Residual Distribution Schemes on Quadrilateral Meshes

We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle     

A Robust Smoother for Convection-Diffusion Problems with Closed Characteristics

In this paper we analyze a model problem for the convection-diffusion equation where the reduced problem has closed characteristics. A full upwinding finite difference scheme is used to discretize the problem. Additionally to the strength of the convection, an arbitrary amount of crosswind-diffusion can be added on the discrete level. We present a smoother which is robust w.r.t. the strength of convection and the amount of crosswind-diffusion. It is of Gauss–Seidel type using a downwind ordering. To handle the cyclic dependencies a frequency-filtering algorithm is used. The algorithm is of nearly optimal complexity ?(n log n). It is proved that it fulfills a robust smoothing property.     

Three-dimensional compressible-incompressible turbulent flow simulation using a pressure-based algorithm

In this work, we extend a finite-volume pressure-based incompressible algorithm to solve three-dimensional compressible and incompressible turbulent flow regimes. To achieve a hybrid algorithm capable of solving either compressible or incompressible flows, the mass flux components instead of the primitive velocity components are chosen as the primary dependent variables in a SIMPLE-based algorithm. This choice warrants to reduce the nonlinearities arose in treating the system of conservative equations. The use of a new Favre-averaging like technique plays a key role to render this benefit. The developed formulations indicate that there is less demand to interpolate the fluxes at the cell faces, which is definitely a merit. To impose the hyperbolic behavior in compressible flow regimes, we introduce an artificial hyperbolicity in pressure correction equation. We choose k-ω turbulence model and incorporate the compressibility effect as a correction. It is shown that the above considerations grant to achieve a robust algorithm with great capabilities in solving both flow regimes with a reasonable range of Mach number applications. To evaluate the ability of the new pressure-based algorithm, three test cases are targeted. They are incompressible backward-facing step problem, compressible flow over a wide range of open to closed cavities, and compressible turbulent flow in a square duct. The current results indicate that there are reliable agreements with those of experiments and other numerical solutions in the entire range of investigation.     

Analysis of a high-order finite difference detector for discontinuities

High-order finite difference discontinuity detectors are essential for the location of discontinuities on discretized functions, especially in the application of high-order numerical methods for high-speed compressible flows for shock detection. The detectors are used mainly for switching between numerical schemes in regions of discontinuity to include artificial dissipation and avoid spurious oscillations. In this work a discontinuity detector is analysed by the construction of a piecewise polynomial function that incorporates jump discontinuities present on the function or its derivatives (up to third order) and the discussion on the selection of a cut-off value required by the detector. The detector function is also compared with other discontinuity detectors through numerical examples.     

Symbolic eigenvalue analysis for adaptive stepsize control in PNS shock stabilization

Parabolized Navier-Stokes (PNS) solution techniques for high speed compressible flow involve marching in a “time like” spatial coordinate but have been observed in practice to be sensitive to flow conditions and require stabilization by numerical dissipation techniques. In this study we develop an eigenvalue analysis for shock stability. Using the algebraic symbolic manipulator MACSYMA, an explicit closed-form solution for the shock system eigenvalues is developed. This is then applied on the evolving shock contour to determine the stability limit on the PNS stepsize. Numerical experiments confirm the validity of the approach and demonstrate that the approach can be implemented explicitly to yield an adaptive stepsize algorithm and thereby a more robust PNS scheme.     

Application of normalized flux in pressure-based algorithm

The paper describes an implementation of normalized flux diagram (NFD) scheme into a pressure-based implicit finite-volume procedure to solve the Euler and Navier-Stokes equations on a non-orthogonal mesh with collocated finite-volume formulation. The newly developed algorithm has two new features: (i) the use of the normalized flux and space formulation (NFSF) methodology to bound the convective fluxes and (ii) the use of a high-resolution scheme in calculating interface density values to enhance the shock-capturing property of the algorithm. The procedure incorporates the k-ε eddy-viscosity turbulence model. The algorithm is first tested for inviscid flows at different Mach numbers ranging from subsonic to supersonic on a bump in channel geometry and inside a planar convergent-divergent nozzle. The results have been compared with those using the same scheme in conjunction with primitive variable limiter (NVD). Also, there has been comparison between the results and predicted data using TVD scheme on the basis of characteristic variable. After these comparison, it was found that the limiter on flux, predicted a sharper shock and there was better boundedness here than limiter on primitive variables in coarse mesh. The method is then validated against experimental data for the case of turbulent transonic flow passing via a gas turbine rotor blade cascade for which wind-tunnel experimental data exist. Findings show a remarkable quality of resolution when NFD scheme is used.     

Real-time NURBS interpolation using FPGA for high speed motion control

Modern motion control adopts NURBS (Non-Uniform Rational B-Spline) interpolation for the purpose of achieving high-speed and high-accuracy performance. However, in conventional control architectures, the computation of the basis functions of a NURBS curve is very time-consuming due to serial computing constraints. In this paper, a novel FPGA (Field Programmable Gate Array) based motion controller utilizing its high-speed parallel computing power is proposed to realize the Cox-de Boor algorithm for second and higher degrees NURBS interpolation. The motion control algorithm is also embedded in the FPGA chip to implement real-time control and NURBS interpolation simultaneously for multi-axis servo systems. The proposed FPGA-based motion controller is capable of performing the Cox-de Boor algorithm and the IIR (Infinite Impulse Response) control algorithm in about 46 clock cycles, as compared to the 1303 clock cycles by the traditional approach. Numerical simulations and experimental tests using an X-Y table verify the outstanding computation performance of the FPGA-based motion controller. The result indicates that shorter sampling time (10 μs) can be achieved for NURBS interpolation which is highly critical to the success of high-speed and high-accuracy motion control.     

Visual simulation of shockwaves

We present an efficient method for visual simulations of shock phenomena in compressible, inviscid fluids. Our algorithm is derived from one class of the finite volume method especially designed for capturing shock propagation, but offers improved efficiency through physically-based simplification and adaptation for graphical rendering. Our technique is capable of handling complex, bidirectional object–shock interactions stably and robustly. We describe its applications to various visual effects, including explosion, sonic booms and turbulent flows. Furthermore, we explore parallelization schemes and demonstrate the scalability of our method on shared-memory, multi-core architectures.     

Computations of gas microflows using pressure correction method with Langmuir slip model

A Langmuir slip model combined with continuum-based compressible Navier-Stokes equations is proposed and implemented for the purpose of analyzing complex microscale gas flows. For our model, an efficient compressible pressure correction algorithm based on an unstructured grid is developed and modified to be applicable to low Reynolds number slip flows in microgeometries. Gaseous slip flows in a uniform microchannel and compressible flow at backward-facing step are computed for the assessment of the adequacy of the method. Separated flow in a T-shaped micro-manifold is also simulated for the Reynolds number ranging from 10 to 60. In the uniform microchannel flow, the pressure increases nonlinearly in Langmuir slip model as the Knudsen number increases, while it drops nonlinearly in Maxwell slip model. The results from Langmuir slip model have been found to be more compatible with physics. From all the simulation cases, nonlinear behavior owing to both compressibility and rarefaction clearly appears in terms of streamwise velocity, pressure profiles and even reattachment length in the separation-associated flows. These results show that the suggested pressure correction method along with the Langmuir slip model may effectively simulate complex microscale gas flows, thereby offering a sound theoretical and numerical basis and an inexpensive computation procedure.     

BDDC preconditioning for high-order Galerkin Least-Squares methods using inexact solvers

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection–diffusion equation and Euler equations for compressible, inviscid flow. A Robin–Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs.     

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.     

A decision-making algorithm for automatic flow pattern identification in high-speed imaging

A digital image processing algorithm was developed to identify flow patterns in high speed imaging. This numerical tool allows to quantify the fluid dynamic features in compressible flows of relevance in aerospace and space related applications. This technique was demonstrated in a harsh environment with poor image quality and illumination fluctuations. This original pattern recognition tool is based on image binarization and object identification. The geometrical properties of the detected elements are obtained by measuring the characteristics of each object in the binary image. In case of multiple shock waves or shock bifurcations, a “decision-making” algorithm chooses the best shock-wave path, based on the original image intensity and local pattern orientation. The algorithm was successfully used for validation on numerical Schlieren images, where the shock-wave fluctuation was triggered by vortex shedding. The applicability of the algorithm was finally evaluated in two Schlieren imaging studies: at the trailing edge of supersonic airfoils and for hypersonic research. The program correctly identified the fuzzy flow features present in all applications.     

Development of a parallelized 2D/2D-axisymmetric Navier–Stokes equation solver for all-speed gas flows

A parallelized 2D/2D-axisymmetric pressure-based, extended SIMPLE finite-volume Navier–Stokes equation solver using Cartesians grids has been developed for simulating compressible, viscous, heat conductive and rarefied gas flows at all speeds with conjugate heat transfer. The discretized equations are solved by the parallel Krylov–Schwarz (KS) algorithm, in which the ILU and BiCGStab or GMRES scheme are used as the preconditioner and linear matrix equation solver, respectively. Developed code was validated by comparing previous published simulations wherever available for both low- and high-speed gas flows. Parallel performance for a typical 2D driven cavity problem is tested on the IBM-1350 at NCHC of Taiwan up to 32 processors. Future applications of this code are discussed briefly at the end.     

An analytical framework for high-speed hardware particle swarm optimization

Engineering optimization techniques are computationally intensive and can challenge implementations on tightly-constrained embedded systems. Particle Swarm Optimization (PSO) is a well-known bio-inspired algorithm that is adopted in various applications, such as, transportation, robotics, energy, etc. In this paper, a high-speed PSO hardware processor is developed with focus on outperforming similar state-of-the-art implementations. In addition, the investigation comprises the development of an analytical framework that captures wide characteristics of optimization algorithm implementations, in hardware and software, using key simple and combined heterogeneous indicators. The framework proposes a combined Optimization Fitness Indicator that can classify the performance of PSO implementations when targeting different evaluation functions. The two targeted processing systems are Field Programmable Gate Arrays for hardware implementations and a high-end multi-core computer for software implementations. The investigation confirms the successful development of a PSO processor with appealing performance characteristics that outperforms recently presented implementations. The proposed hardware implementation attains 23,300 improvement ratio of execution times with an elliptic evaluation function. In addition, a speedup of 1777 times is achieved with a Shifted Schwefels function. Indeed, the developed framework successfully classifies PSO implementations according to multiple and heterogeneous properties for a variety of benchmark functions.     

Direct numerical simulation of transitional flow at high Mach number coupled with a thermal wall model

In transitional and turbulent high speed boundary-layer flows the wall thermal boundary conditions play an important role and in many cases an assumption of a constant temperature or a specified heat flux may not be appropriate for numerical simulations. In this paper we extend a formulation for direct numerical simulation of compressible flows to include a thin plate that is thermally fully coupled to the flow. Even without such thermal coupling compressible flows with shock waves and turbulence represent a challenge for numerical methods. In this paper we review the scaling properties of algorithms, based on explicit high-order finite differencing combined with shock capturing, that are suitable for dealing with such flows. An application is then considered in which an isolated roughness element is of sufficient height to trigger transition in the presence of acoustic forcing. With the thermal wall model included it is observed that the plate heats up sufficiently during the simulation for the transition process to be halted and the flow consequently re-laminarises.     

Discontinuous Galerkin Methods Applied to Shock and Blast Problems

We describe procedures to model transient shock interaction problems using discontinuous Galerkin methods to solve the compressible Euler equations. The problems are motivated by blast flows surrounding cannons with perforated muzzle brakes. The goal is to predict shock strengths and blast over pressure. This application illustrates several computational difficulties. The software must handle complex geometries. The problems feature strong interacting shocks, with pressure ratios on the order of 1000 as well as weaker precursor shocks traveling rearward that also must be accurately captured. These aspects are addressed using anisotropic mesh adaptation. A shock detector is used to control the adaptation and limiting. We also describe procedures to track projectile motion in the flow by a level-set procedure.This revised version was published online in July 2005 with corrected volume and issue numbers.     

A note on compressed sensing and the complexity of matrix multiplication

We consider the conjectured O(N2+?) time complexity of multiplying any two N×N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A⋅B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given N×N matrix B, which allows the exact computation of A⋅B to be carried out using the conjectured O(N2+?) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk's recently proposed extractor-based CS algorithm [P. Indyk, Explicit constructions for compressed sensing of sparse signals, in: SODA, 2008] which is resilient to noise.     

A DNS algorithm using B-spline collocation method for compressible turbulent channel flow

Direct numerical simulation (DNS) offers useful information about the understanding and modeling of turbulent flow. However, few DNSs of wall-bounded compressible turbulent flows have been performed. The objective of this paper is to construct a DNS algorithm which can simulate the compressible turbulent flow between the adiabatic and isothermal walls accurately and efficiently. Since this flow is the simplest turbulent flow with adiabatic and isothermal walls, it is ideal for the modeling of compressible turbulent flow near the adiabatic and isothermal walls. The present DNS algorithm for wall-bounded compressible turbulent flow is based on the B-spline collocation method in the wall-normal direction. In addition, the skew-symmetric form for convection term is used in the DNS algorithm to maintain numerical stability. The validity of the DNS algorithm is confirmed by comparing our results with those of an existing DNS of the compressible turbulent flow between isothermal walls [J. Fluid Mech. 305 (1995) 159]. The applicability and usefulness of the DNS algorithm are demonstrated by the stable computation of the DNS of compressible turbulent flow between adiabatic and isothermal walls.     

A Combined Large-Eddy Simulation and Time-Dependent RANS Capability for High-Speed Compressible Flows

An entirely new approach to the large-eddy simulation (LES) of high-speed compressible turbulent flows is presented. Subgrid scale stress models are proposed that are dimensionless functions of the computational mesh size times a Reynolds stress model. This allows a DNS to go continuously to an LES and then a Reynolds-averaged Navier–Stokes (RANS) computation as the mesh becomes successively more coarse or the Reynolds number becomes much larger. Here, the level of discretization is parameterized by the nondimensional ratio of the computational mesh size to the Kolmogorov length scale. The Reynolds stress model is based on a state-of-the-art two-equation model whose enhanced performance is documented in detail in a variety of benchmark flows. It contains many of the most recent advances in compressible turbulence modeling. Applications to the high-speed aerodynamic flows of technological importance are briefly discussed.     

SUPG finite element computation of inviscid supersonic flows with YZβ shock-Capturing

Stabilization and shock-capturing parameters introduced recently for the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation of compressible flows based on conservation variables are assessed in test computations with inviscid supersonic flows and different types of finite element meshes. The new shock-capturing parameters, categorized as “YZβ Shock-Capturing” in this paper, are compared to earlier parameters derived based on the entropy variables. In addition to being much simpler, the new shock-capturing parameters yield better shock quality in the test computations, with more substantial improvements seen for triangular elements.