Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

Application of artificial viscosity for suppressing the carbuncle phenomenon in Godunov-type schemes

A new approach for solving the carbuncle phenomenon that occurs in Godunov-type schemes when applied to hypersonic flow simulations is proposed. The approach suppresses carbuncle-instability by additional dissipative terms in the form of the right-hand side of Navier-Stokes equations, but with artificial instead of molecular viscosity. The efficiency of the proposed approach is demonstrated on several test problems.     

Numerical analysis of unsteady compressible laminar boundary layer flow

The unsteady compressible laminar boundary layer flow on an arbitrary cylinder due to an incident stream whose velocity varies arbitrarily with time is considered. The method presented is based on the separation between the convective and diffusive quantities. By defining some new variables, the splitting appears rather naturally, and the initialisation problem can be solved without difficulty. The transformed equations are solved with the help of a semi-implicit finite difference scheme which is unconditionally linearly stable. The computations have been applied to flows past a cylinder with constant and fluctuating free-stream velocities.     

Gas-kinetic schemes for compressible flow of real gases

This paper extends gas-kinetic schemes to the Euler equations for real gases. In the current scheme, the maxwell-Boltzmann gas distribution function is modified to recover macroscopic flow equations. More specifically, the internal degree of freedom of the gas distribution function becomes a function of flow variables according to the general equation of state. The numerical results confirm the accuracy and robustness of the gas-kinetic approach.     

A review and comparative study of upwind biased schemes for compressible flow computation. Part II: 1-D higher-order schemes

Summary Higher-order upwind biased procedures for solving the equations of 1-D compressible unsteady flow are surveyed. Approaches based upon the use of either switched artificial viscosity, flux-limiting or slopelimiting are considered and described within a unified framework. The approaches are implemented within the context of an edge-based finite element solution algorithm, which represents the basis for a future multi-dimensional extension on general grids. The performance of the different approaches is illustrated by application to the solution of the shock tube problem in different flow regimes.     

Analysis of unsteady compressible boundary layer flow via finite elements

This paper is concerned with the discrete formulation and numerical solution of unsteady compressible boundary layer flows using the Galerkin-finite element method. Linear interpolation functions for the velocity, density, temperature and pressure are used in the momentum equation and equations of continuity, energy and state. The coupled nonlinear finite element equations are approximated by a third order Taylor series expansion as temporal operator to integrate in time with Newton-Raphson type iterations performed until convergence within each time step. As an example, a boundary layer problem of a perfect gas behind a normal shock wave is solved. A comparison of the results with those by other method indicates a favorable agreement.     

Spectral simulations of an unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder has been successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for a periodic problem is introduced. We have further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It has been shown that the solution of a model problem has exponential growth in time if an ‘improper’ boundary conditions procedure is used. A characteristic boundary conditions, which is based on the characteristics of the Euler equations of gas dynamics, has been derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.     

A review and comparative study of upwind biased schemes for compressible flow computation. Part I: 1—D first—order schemes

Summary First-order upwind biased formulations for simulating 1-D compressible flows are presented within a unified framework. A detailed study and comparison of the different formulations is important, as high-resolution extensions are known to inherit both the good and bad features of the first-order formulations used for their construction. The most popular flux difference splitting, flux vector splitting and some recently proposed hybrid splitting methodologies are considered. A finite element solution approach is adopted, as this provides a framework for the multidimensional extension of the solvers. Representative one-dimensional test cases are considered in order to provide evidence of the effectiveness and performance of the formulations. The results that are presented, together with the corresponding exact solutions, provide a set of benchmark test cases for comparison purposes.     

Accuracy evaluation of unsteady CFD numerical schemes by vortex preservation

Numerical uncertainty is an important but sensitive subject in computational fluid dynamics and there is a need for improved methods to quantify calculation accuracy. A known analytical solution, a Lamb-type vortex unsteady movement in a free stream, is compared to the numerical solutions obtained from different numerical schemes to assess their temporal accuracies. Solving the Navier-Stokes equations and using the standard Linearized Block Implicit ADI scheme, with first order accuracy in time second order in space, a vortex is convected and results show the rapid diffusion of the vortex. These calculations were repeated with the iterative implicit ADI scheme which has second-order time accuracy. A considerable improvement was noticed. The results of a similar calculation using an iterative fifth-order spatial upwind-biased scheme is also considered. The findings of the present paper demonstrate the needs and provide a means for quantification of both distribution and absolute values of numerical error.     

A review and comparative study of upwind biased schemes for compressible flow computation. Part III: Multidimensional extension on unstructured grids

Summary The edge based Galerkin finite element formulation is used as the basic building block for the construction of multidimensional generalizations, on unstructured grids, of several higher order upwind biased procedures originally designed for the solution of the 1D compressible Euler system of equations. The use of a central type discretization for the viscous flux terms enables the simulation of multidimensional flows governed by the laminar compressible Navier Stokes equations. Numerical issues related to the development and implementation of multidimensional solution algorithms are considered. A number of inviscid and viscous flow simulations, in different flow regimes, are analyzed to enable the reader to assess the performance of the surveyed formulations.     

Comparative analysis of line-drawing modeling schemes

The computer processing of line drawings necessarily requires that the line drawings be first quantized and then encoded. Invariably, the quantization process forces the computer approximation to connect nodes lying on a lattice. The lattice normally is uniform square (though it could also be rectangular, logarithmic, or curvilinear) and is either explicitly or implicitly defined; its size is determined by the limit in the ability to resolve neighboring points in the quantization process. The coded representation of the drawing may be based on the use of straight or curved segments (approximants). The segments may be restricted either to a small set of fixed lengths or their lengths may take on any of the discrete values permissible on the lattice field. The paper examines some of the possibilities for line drawing modeling and shows that in many cases they may be regarded as special cases of the so-called generalized chain coding scheme. A procedure is developed for computing the relative probabilities of the approximating segments in a line drawing representation. The latter are necessary for any efficient (compact) line-drawing modeling scheme.     

A pressure-correction method and its applications on an unstructured Chimera grid

In this paper, an unstructured Chimera mesh method is used to compute incompressible flow around a rotating body. To implement the pressure correction algorithm on unstructured overlapping sub-grids, a novel interpolation scheme for pressure correction is proposed. This indirect interpolation scheme can ensure a tight coupling of pressure between sub-domains. A moving-mesh finite volume approach is used to treat the rotating sub-domain and the governing equations are formulated in an inertial reference frame. Since the mesh that surrounds the rotating body undergoes only solid body rotation and the background mesh remains stationary, no mesh deformation is encountered in the computation. As a benefit from the utilization of an inertial frame, tensorial transformation for velocity is not needed. Three numerical simulations are successfully performed. They include flow over a fixed circular cylinder, flow over a rotating circular cylinder and flow over a rotating elliptic cylinder. These numerical examples demonstrate the capability of the current scheme in handling moving boundaries. The numerical results are in good agreement with experimental and computational data in literature.     

Simulation of unsteady compressible flow in a channel with vibrating walls - Influence of the frequency

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract.The flow is described by the system of Navier–Stokes equations for laminar flows. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is considered in the Arbitrary Lagrangian–Eulerian (ALE) form.The authors present the numerical simulations of flow fields in the channel, acquired from a program developed exclusively for this purpose. The numerical results for unsteady flows in the channel are presented for inlet Mach number M_∞ = 0.012, Reynolds number Re_∞ = 4.5 × 10³ and the wall motion frequency 20 and 100 Hz.     

A two-level computational graph method for the adjoint of a finite volume based compressible unsteady flow solver

The adjoint method is a useful tool for finding gradients of design objectives with respect to system parameters for fluid dynamics simulations. But the utility of this method is hampered by the difficulty in writing an efficient implementation for the adjoint flow solver, especially one that scales to thousands of cores. This paper demonstrates a Python library, called adFVM, that can be used to construct an explicit unsteady flow solver and derive the corresponding discrete adjoint flow solver using automatic differentiation (AD). The library uses a two-level computational graph method for representing the structure of both solvers. The library translates this structure into a sequence of optimized kernels, significantly reducing its execution time and memory footprint. Kernels can be generated for heterogeneous architectures including distributed memory, shared memory and accelerator based systems. The library is used to write a finite volume based compressible flow solver. A wall clock time comparison between different flow solvers and adjoint flow solvers built using this library and state of the art graph based AD libraries is presented on a turbomachinery flow problem. Performance analysis of the flow solvers is carried out for CPUs and GPUs. Results of strong and weak scaling of the flow solver and its adjoint are demonstrated on subsonic flow in a periodic box.     

Comparative study of discretization zero dynamics behaviors in two multirate cases

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than two is of minimum phase. This paper investigates the zero dynamics, as the sampling period tends to zero, of sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time nonlinear plant and a sampler in cascade. More precisely, we show how an approximate sampled-data model can be obtained for nonlinear systems with two special GSHF cases such that sampled zero dynamics of the resulting model can be arbitrarily placed. Further, two GSHFs with appropriate parameters provide nonlinear zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time plant. It is also shown that the intersample behavior arising from the multirate input can be localised by appropriately selecting the design parameters based on the stability condition of the zero dynamics. The results presented here generalize well-known ideas from the linear to nonlinear cases.     

Inter-destination multimedia synchronization: schemes, use cases and standardization

Traditionally, the media consumption model has been a passive and isolated activity. However, the advent of media streaming technologies, interactive social applications, and synchronous communications, as well as the convergence between these three developments, point to an evolution towards dynamic shared media experiences. In this new model, geographically distributed groups of consumers, independently of their location and the nature of their end-devices, can be immersed in a common virtual networked environment in which they can share multimedia services, interact and collaborate in real-time within the context of simultaneous media content consumption. In most of these multimedia services and applications, apart from the well-known intra and inter-stream synchronization techniques that are important inside the consumers’ playout devices, also the synchronization of the playout processes between several distributed receivers, known as multipoint, group or Inter-destination multimedia synchronization (IDMS), becomes essential. Due to the increasing popularity of social networking, this type of multimedia synchronization has gained in popularity in recent years. Although Social TV is perhaps the most prominent use case in which IDMS is useful, in this paper we present up to 19 use cases for IDMS, each one having its own synchronization requirements. Different approaches used in the (recent) past by researchers to achieve IDMS are described and compared. As further proof of the significance of IDMS nowadays, relevant organizations’ (such as ETSI TISPAN and IETF AVTCORE Group) efforts on IDMS standardization (in which authors have been and are participating actively), defining architectures and protocols, are summarized.     

Large stencil viscous flux linearization for the simulation of 3D compressible turbulent flows with backward-Euler schemes

The purpose of this article is to study different approximate linearizations of the RANS equations viscous fluxes, for numerical simulations of compressible turbulent flows with backward-Euler schemes. The explicit convective flux under consideration is centred with artificial dissipation. The discrete viscous flux, calculated from cell-centred evaluation of the gradients, needs less computations and memory storage than other discretizations. But, in other respects, the balance of this numerical flux has a large stencil, which is not coherent with the 3-point per mesh direction stencil of classical implicit stages. Therefore 3-point and 5-point per mesh direction approximate linearizations are built from the thin layer flux formula. The stability condition of the corresponding backward-Euler schemes is given for a scalar linear equation (for the basic non-factored version of scheme and with LU-relaxation). Multigrid and monogrid computations of turbulent flow around two external configurations are performed with Wilcox’s k-ω turbulence model. The 5-point per mesh direction linearizations, coherent with the differential of the fluxes balance of thin layer approximation of explicit viscous fluxes, leads to the most efficient implicit stages.     

High-resolution finite difference schemes using curvilinear coordinate grids for DNS of compressible turbulent flow over wavy walls

Direct numerical simulations of compressible turbulent flow over wavy wall geometries have been carried out by solving N–S equations on general curvilinear coordinates. A 6th order WENO scheme with minimized dispersion and controllable dissipation is employed to compute the inviscid fluxes, a 4th order central difference scheme is applied to compute the viscous fluxes, and a 6th order conservative compact scheme is used for computing the geometrical metrics. An implicit LU-SGS method is used for time integration to improve computational efficiency over the explicit schemes such as the Runge–Kutta approach. The validity and applicability of the present algorithm is confirmed by comparing our results with laboratory experimental measurements and DNS results in the literature.     

The stability of explicit difference schemes for solving the problem of interaction between a compressible fluid and an elastic shell

A mathematical model for describing the interaction between a compressible fluid and an elastic shell is formulated as an initial boundary value problem. The partial differential equations of the model are discretized both in time and space by a finite-difference method. The stability of the resulting explicit difference schemes is analyzed by Kreiss' theory for the stability analysis of difference schemes in initial boundary value problems. It is shown that the stability properties of the schemes for the interaction problem may be influenced by the type of discretization in space used for the contact condition on the interface between the fluid and the shell and also by the approximation of the hydrodynamic pressure on the surface of the shell. A simple sufficient condition is found that will ensure the best possible stability properties of the schemes. Several of these, which are of practical interest, are analyzed.     

基于ABS的干扰协调方案研究

异构网通过运用不同功率的基站同频部署,扩大了网络覆盖率及系统容量。但同时这种多层结构也带来了小区间干扰问题。ABS是一种时域上的小区间干扰协调技术,解决了下行传输中高功率宏基站和低功率小基站共享频域资源时的干扰问题,但该技术的缺点是降低了宏基站的用户容量。文中研究了一种从频域、时域、功率三个维度联合进行干扰协调的新型ABS解决方案。仿真结果证明该方案可以显著增加宏基站用户的吞吐量,同时不对小基站产生过多干扰。