Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers

The direct and adjoint operators play an undeniably important role in a vast number of theoretical and practical studies that range from linear stability to flow control and nonlinear optimization. Based on an existing nonlinear flow solver, the design of efficient and straightforward procedures to access these operators is thus highly desirable. In the case of compressible solvers, the use of high-order numerical schemes combined with complicated governing equations makes the derivation of efficient procedures a challenging and often tedious undertaking. In this work, a novel technique for the evaluation of the direct and adjoint operators directly from compressible flow solvers is presented and extended to include nonlinear differentiation schemes and turbulence models. The application to the incompressible counterpart is also discussed. The presented method requires minimal additional programming effort and automatically takes into account subsequent modifications in the governing equations and boundary conditions. The introduced methodology is demonstrated on existing numerical codes, and direct and adjoint global modes are calculated for three typical flow configurations. Implementation issues and the performance measures are also discussed. The proposed algorithm presents an easy-to-implement and efficient technique to extract valuable information for the quantitative analysis of complex flows.     

三维可压缩绕球流的整体线性稳定性研究

研究基于三维可压缩Navier-Stokes方程拟线化方法的整体稳定性问题的数值求解,采用隐式重启的Arnoldi方法求解其特征值问题.针对三维可压缩绕球基本流,研究其在亚临界参数Reynolds数Re=200,马赫数M=0.2,以及超临界参数Re=300,M=0.6下的整体稳定性问题.结果表明,Mach数的增加(直至M=0.6)对流场模态的转变没有定性影响.     

Stable Galerkin reduced order models for linearized compressible flow

The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.     

相容拉氏方法中的ADRS近似Riemann解算器

基于可压缩多介质流动问题，分析AC（acoustic），MFCAV（multi fluid channel on averaged volume）和HLLC等近似Riemann解算器的优缺点，通过加权组合的方式设计一种自适应近似Riemann解算器ADRS（adaptive Riemann solver），详细介绍加权组合的自适应选取原则.将ADRS写成AC解算器的修正形式应用于健壮性好的相容中心型拉氏方法.给出Taylor Green vortex稳态流问题的误差分析等数值算例.     

高压联合进汽阀门三维粘性流场数值分析

本文采用多分区 ＳＩＭＰＬＥＣ算法对 ６００ ＭＷ汽轮机高压联合进汽阀门内部复杂流动进行了三维粘性数值分析。流场计算结果表明总压损失主要集中在调节阀喉口附近。调节阀阀碟前后的两股高速气流在阀碟的正上方位置相碰引起较大的掺混损失;气流经过调节阀喉口后,附面层急剧增厚,形成一层较厚的沿管壁的低总压区。计算得到的流场特性为即将开展的阀门结构优化设计提供了重要依据。     

A symmetry-preserving discretisation and regularisation model for compressible flow with application to turbulent channel flow

Most simulation methods for compressible flow attain numerical stability at the cost of swamping the fine turbulent flow structures by artificial dissipation. This article demonstrates that numerical stability can also be attained by preserving conservation laws at the discrete level. A new mathematical explanation of conservation in compressible flow reveals that many conservation properties of convection are due to the skew-symmetry of the convection operator. By preserving this skew-symmetry at the discrete level, a fourth-order accurate collocated symmetry-preserving discretisation with excellent conservation properties is obtained. Also a new symmetry-preserving regularisation subgrid-scale model is proposed. The proposed techniques are assessed in simulations of compressible turbulent channel flow. The symmetry-preserving discretisation for compressible flow has good stability without artificial dissipation and yields acceptable results already on coarse grids. Regularisation does not consistently improve upon no-model results, but often compares favourably with eddy-viscosity models.     

Characterization of acoustic disturbances in linearly sheared flows

The equation describing the plane wave propagation, the stability, or the rectangular duct mode characteristics in a compressible inviscid linearly sheared parallel, but otherwise homogeneous, flow, is shown to be reducible to Whittaker's equation. The resulting solutions, which are real, viewed as functions of two variables, depend on a parameter and an argument the values of which have precise physical meanings depending on the problem. The exact solutions in terms of Whittaker functions are used to obtain a number of known results of plane wave propagation and stability in linearly sheared flows as limiting cases in which the speed of sound goes to infinity (incompressible limit) or the shear layer thickness, or wave number, goes to zero (vortex sheet limit). The usefulness of the exact solutions is then discussed in connection with the problems of plane wave propagation and stability of a finite thickness shear layer with a linear velocity profile. With respect to the plane wave propagation it is shown that, unlike the compressible vortex sheet, the shear layer possesses no resonances and no Brewster angles, whereas with respect to the stability problem it is shown that, again unlike the compressible vortex sheet, the thin layer is unstable to long wavelength disturbances for all Mach numbers. These results imply that the reflection and stability characteristics of a non-zero thickness but thin shear layer (i.e., the long wavelength characteristics) do not go over smoothly into the results of the compressible vortex sheet as the wave number approaches zero, except for a limited range of generally subsonic relative flow of the two parallel streams bounding the shear layer.     

HLLC-type Riemann solver for the Baer–Nunziato equations of compressible two-phase flow

We first construct an approximate Riemann solver of the HLLC-type for the Baer–Nunziato equations of compressible two-phase flow for the “subsonic” wave configuration. The solver is fully nonlinear. It is also complete, that is, it contains all the characteristic fields present in the exact solution of the Riemann problem. In particular, stationary contact waves are resolved exactly. We then implement and test a new upwind variant of the path-conservative approach; such schemes are suitable for solving numerically nonconservative systems. Finally, we use locally the new HLLC solver for the Baer–Nunziato equations in the framework of finite volume, discontinuous Galerkin finite element and path-conservative schemes. We systematically assess the solver on a series of carefully chosen test problems.     

A new formulation of Kapila’s five-equation model for compressible two-fluid flow,and its numerical treatment

A new formulation of Kapila’s five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.     

一种适用于磁流体力学切向间断的HLLC黎曼算子

磁场的存在使得磁流体力学特征波模不同于流体力学, 因此直接由流体力学HLLC黎曼算子导出的HLLC双中间态在交界的间断面会出现不守恒的问题。通常降级采用HLL单磁场中间态代替HLLC双磁场中间态以实现守恒和计算稳定, 代价是切向间断的模拟精度不足。本文对此进行改进, 在模拟切向间断时仍然保留原有的HLLC双磁场中间态, 同时各守恒量仍然能够满足Toro相容条件; 改进型HLLC算子在间断两侧的磁场分量存在差异, 因此能够更精确还原切向间断面。基于数值测试, 包括一维激波管和切向间断的时变模拟, 以及地球磁层三维数值模拟, 将模拟结果进行对比, 结果表明: 相比于已发展的HLLC算子, 改进型HLLC算子对切向间断具有更好的捕捉精度, 能够达到或接近耗时更多的HLLD算子的模拟精度。     

Accelerating turbulent reacting flow simulations on many-core/GPUs using matrix-based kinetics

The present work assesses the impact - in terms of time to solution, throughput analysis, and hardware scalability - of transferring computationally intensive tasks, found in compressible reacting flow solvers, to the GPU. Attention is focused on outlining the workflow and data transfer penalties associated with “plugging in” a recently developed GPU-based chemistry library into (a) a purely CPU-based solver and (b) a GPU-based solver, where, except for the chemistry, all other variables are computed on the GPU. This comparison allows quantification of host-to-device (and vice versa) data transfer penalties on the overall solver speedup as a function of mesh and reaction mechanism size. To this end, a recently developed GPU-based chemistry library known as UMChemGPU is employed to treat the kinetics in the flow solver KARFS. UMChemGPU replaces conventional CPU-based Cantera routines using a matrix-based formulation. The impact of i) data transfer times, ii) chemistry acceleration, and iii) the hardware architecture is studied in detail in the context of GPU saturation limits. Hydrogen and dimethyl ether (DME) reaction mechanisms are used to assess the impact of the number of species/reactions on overall/chemistry-only speedup. It was found that offloading the source term computation to UMChemGPU results in up to 7X reduction in overall time to solution and four orders of magnitude faster source term computation compared to conventional CPU-based methods. Furthermore, the metrics for achieving maximum performance gain using GPU chemistry with an MPI + CUDA solver are explained using the Roofline model. Integrating the UMChemGPU with an MPI + OpenMP solver does not improve the overall performance due to the associated data copy time between the device (GPU) and host (CPU) memory spaces. The performance portability was demonstrated using three different GPU architectures, and the findings are expected to translate to a wide variety of high-performance codes in the combustion community.     

Multiple-relaxation-time lattice Boltzmann model for compressible fluids

We present an energy-conserving multiple-relaxation-time finite difference lattice Boltzmann model for compressible flows. The collision step is first calculated in the moment space and then mapped back to the velocity space. The moment space and corresponding transformation matrix are constructed according to the group representation theory. Equilibria of the nonconserved moments are chosen according to the need of recovering compressible Navier-Stokes equations through the Chapman-Enskog expansion. Numerical experiments showed that compressible flows with strong shocks can be well simulated by the present model. The new model works for both low and high speeds compressible flows. It contains more physical information and has better numerical stability and accuracy than its single-relaxation-time version.     

Stabilization of thermal lattice Boltzmann models

A three-dimensional thermal lattice-Boltzmann model with two relaxation times to separately control viscosity and thermal diffusion is developed. Numerical stability of the model is significantly improved using Lax-Wendroff advection to provide and adjustable time step. Good agreement with a conventional fiitedifference Navier-Stokes solver is obtained in modeling compressible Rayleigh-Bénard convestion when boundary conditions are treated similarly.     

A quadrature-based LES/transported probability density function approach for modeling supersonic combustion

The joint-scalar probability density function (PDF) approach provides a comprehensive framework for large eddy simulation (LES) based combustion modeling. However, currently available stochastic approaches for solving the high-dimensional PDF transport equation can be error prone and numerically unstable in highly compressible shock-containing flows. In this work, a novel Eulerian approach called the direct quadrature method of moments (DQMOM) is developed for evolving the PDF-based supersonic combustion model. The DQMOM technique uses a set of scalar transport equations with specific source terms to recover the PDF. The new technique is coupled to a compressible LES solver through the energy equation. The DQMOM approach is then used to simulate two practical flow configurations: a supersonic reacting jet and a cavity-stabilized supersonic combustor. Comparisons with experimental data demonstrate the predictive accuracy of the method.     

Modeling and optimization of non-periodic grating couplers

The higher degree of freedom available for non-periodic gratings (as compared with their periodic counterparts) is investigated. These non-periodic structures may be employed to design novel light couplers with increased functionality. Optimizing such devices requires a complex search in a huge parameter space. The success in the solution of this task depends on the availability of a fast forward solver and a reliable search algorithm. Here, a fast forward solver based on the multiple multipole (MMP) method together with a near-to-far field transformation and a multiple scattering calculation is presented. Thanks to the efficiency of our approach, non-periodic gratings are evaluated with a speed comparable to commonly used periodic grating approximations. This allows our solver to be combined with a heuristic global search scheme, namely an evolutionary algorithm. The procedure is demonstrated with the optimization of a non-periodic grating output coupler that suppresses an unwanted second diffracted order.     

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin(RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ?ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ?ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

All Eulerian method of computing elastic response of explosively pressurised metal tube

We present an all Eulerian approach to simulate the elastic response of a metal tube loaded explosively by a gaseous detonation. The high strain rate deformation of the metal tube subjected to high explosive detonation is mathematically described by hyperbolic processes where the characteristics of existing wave motions were correlated with the local particle velocities through the speed of sound in the metal. This is a favourable case for the hydrocode which is based on a compressible gas dynamics solver and for simulating a high strain rate and dominantly plastic response of a material subject to an explosive loading. The hydrocodes fall substantially short of predicting elastic motion without the plastic flow of the confining material, for relatively minor pressure loadings due to a gaseous explosion as opposed to a high explosive detonation of a charged tube. The corresponding loading pressure due to gaseous explosion is a few orders of magnitude lower than those resulting from high explosive loadings. Utilising a hydrocode designed to handle the reactive process leading to a plastic flow of the confining materials is of great interest and a significant challenge. The new technique, based on the Eulerian framework, preserves the feature of a Lagrangian code while utilising all the benefits of an Eulerian solver that uses fixed grids with the level-sets for defining the multi-material interfaces. The hybrid particle level-set algorithm is combined with a hydrodynamic solver that adds an elasticity correction when handling the structural response while the overall scheme remained hyperbolic during the entire reactive flow. Several unseen dynamics of detonation flow associated with the elastically loaded tube of finite thickness are reported by using the present method for analysing the highly pressurised vessel.     

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

Coupled fluid–structure solver: The case of shock wave impact on monolithic and composite material plates

An unstructured adaptive mesh flow solver, a finite element structure solver and a moving mesh algorithm were implemented in the numerical simulation of the interaction between a shock wave and a structure. In the past, this interaction is mostly considered as one-way in the sense that the shock causes a transient load on the structure while it is reflected uneffected by the impact. A fully coupled approach was implemented in the present work which can account for the effects associated with a mutual interaction. This approach included a compressible flow Eulerian solver of second order accuracy in finite volume formulation for the fluid and a Langargian solver in finite element formulation for the solid structure. A novel implementation of advancing front moving mesh algorithm was made possible with the introduction of a flexible and efficient quad-edge data structure. Adaptive mesh refinement was introduced into the flow solver for improved accuracy as well. Numerical results are further validated by theoretical analysis, experimental data and results from other numerical simulations. Grid dependency study was performed and results showed that the physical phenomena and quantities were independent of the numerical grid chosen in the simulations. The results illuminated complicated flow phenomena and structure vibration patterns, which in order to be detected experimentally require capabilities beyond those of the current experimental techniques. The numerical simulations also successfully modelled the aero-acoustic damping effects on the structure, which do not exist in previous numerical models. Further analysis of the results showed that the mutual interaction is not linear and that the non-linearity arises because the wave propagation in the fluid is not linear and it cascades a non-linear and non-uniform loading on the plate. Non-linearity intensifies when the plate is vibrating at high frequency while the wave propagation speed is low.     

二维多相反应流的欧拉数值方法

本文研究二维多相反应流的数值模拟方法.对于非完全流体,考虑两种状态方程：刚性气体状态方程和JWL（Jones-Wilkins-Lee）型的状态方程.针对这些非完全流体,研究一种混合单元物理量重构法,它由单元内的已知物理量出发,应用单元内不同相的物理量之间的关系：混合密度与各自密度之间的关系、及混合内能与各物质内能之间的关系,再由温度与压强平衡条件,建立关于单变量的非线性方程,对所得方程通过"移动跟踪法"求解,得到单元内的压力、温度及声速等.再利用HLLC解法器,求解各个单元边界数值通量,从而构造一种高效的数值方法.数值结果表明：该算法能清晰地捕捉爆轰波的结构特征,胞格排列有序,三波点的特征明显.