term of Boltzmann equation with internal energy excitation is divided into elastic and inelastic collision terms.The inelastic collision is decomposed into translational-rotational energy relaxation and translational-rotationalvibrational energy relaxation according to a certain relaxation rate obtained from the reciprocalities of rotational and vibrational collisions numbers per one elastic collision.Then a kind of Boltzmann model equation considering the excitation of vibrational energy is constructed.For showing the consistency between the present model equation and Boltzmann equation,the conservation of summational invariants and the H-theorem of this model are proved.When solving the present model equation with numerical methods,because of the continuous energy modes,it is difficult to simulate this model equation directly.In this paper,three control equations are derived and solved by the LU-SGS (lower-upper symmetric Gauss-Seidel) method,and the cell-centered finite volume method with multi-block patched grid technique in physical space.As a result,these gas-kinetic unified algorithm (GKUA) with vibrational energy excitation has been developed.Results are presented for N2 with different Knudsen numbers around cylinder from continuum to rarefied gas flow by using the present Boltzmann model equation,GKUA with simple gas model,and DSMC method. Very good agreement between the present model and DSMC results is obtained,which shows that the accuracy and reliability of the present model.Comparing the translational,rotational,vibrational,and total temperatures computed by different methods,the effects of the rotational and vibrational degrees of freedom are demonstrated.For the simple gas model,the translational temperature is much higher than those for the other two models with internal energy excitation. At the same time,the distance from shock wave to wall for the simple gas model is about twice those for the other two models.On the other hand,the obtained aerodynamic force coefficients of the cylinder are increasing according to the sequence from the simple gas model to the rotational energy excitation model to the vibrational energy excitation model, but the variation range is very small.By reducing the gas characteristic vibrational temperature,the temperature after the shock wave is much lower,and the heat flux declines evidently at the stagnation point with the same temperature as the wall temperature.This implies that with the wall temperature increasing the heat flux declines.

On the implicit density based OpenFOAM solver for turbulent compressible flows

The contribution deals with the development of coupled implicit density based solver for compressible flows in the framework of open source package OpenFOAM. However the standard distribution of OpenFOAM contains several ready-made segregated solvers for compressible flows, the performance of those solvers is rather week in the case of transonic flows. Therefore we extend the work of Shen [15] and we develop an implicit semi-coupled solver. The main flow field variables are updated using lower-upper symmetric Gauss-Seidel method (LU-SGS) whereas the turbulence model variables are updated using implicit Euler method.

Performance modeling and optimization of parallel LU-SGS on many-core processors for 3D high-order CFD simulations

As a typical Gauss---Seidel method, the inherent strong data dependency of lower-upper symmetric Gauss---Seidel (LU-SGS) poses tough challenges for shared-memory parallelization. On early multi-core processors, the pipelined parallel LU-SGS approach achieves promising scalability. However, on emerging many-core processors such as Xeon Phi, experience from our in-house high-order CFD program show that the parallel efficiency drops dramatically to less than 25%. In this paper, we model and analyze the performance of the pipelined parallel LU-SGS algorithm, present a two-level pipeline (TL-Pipeline) approach using nested OpenMP to further exploit fine-grained parallelisms and mitigate the parallel performance bottlenecks. Our TL-Pipeline approach achieves 20% performance gains for a regular problem $$(256\times 256\times 256)$$(256×256×256) on Xeon Phi. We also discuss some practical problems including domain decomposition and algorithm parameters tuning for realistic CFD simulations. Generally, our work is applicable to the shared-memory parallelization of all Gauss---Seidel like methods with intrinsic strong data dependency.

A Jacobian-free Edge-based Galerkin Formulation for Compressible Flows

A parallel formulation of a Jacobian-free all Mach numbers solver on unstructured hybrid meshes is proposed. The Finite Element formulation is edge-based with flow stabilization obtained with either AUSM+ -up or Roe scheme. The linear system is solved via a Jacobian-Free Newton-Krylov (JFNK) method with Lower-Upper Symmetric GaussSeidel (LU-SGS) used as matrix-free preconditioner. The traditional formulation of LU-SGS is enriched by including the contributions from viscous fluxes and boundary conditions. The accuracy and efficiency of the proposed approach are demonstrated over cases ranging from low to high Mach numbers: subsonic flow over the Trap Wing, transonic flow over the ONERA M6 wing, supersonic flow over a sphere, supersonic flow over a waverider and finally hypersonic flow over a sphere.

Stability Analysis of a Fully Coupled Implicit Scheme for Inviscid Chemical Non-Equilibrium Flows

AbstractVon Neumann stability theory is applied to analyze the stability of a fully coupled implicit (FCI) scheme based on the lower-upper symmetric Gauss-Seidel (LU-SGS) method for inviscid chemical non-equilibrium flows. The FCI scheme shows excellent stability except the case of the flows involving strong recombination reactions, and can weaken or even eliminate the instability resulting from the stiffness problem, which occurs in the subsonic high-temperature region of the hypersonic flow field. In addition, when the full Jacobian of chemical source term is diagonalized, the stability of the FCI scheme relies heavily on the flow conditions. Especially in the case of high temperature and subsonic state, theCFLnumber satisfying the stability is very small. Moreover, we also consider the effect of the space step, and demonstrate that the stability of the FCI scheme with the diagonalized Jacobian can be improved by reducing the space step. Therefore, we propose an improved method on the grid distribution according to the flow conditions. Numerical tests validate sufficiently the foregoing analyses. Based on the improved grid, theCFLnumber can be quickly ramped up to large values for convergence acceleration.

A new method of adjustment of inlet boundary for improving heat transfer of mixed convection in a vertical channel

This article is dedicated to a new method for improving heat transfer of mixed convection in a vertical channel from the numerical perspective. An adjustable inlet boundary is proposed in the new method to substitute an inlet that is completely occupied by the amount of fluid provided by forced convection included in mixed convection. According to this new method, the flow reversal occurring in the large Richardson number situation can be eliminated significantly, which results in an improvement in the heat transfer rate of mixed convection. In order to investigate a compressible flow problem, this study focuses on four methods, i.e., Roe scheme, preconditioning, dual time stepping, and the LUSGS (Lower–Upper Symmetric-Gauss–Seidel), to solve the governing equations. Except for the aperture occupied by the mount of fluid provided by forced convection, this study simulates the non-reflecting boundary conditions in other open boundaries. Based on our previous study which has successfully explored a phenomenon of flow reversal, this study intends to enhance the heat transfer rate under this circumstance, and the result has shown a significant improvement, by 29 percentage, comparing with the previous study. Also, a criterion of estimation of the necessity of forced convection included in mixed convection is proposed for enhancing heat transfer rate. Present results and existing results have good agreement.

Highly scalable implementation of an implicit matrix-free solver for gas dynamics on GPU-accelerated clusters

A numerical approach for solving gas dynamics on Cartesian grids is considered which employs an implicit time marching scheme with the matrix-free Lower-Upper Symmetric Gauss---Seidel (LU-SGS) method for solving discrete equations. Boundary conditions are treated with an embedded-boundary method. The method has two attractive features--(1) algorithmic uniformity of calculations and (2) structured memory accesses that well fit massively parallel architectures with GPU accelerators. We propose a novel CUDA+MPI computational algorithm scalable up to hundreds of GPUs and give in-depth analysis of its implementation (interoperability issues, libraries tuning).

Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes

This paper presents an implicit unified gas-kinetic scheme (UGKS) for non-equilibrium steady state flow computation. The UGKS is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function. By solving the macroscopic equations implicitly, a predicted equilibrium state can be obtained first through iterations. With the newly predicted equilibrium state, the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well. The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization method is implemented to solve both macroscopic and microscopic equations, which improves the efficiency of the scheme. Since the UGKS is a direct modeling method and its physical solution depends on the mesh resolution and the local time step, a physical time step needs to be fixed before using an implicit iterative technique with a pseudo-time marching step. Therefore, the physical time step in the current implicit scheme is determined by the same way as that in the explicit UGKS for capturing the physical solution in all flow regimes, but the convergence to a steady state speeds up through the adoption of a numerical time step with large CFL number. Many numerical test cases in different flow regimes from low speed to hypersonic ones, such as the Couette flow, cavity flow, and the flow passing over a cylinder, are computed to validate the current implicit method. The overall efficiency of the implicit UGKS can be improved by one or two orders of magnitude in comparison with the explicit one.

Computational-Fluid-Dynamics Solver with Preconditioned Method for Aerodynamic Simulation of High-Lift Configuration

A preconditioned method with characteristic boundary conditions for Navier–Stokes equations is implemented for flowfield analysis of a high-lift configuration at low Mach number. A preconditioned Harten-Lax-van Leer-Einfeldt-Wada scheme is adopted for the spatial discretization, and the dissipation term of which is rededuced to reduce the overlarge numerical dissipation. Preconditioned characteristic boundary conditions for the far field are derived based on the Weiss–Smith preconditioner and are demonstrated to be more reliable and accurate when coupled with the preconditioned method. A preconditioned Lower-Upper Symmetric Gauss-Seidel implicit time-marching method is modified according to the preconditioned Jacobian matrix. The applications of the current method on both two- and three-dimensional high-lift configurations indicate that the use of the preconditioned spatial and time discrete schemes and the derived preconditioned characteristic boundary conditions is capable of improving the robustness, efficiency, and accuracy of the computational-fluid-dynamics solver for low-Mach-number aerodynamic simulations.

Simulation of Two-Dimensional Scramjet Combustor Reacting Flow Field Using Reynolds Averaged Navier-Stokes WENO Solver

AbstractThe non-equilibrium chemical reacting combustion flows of a proposed long slender scramjet system were numerically studied by solving the turbulent Reynolds averaged Navier-Stokes (RANS) equations. The Spalart-Allmaras one equation turbulence model is used which produces better results for near wall and boundary layer flow field problems. The lower-upper symmetric Gauss-Seidel implicit scheme, which enables results converge efficiently under steady state condition, is combined with the weighted essentially non-oscillatory (WENO) scheme to yield an accurate simulation tool for scramjet combustion flow field analysis. Using the WENO schemes high-order accuracy and its non-oscillatory solution at flow discontinuities, better resolution of the hypersonic flow problems involving complex shock-shock/shock-boundary layer interactions inside the flow path, can be achieved. Two types of scramjet combustor with cavity-based and strut-based fuel injector were considered as the testing models. The flow characteristics with and without combustion reactions of the two types combustor model were studied with a transient hydrogen/oxygen combustion model. The detailed results of aerodynamic data are obtained and discussed, moreover, the combustion properties of varying the equivalent ratio of hydrogen, including the concentration of reacting species, hydrogen and oxygen, and the reacting products, water, are demonstrated to study the combustion process and performance of the combustor. The comparisons of flow field structures, pressure on wall and velocity profiles between the experimental data and the solutions of the present algorithms, showed qualitatively as well as the quantitatively in good agreement, and validated the adequacy of the present simulation tool for hypersonic scramjet reacting flow analysis.

Influence of ribs on train aerodynamic performances

The influence of ribs on the train aerodynamic performance was computed using detached eddy simulation (DES), and the transient iteration was solved by the dual-time step lower-upper symmetric Gauss-Seidel (LU-SGS) method. The results show that the ribs installed on the roof have a great effect on the train aerodynamic performance. Compared with trains without ribs, the lift force coefficient of the train with convex ribs changes from negative to positive, while the side force coefficient increases by 110% and 88%, respectively. Due to the combined effect of the lift force and side force, the overturning moment of the train with convex ribs and cutting ribs increases by 140% and 106%, respectively. There is larger negative pressure on the roof of the train without ribs than that with ribs. The ribs on the train would disturb the flow structure and contribute to the air separation, so the separation starts from the roof, while there is no air separation on the roof of the train without ribs. The ribs can also slow down the flow speed above the roof and make the air easily sucked back to the train surface. The vortices at the leeward side of the train without ribs are small and messy compared with those of the train with convex or cutting ribs.

Simulation of axisymmetric scramjet inlet flow fields using anti-diffusive WENO Navier-Stokes solver

This study is based on the theory of gas dynamics and uses the Navier-Stokes equations as the governing equations to solve the hypersonic flow problems of the scramjet. For the turbulent flow, we use the Spalart-Allmaras one equation turbulence model, which produces better results for near wall and boundary layer flow field problems. The lower-upper symmetric Gauss-Seidel implicit scheme, which enables results to converge efficiently under steady state conditions, is combined with the anti-diffusive weighted essentially nonoscillatory (WENO) scheme to yield an accurate and efficient simulation tool for an axisymmetric scramjet flow field analysis. Using the WENO scheme’s high-order accuracy and its non-oscillatory solution at discontinuous regions, we can solve the hypersonic flow problems involving complex shock-shock/shock-boundary layer interactions inside the flow path. This simulation procedure is first verified against two existing partial examples to ensure its accuracy, and is then applied to a complete scramjet model with different initial conditions for a full flow field analysis. The aerodynamic data of Mach number, density, static temperature and pressure are obtained and the results discussed. The anti-diffusive WENO scheme produces more accurate resolution of shock and slip lines and their complex multiple interactions than other numerical approaches. This is of crucial importance for the scramjet complete flow field analysis as multiple shock-boundary layer and discontinuities interactions often occur within the long flow path.

A hybrid implicit scheme for solving Navier–Stokes equations

SummaryA hybrid implicit scheme is developed in this study to solve 3D compressible Navier–Stokes equations on a hybrid unstructured mesh. An approximate evaluation technique for the residual term at a new time level is presented according to the analysis of the main physical mechanism introduced by the lower–upper symmetric Gauss–Seidel (LU‐SGS) relaxation‐based backward Euler implicit method. The residual evaluation method is computationally inexpensive because it does not require flux calculations for every cell surface. The approximated new time residual can be obtained without considerable computational effort; thus, a point‐implicit scheme with a multigrid property is designed to march a further time step after the previous LU‐SGS‐based backward Euler stage. This step leads to the principal ingredient of the presented two‐stage hybrid implicit scheme.To investigate the convergence behavior and performance of the two‐stage hybrid implicit scheme, 2D and 3D viscous flows around the NACA0012 airfoil, NHLP 2D multi‐airfoil, and DLR‐F6 wing–body–pylon–nacelle configuration are simulated by using different grid types. The convergence histories of the single LU‐SGS‐based backward Euler scheme and the presented two‐stage hybrid implicit scheme are analyzed and compared for each test case. The test results show that the hybrid implicit scheme is more efficient and robust than the single‐stage LU‐SGS scheme. Copyright © 2015 John Wiley & Sons, Ltd.

Parameter investigation with line-implicit lower–upper symmetric Gauss–Seidel on 3D stretched grids

An implicit lower–upper symmetric Gauss–Seidel (LU-SGS) solver has been implemented as a multigrid smoother combined with a line-implicit method as an acceleration technique for Reynolds-averaged Navier–Stokes (RANS) simulation on stretched meshes. The computational fluid dynamics code concerned is Edge, an edge-based finite volume Navier–Stokes flow solver for structured and unstructured grids. The paper focuses on the investigation of the parameters related to our novel line-implicit LU-SGSsolver for convergence acceleration on 3D RANS meshes. The LU-SGS parameters are defined as the Courant–Friedrichs–Lewy number, the left-hand side dissipation, and the convergence of iterative solution of the linear problem arising from the linearisation of the implicit scheme. The influence of these parameters on the overall convergence is presented and default values are defined for maximum convergence acceleration. The optimised settings are applied to 3D RANScomputations for comparison with explicit and line-implicit Runge–Kutta smoothing. For most of the cases, a computing time acceleration of the order of 2 is found depending on the mesh type, namely the boundary layer and the magnitude of residual reduction.

Aerodynamic shape optimization for alleviating dynamic stall characteristics of helicopter rotor airfoil

In order to alleviate the dynamic stall effects in helicopter rotor, the sequential quadratic programming (SQP) method is employed to optimize the characteristics of airfoil under dynamic stall conditions based on the SC1095 airfoil. The geometry of airfoil is parameterized by the class-shape-transformation (CST) method, and the C-topology body-fitted mesh is then automatically generated around the airfoil by solving the Poisson equations. Based on the grid generation technology, the unsteady Reynolds-averaged Navier-Stokes (RANS) equations are chosen as the governing equations for predicting airfoil flow field and the highly-efficient implicit scheme of lower–upper symmetric Gauss–Seidel (LU-SGS) is adopted for temporal discretization. To capture the dynamic stall phenomenon of the rotor more accurately, the Spalart–Allmaras turbulence model is employed to close the RANS equations. The optimized airfoil with a larger leading edge radius and camber is obtained. The leading edge vortex and trailing edge separation of the optimized airfoil under unsteady conditions are obviously weakened, and the dynamic stall characteristics of optimized airfoil at different Mach numbers, reduced frequencies and angles of attack are also obviously improved compared with the baseline SC1095 airfoil. It is demonstrated that the optimized method is effective and the optimized airfoil is suitable as the helicopter rotor airfoil.

Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation

The implicit lower–upper symmetric Gauss–Seidel (LU-SGS) solver is combined with the line-implicit technique to improve convergence on the very anisotropic grids necessary for resolving the boundary layers. The computational fluid dynamics code used is Edge, a Navier–Stokes flow solver for unstructured grids based on a dual grid and edge-based formulation. Multigrid acceleration is applied with the intention to accelerate the convergence to steady state. LU-SGS works in parallel and gives better linear scaling with respect to the number of processors, than the explicit scheme. The ordering techniques investigated have shown that node numbering does influence the convergence and that the orderings from Delaunay and advancing front generation were among the best tested. 2D Reynolds-averaged Navier–Stokes computations have clearly shown the strong efficiency of our novel approach line-implicit LU-SGS which is four times faster than implicit LU-SGS and line-implicit Runge–Kutta. Implicit LU-SGS for Euler and line-implicit LU-SGS for Reynolds-averaged Navier–Stokes are at least twice faster than explicit and line-implicit Runge–Kutta, respectively, for 2D and 3D cases. For 3D Reynolds-averaged Navier–Stokes, multigrid did not accelerate the convergence and therefore may not be needed.

Efficient Finite-Volume Model for Shallow-Water Flows Using an Implicit Dual Time-Stepping Method

AbstractAn implicit dual time-stepping method (DTS) is applied to a Godunov-type finite-volume model for two-dimensional (2D) shallow-water flows on unstructured grids to improve run-time efficiency. In this model, an implicit nonlinear lower–upper symmetric Gauss–Seidel (LU-SGS) solution algorithm is used as an inner iteration solver for DTS. To relieve the quantity nonconservation problem of DTS, a water quantity conservation correction method is presented. Five extensive test cases including two analytical benchmark cases and experimental and actual dam-break cases have been applied to validate the proposed model and to demonstrate its performance by comparison with an explicit scheme. The presented results show that DTS can reduce the run time from 55 to 78% without or with a minimal loss of accuracy. The overall performance demonstrates that the proposed model is accurate and efficient for simulating shallow water in practical applications.

Numerical study on flow fields and aerodynamics of tilt rotor aircraft in conversion mode based on embedded grid and actuator model

A method combining rotor actuator disk model and embedded grid technique is presented in this paper, aimed at predicting the flow fields and aerodynamic characteristics of tilt rotor aircraft in conversion mode more efficiently and effectively. In this method, rotor’s influence is considered in terms of the momentum it impacts to the fluid around it; transformation matrixes among different coordinate systems are deduced to extend actuator method’s utility to conversion mode flow fields’ calculation. Meanwhile, an embedded grid system is designed, in which grids generated around fuselage and actuator disk are regarded as background grid and minor grid respectively, and a new method is presented for ‘donor searching’ and ‘hole cutting’ during grid assembling. Based on the above methods, flow fields of tilt rotor aircraft in conversion mode are simulated, with three-dimensional Navier–Stokes equations discretized by a second-order upwind finite-volume scheme and an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) time-stepping scheme. Numerical results demonstrate that the proposed CFD method is very effective in simulating the conversion mode flow fields of tilt rotor aircraft.