Roe's Flux-difference Splitting Research Articles

Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids.

In this paper, a finite-volume discrete Boltzmann method based on a cell-centered scheme for inviscid compressible flows on unstructured grids is presented. In the new method, the equilibrium distribution functions are obtained from the circle function in two-dimensions (2D) and the spherical function in three-dimensions (3D). Moreover, the advective fluxes are evaluated by Roe's flux-difference splitting scheme, the gradients of the density and total energy distribution functions are computed with a least-squares method, and the Venkatakrishnan limiter is employed to prevent oscillations. To parallelize the method we use a graph-based partitioning approach that also guarantees the load balancing. The method is validated by seven benchmark problems: (a) a 2D flow pasting a bump, (b) a 2D Riemann problem, (c) a 2D flow passing the RAE2822 airfoil, (d) flows passing the NACA0012 airfoil, (e) 2D supersonic flows around a cylinder, (f) an explosion in a 3D box, and (g) a 3D flow around the ONERA M6 wing. The benchmark tests show that the results obtained by the proposed method match well with the published results, and the parallel numerical experiments show that the proposed parallel implementation has close to linear strong scalability, and parallel efficiencies of 95.31% and 94.56% are achieved for 2D and 3D problems on a supercomputer with up to 4800 processor cores, respectively.

An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on Hybrid Grids

AbstractA hybrid grid based second-order finite volume algorithm has been developed for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the effect caused by the numerical dissipation of the commonly used second order upwind schemes in implementing DES with unstructured computational fluid dynamics (CFD) algorithms, an improved second-order hybrid scheme is established through modifying the dissipation term of the standard Roe's flux-difference splitting scheme and the numerical dissipation of the scheme can be self-adapted according to the DES flow field information. By Fourier analysis, the dissipative and dispersive features of the new scheme are discussed. To validate the numerical method, DES formulations based on the two most popular background turbulence models, namely, the one equation Spalart-Allmaras (SA) turbulence model and the two equationk–ωShear Stress Transport model (SST), have been calibrated and tested with three typical numerical examples (decay of isotropic turbulence, NACA0021 airfoil at 60° incidence and 65° swept delta wing). Computational results indicate that the issue of numerical dissipation in implementing DES can be alleviated with the hybrid scheme, the resolution for turbulence structures is significantly improved and the corresponding solutions match the experimental data better. The results demonstrate the potentiality of the present DES solver for complex geometries.

A Comparative Study of Accuracy of Shock Capturing Schemes for Simulation of Shock/Acoustic Wave Interactions

We simulate transmission of a small-amplitude disturbance wave through a shock wave. Results of our numerical experiments performed with different high-order shock-capturing schemes show that the capability of a scheme to correctly predict the amplification of the disturbances crucially depends on the Riemann solver used in evaluation of numerical fluxes. Incorrectly high amplification rates are produced by the solvers resolving shock waves sharply, with no interior points in numerical profiles of steady shock waves. In particular, both the exact Riemann solver and the popular Roe flux difference splitting demonstrate such unphysical behavior. A possible explanation of such behavior is proposed. More dissipative solvers, such as the global Lax–Friedrichs splitting, produce transmission coefficients close to the predictions of linear theory.

A study of convective flux schemes for aerospace flows

This paper presents the effects of some convective flux computation schemes on boundary layer and shocked flow solutions. Second-order accurate centered and upwind convective flux computation schemes are discussed. The centered Jameson scheme, plus explicitly added artificial dissipation terms are considered. Three artificial dissipation models, namely a scalar and a matrix version of a switched model, and the CUSP scheme are available. Some implementation options regarding these methods are proposed and addressed in the paper. For the upwind option, the Roe flux-difference splitting scheme is used. The CUSP and Roe schemes require property reconstructions to achieve second-order accuracy in space. A multidimensional limited MUSCL interpolation method is used to perform property reconstruction. Extended multidimensional limiter formulation and implementation are here proposed and verified. Theoretical flow solutions are used in order to provide a representative testbed for the current study. It is observed that explicitly added artificial dissipation terms of the centered scheme may nonphysically modify the numerical solution, whereas upwind schemes seem to better represent the flow structure.

Numerical Simulation of Two-Dimensional Parallel Blade-Vortex Interactions Using Large Eddy Simulation

In this paper, an unsteady, two-dimensional, parallel blade-vortex interaction (BVI) has been calculated using large eddy simulation (LES) and the predicted pressures are compared with experimental measurements. The BVI is a primary source of rotor vibratory loading and rotorcraft noise. The LES technique provides a promising tool for obtaining the unsteady wall-pressure fields, aerodynamic coefficients, and acoustic functions. The dynamic sub-grid scale stress model, the low diffusion Roe flux-difference splitting scheme, and second order accuracy in the time and space are used. The simulation was performed for Mtip= 0.714 and Re = 2.2e6, on a case where the miss distance equals 0, which will be referred to as a strong BVI. The simulation was first run until a unsteady compressible fully-developed turbulent flow field was achieved for the non-lifting NACA 0012 airfoil. Then, the Scully vortex, which has been widely used, was added to the fully-developed turbulent flow field as a perturbation. In the present computation, the spatial resolution close to the surface was very good. The wall y+ was smaller than 0.6. The LES method can accurately preserve the characteristics of the vortex. The simulation which was low dissipative had a good agreement with the experimental results (C. Kitaplioglu, 1999). Moreover, the vortex had an important influence on the aerodynamic performance of the blade.

Roe의 Flux-difference Splitting 기법을 이용한 자유표면 유동 모사

A code is developed to simulate incompressible free surface flows using the Roe's flux-difference splitting scheme. An interface of two fluids is considered as a moving contact discontinuity. The continuities of pressure and normal velocity across the interface are enforced by the conservation law in the integral sense. The fluxes are computed using the Roe's flux-difference splitting scheme for two incompressible fluids. The interface can be identified based on the computed density distribution. However, no additional treatment is required along the interface during the whole computations. Complicated time evolution of the interface including topological change can be captured without any difficulties. The developed code is applied to simulate the Rayleigh-Taylor instability of two incompressible fluids in the density ratio of 7.2:1 and the broken dam problem of water-air. The present results are compared with other available results and good agreements are achieved for the both cases.

Experimental and numerical study of combusting solid propellant in an axisymmetric flow field with two moving boundaries

Simulation of compressible gases produced by combustion of solid propellant inside a cylinder with a moving boundary experimentally and numerically is considered. One-dimensional combustion model is used for solid propellant while, axisymmetrical Euler Equation can be used to simulate combusting gases. A finite volume upwind scheme based on the Roe's flux-difference splitting method is employed. VOF technique is applied to capture moving boundary. Test technique applied to this case measures chamber pressure vs. time with piezoelectric transducer and final velocity with two light screens. The numerical results compared well with experimental results.

폐쇄형 풍동 시험부내의 원형 실린더 유동에 대한 비정상 벽면효과 연구

풍동 시험부 비정상 벽면효과에 대한 연구를 위해 폐쇄형 시험부내의 원형 실린더 주위 유동장에 대한 수치적 연구를 수행하였다. 수치기법은 Roe의 flux-difference-splitting을 사용한 격자점 중심 유한체적법과 이중시간 전진 기법을 사용하는 내재적 시간적분법을 사용하였다. 계산 결과 폐쇄형 시험부에는 실린더 주위 비정상 유동장에 압력구배를 강화시켜 실린더의 양력 및 항력의 진폭을 크게 하고, 실린더 뒷전에서의 기저압력을 작게 하여 항력을 증가시키는 벽면효과가 있음을 확인하였다. 또한, 이러한 시험부 벽면은 실린더 와류 shedding 주파수를 커지게 하는 효과가 있다. 시험부 벽면에서의 압력은 벽면효과가 포함된 shedding 주파수를 기본으로 하는 고조파 현상을 보인다. For study on the unsteady wall interference effect, flows around a circular cylinder in closed test-section wind tunnels have been numerically investigated by solving compressible Navier-Stokes equations. The numerical scheme is based on a node-based finite-volume method with the Roe's flux-difference splitting and an implicit time-integration method coupled with dual time-step sub-iteration. The computed results showed that the unsteady pressure gradient over the cylinder is enhanced by the wall interference, and as a result the fluctuations of lift and drag are augmented. The drag is further increased because of the lower base pressure. The vortex shedding frequency is also increased by the wall interference. The pressure on the test section wall shows the harmonics having the shedding frequency contained in the wall effect.

A parallel unstructured dynamic mesh adaptation algorithm for 3-D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three-dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell-centred finite-volume method and the Roe's flux-difference splitting. Second-order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time-dependent solutions was enhanced by adopting an h-refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed-up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock-tube problem. The flows around oscillating NACA0012 wing and F-5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.

Modified Multidimensional Dissipation Scheme on Unstructured Meshes for High-speed Compressible Flow Analysis

A Roe's flux-difference splitting scheme, combining with the entropy fix method according to Van Leer et al., and the H-correction entropy fix method by Pandolfi and D'Ambrosio, is proposed. The presented scheme eliminates unphysical flow behaviors such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the expansion shock generated from flow over a forward facing step. The proposed scheme is further extended to obtain high-order spatial and temporal solution accuracy. The scheme is, in addition, combined with an adaptive meshing technique that generates unstructured triangular meshes to resemble the flow phenomena for reducing computational effort. The entire procedure is evaluated by solving several benchmarks as well as complex steady-state and transient high-speed compressible flow problems.

Implicit Gas-Kinetic Bhatnagar-Gross-Krook Scheme for Compressible Flows

A gas-kinetic scheme based on the Bhatnagar‐Gross‐Krook (BGK) model is developed for two-dimensional compressible inviscid flowfields. The BGK scheme is an approximate Riemann solver that uses the collisional Boltzmann equation as the governing equation for flow evolutions. For efficient computations, particle distribution functions in the general solution of the BGK model are simplified and used for the flow simulations. High-order accuracy is achieved via the reconstruction of flow variables using the MUSCL interpolation technique. For steadystate problems, the approximate factorization‐alternating direction implicit implicit time integration method is adopted, which is preferable to a multistage Runge‐Kutta method. To investigate the computational characteristics of BGK scheme in detail, it has been applied to various two-dimensional numerical experiments. The results, compared with some typical schemes, such as the Steger‐Warming flux vector splitting and Roe flux difference splitting, as well as exact solutions, are found to be robust and accurate, and to have high resolution at discontinuities such as shock waves.

비정렬 동적격자를 이용한 블레이드-와류 간섭에 따른 공탄성 변위예측

본 연구에서는 이차원에서 비정상 비점성 유동해석을 위한 비정렬 동적 편자 기법을 개발하였다. 유동해석 기법은 시간에 대해 2차의 정확도를 갖는 내재적인 시간적분법을 사용하였으며, 격자중심의 유한 체적법과 Roe의 풍상차분법을 이용하여 공간에 대한 차분화를 하였다. 시간과 공간에 대한 정확도를 증가시키기 위해서는 해에 따라 원하는 위치에 격자점들을 임의로 추가할 수 있는 비정상 동적 적응격자 기법을 사용하였다. 이를 이용하여 이차원의 2자유도를 갖는 스프링 에어포일 시스템의 와류와의 간섭현상에 따른 공탄성적 변위를 예측하였다. A two-dimensional unsteady, inviscid flow solver has been developed for the simulation of airfoil-vortex interactions on unstructured dynamically adapted meshes. The Euler solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with the Roe's flux-difference splitting. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of both spatial and temporal variations of the flow field. The effect of vortex interaction on the aeroelastic displacement of an airfoil attached to the idealized two degree-of-freedom spring system is investigated.

3-D Multiple-level simulation of free surface flows

A 3-D multiple-layer model to describe and compute free surface flows has been applied to rivers and seashore parts. Vertical convective fluxes and shear stress at the interface are kept in the x-y momentum equation. The hydrostatic pressure of the top layer, is taken into account. The flow is calculated via a finite volume numerical scheme. The solver is based on Roe's Flux Difference-Splitting (FDS) method. The spatial discretization uses triangular elements as the basic control-volume cells. Data obtained from references, analytical solutions and real configurations have been used to evaluate the performance of the proposed method. The numerical scheme has been shown to be capable of predicting the behavior of Ekman's classical surface layer or of a simple tide wave fixed on an open boundary of a rectangular domain, for example. The evolution of a tidal constituent on the St.Lawrence estuary is also presented.

A numerical study of an unsteady laminar flow in a doubly constricted 3D vessel

AbstractUnsteady flow dynamics in doubly constricted 3D vessels have been investigated under pulsatile flow conditions for a full cycle of period T. The coupled non‐linear partial differential equations governing the mass and momentum of a viscous incompressible fluid has been numerically analyzed by a time accurate Finite Volume Scheme in an implicit Euler time marching setting. Roe's flux difference splitting of non‐linear terms and the pseudo‐compressibility technique employed in the current numerical scheme makes it robust both in space and time. Computational experiments are carried out to assess the influence of Reynolds' number and the spacing between two mild constrictions on the pressure drop across the constrictions. The study reveals that the pressure drop across a series of mild constrictions can get physiologically critical and is also found to be sensitive both to the spacing between the constrictions and the oscillatory nature of the inflow profile. The flow separation zone on the downstream constriction is seen to detach from the diverging wall of the constriction leading to vortex shedding with 3D features earlier than that on the wall in the spacing between the two constrictions. Copyright © 2002 John Wiley & Sons, Ltd.

A Parallel 3D Unsteady Flow Analysis in an Asymmetrically Constricted Vessel

In this study, parallel computation of unsteady incompressible flow in an asymmetrically constricted 3D vessel has been presented. A time accurate cell centered finite volume method (FVM) in conjunction with pseudo-compressibility technique and Roe's flux difference splitting of nonlinear terms has been employed for solving the Navier-Stokes (NS) equations on the multiple instruction multiple data (MIMD) machine VPP700. The influence of Reynolds' number ( Re ) and the Strouhal number ( St ) on flow dynamic factors like wall pressure (WP), wall shear stress (WSS), central axis velocity (CAV), etc., have been analyzed. Three-dimensional (3D) features in the formation and detachment of separation zones, which are sensitive to both Re and St have been noticed on the diverging wall of the constriction.

A parallel 3D unsteady incompressible flow solver on VPP700

In this study, MIMD parallel computation of coupled non-linear partial differential equations governing the mass and momentum conservation of a 3D viscous incompressible unsteady flow on distributed memory-based vector parallel processing system (VPP700) under UXP/VPPF90 has been discussed. The parallel numerical implementation is based on a time accurate cell centered finite volume method in conjunction with Roe's flux difference splitting of non-linear terms and pseudo-compressibility technique. Data flow graphs necessary for combined parallelization and vectorization have been generated to provide a logical basis for parallel code development. The performance of the parallel implementation measured in terms of speedup and efficiency factors is found to be good. The parallel code is tested on three test problems of practical significance.

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.

A Flux-Splitting Scheme for Compressible and Incompressible Flows

In the present work, a recently proposed flux-splitting scheme suitable for compressible flow is extended to incompressible flows. Appropriate dissipation terms for both incompressible and compressible flows are determined by expanding the Roe flux-difference splitting in terms of Mach number. Analysis of the dissipation terms in this form for both the Roe scheme and the basic (current) scheme leads to the incorporation of certain terms into the basic scheme to establish the transition from compressible flow toward the incompressible limit. Using the proper terms in the dissipation formulation, convergence rates for airfoil flows became nearly independent of the freestream Mach number.

A parallel, finite-volume algorithm for large-eddy simulation of turbulent flows

A parallel, finite-volume algorithm has been developed for large-eddy simulation (LES) of compressible turbulent flows. This algorithm includes piecewise linear least-square reconstruction, trilinear finite-element interpolation, Roe flux-difference splitting (FDS), and second-order MacCormack time marching. A systematic and consistent means of evaluating the surface and volume integrals of the control volume is described. Parallel implementation is done using the message-passing programming model. To validate the numerical method for turbulence simulation, LES of fully developed turbulent flow in a square duct is performed for a Reynolds number of 320 based on the average friction velocity and the hydraulic diameter of the duct. Direct numerical simulation (DNS) results are available for this test case, and the accuracy of this algorithm for turbulence simulations can be ascertained by comparing the LES solutions with the DNS results. For the first time, a finite volume method with Roe FDS was used for LES of turbulent flow in a square duct, and the effects of grid resolution, upwind numerical dissipation, and subgrid-scale dissipation on the accuracy of the LES are examined. Comparison with DNS results shows that the standard Roe FDS adversely affects the accuracy of the turbulence simulation. For accurate turbulence simulations, only 3–5% of the standard Roe FDS dissipation is needed.

Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier–Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourth-order differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.