Roe Method Research Articles

圧縮性流れの陰的非構造解法における前処理付反復法の比較

Preconditioned iterative methods in an implicit unstructured grid solver of compressible flows are compared. The three-dimensional Euler and Navier-Stokes equations are discretized by the edge-based finite volume method. The artificial dissipation scheme for the inviscid numerical fluxes is developed for the unstructured grid from Jameson's scheme for the structured grid. The following methods are compared to the supersonic and transonic inviscid flows and subsonic viscous laminar and turbulent flows : the central difference methods with the scalar and matrix dissipation scheme and Roe's upwind method with the MUSCL sheme for the inviscid fluxes ; the scalar and matrix dissipation method for the left-hand side operator in the implicit scheme ; the ILU, DILU and SGS factorizations for the preconditioner ; and GMRES and Bi-CGSTAB methods for the linear system solver. The results show that a combination of the matrix dissipation in the inviscid fluxes and the scalar dissipation in the left-hand side works better than the other shemes. In the preconditioned iterative methods, a combination of DILU or SGS and the Bi-CGSTAB is recommended for the aspect of required memory.

TWO‐DIMENSIONAL SOLUTIONS OF MHD EQUATIONS WITH AN ADAPTED ROE METHOD

TWO‐DIMENSIONAL SOLUTIONS OF MHD EQUATIONS WITH AN ADAPTED ROE METHOD

TWO-DIMENSIONAL SOLUTIONS OF MHD EQUATIONS WITH AN ADAPTED ROE METHOD

A higher-order Godunov method for two-dimensional solutions of the ideal MHD (magnetohydrodynamic) equations is presented. The method utilizes the finite volume approach with quadrilateral cells. In Section 2 the MHD equations (including flux and source terms) in conservative form are given. The momentum flux is rearranged such that while a source vector is produced, the eigenstructure of the Jacobian matrix does not change. This rearrangement allows a full Roe averaging of the density, velocity and pressure for any value of adiabatic index. Full Roe averaging for the magnetic field is possible only when the normal gradient of the magnetic field is negligible ; otherwise an arithmetic averaging can be used. This new procedure to get Roe-averaged MHD fields at the interfaces between left and right states has been presented by Aslan. This section also includes the shock structure and an eigensystem for MHD problems. The eigenvalues, right eigenvectors and wave strengths for MHD are given in detail to provide the reader with a full description. The second-order, limited finite volume approach which utilizes quadrilateral cells is given in full detail in Section 3.

On the Use of Polycrystal and Individual Orientation Texture Analysis Methods for BCC Materials

The reliability of the harmonic method of the ODF calculation from X‐ray pole figures was estimated for cubic symmetry materials. For this purpose simulated textures with a preset scattering value of the components were used. Some examples using the Roe method to the study of the secondary recrystallization process in the Fe‐3% Si alloy are given. The capabilities of the etch‐pits methods as the simplest discrete method used to determine the orientation density are discussed.

PARALLEL COMPUTATION OF THE 2-D NAVIER-STOKES FLOWFIELD OF A PITCHING AIRFOIL

SUMMARY An unsteady, compressible, two dimensional Navier-Stokes solver has been developed for the Connection Machine (CM-2 and CM-5). The solver has been applied to the computations of subsonic viscous flow past a pitching airfoil at Rec = 1044, (based on airfoil chord length) M∞ = 0.2, non-dimensional pitching rate Ωυ + = 0.2 and 0.4. The algorithm for solving the Navier-Stokes equations is based on an unstructured grid of triangles. Roe's flux difference split method is employed for the inviseid fluxes. A discrete representation of Gauss' theorem is used for the viscous fluxes and heat transfer. A four stage modified Runge-Kutta algorithm is employed for the temporal integration. The algorithm is second order accurate in space and time. The implementation of the algorilhm on the Connection Machine and the investigation of the incipient boundary layer separation for a pitching airfoil are presented. The investigation of the effect of pitching rate on incipient boundary layer separation for a pitching ai...

A fully implicit Navier-Stokes algorithm using an unstructured grid and flux difference splitting

An implicit algorithm is developed for the two-dimensional, compressible, laminar Navier-Stokes equations using an unstructured grid of triangles. A sell-centered data structure is employed with the flow variables stored at the centroids of the triangles. The algorithm is based on Roe's flux difference split method for the inviscid fluxes, and a discrete representation of the viscous fluxes and heat transfer using Gauss' Theorem. Linear reconstruction of the flow variables to the cell faces, employed for the inviscid terms, provides second-order spatial accuracy. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, trapezoidal or 3-point backward differencing. The complete, exact Jacobian of the inviscid and viscous terms is derived. The algorithm is applied to the Riemann Shock tube problem, a supersonic laminar boundary layer on a flat plate, and subsonic viscous flow past an NACA0012 airfoil. Results are in excellent agreement with theory and previous computations.

Flux difference splittings and limiters for the resolution of contact discontinuities

Harten, Lax, and van Leer developed a theory of approximate Riemann solutions and used it to derive numerical methods for systems of conservation laws. In the first part of this paper, we reformulate this theory so that it can be used to compare various flux difference splitting formulae. In particular we show how to compare the resolution of contact discontinuities by various methods with that of the Roe method. Since the Roe method can resolve an isolated steady contact perfectly, it is a good standard for comparison. In the second part of the paper, we show that Harten's subcell resolution method is the ultimate compressive limiter. We discuss the differences between compressive and noncompressive limiters and show why it is important to choose the limiter to suit the problem at hand.

High-resolution, nonoscillatory schemes for unsteady compressible flows

High-resolution explicit finite difference nonoscillatory shock-capturing schemes based on Harten's essentially nonoscillatory interpolation using reconstruction via primitive function approach with N=3 are described for simulating unsteady compressible flows. The extension to nonlinear system is done by using Roe's method, which permits the use of different scalar schemes for different characteristic fields. For multidimensional problems, Strang-type dimensional splitting is adopted

Numerical simulation of macroscopic continuum traffic models

This paper presents the derived Roe's flux difference splitting method for Payne's formulation of the macroscopic model. In recent years, this finite difference method has generated much interest and been used successfully in gas dynamics. Payne's equations are actually those of the isentropic gas having constant speed of sound and in homogeneous terms on the right hand side. Our motivation for deriving Roe's flux difference splitting algorithm for Payne's model and also highlighting the scalar equivalent of Roe's method (the Murman scheme) for the Lighthill-Whitham (LW) model stems from the reports of numerical simulation difficulties pertaining to two models. The numerical schemes were used on Payne's and LW models to simulate three traffic scenarios with satisfactory results. We were able to simulate the propagation of congested density upstream in one freeway bottleneck scenario at a finer discretization. Another scenario did not manage to obtain realistic results with Payne's model, we did.

Numerical simulation of the gas flow in a circuit‐breaker

AbstractA computational methodology for the solution of unsteady two‐dimensional/axisymmetric Euler equations within geometries with moving boundaries is presented. The flow simulation is carried out by applying a finite‐volume method which makes use of a Lagrangian‐Eulerian version of Roe's approximate Riemann solver. The domain discretization is handled via unstructured triangular grids. Grid adaptation is applied on the basis of geometric and physical requirements. The importance of the implicit treatment of the space conservation laws, based on geometric analysis, is evoked. The procedure for reconstructing Roe's method for moving meshes is described and validated. Finally, the ability of the method for the prediction of the transient flow in a circuit‐breaker during its opening phase is illustrated.

Numerical Analyses of Reflection and Diffraction of Weak Shock Wave.

In order to investigate the accuracy of finite difference methods for weak shock waves, comparisons for the reflection and the diffraction of weak shock waves around wedges and corners between simulations and analyses are made. Analytical solutions are obtained for the wave equations by employing the conical flow method of Keller and Blank. Finite difference simulations are conducted by applying such finite difference methods as the total variation diminishing (TVD) method, the flux-corrected transport (FCT) method, the flux vector splitting (FVS) method and the method of Roe to numerically solve the Euler equations. Computed flow fields by the finite difference methods are almost equivalent to each other in the present investigation. The approximate solutions obtained from finite difference calculations deviate from the analytical exact solutions near the apexes of the wedges or the corners that are singular points in grid systems. The weaker the shock waves passing through the apexes, the more noticeable numerical errors originating from the singular points. Among the finite difference schemes examined in the present investigation, the TVD method is the most reliable because of its stability and smooth results. The FVS method is almost comparable to the TVD method. In the results of the FCT method, small noises are observed to be superposed. In the results of Roe's method, shock waves are simulated most sharply, but instabilities near the singular points are observed to be great.

A finite-volume method for the Euler equations on arbitrary lagrangian-eulerian grids

This study presents a finite-volume method for the solution of 2-D/axisymmetric Euler equations using triangular moving grids. The flow simulation is carried out using Roe's approximate Riemann solver. The importance of the implicit treatment of the space conversation laws, based on geometric analysis, is emphasized. The procedure for reconstructing Roe's method for moving meshes is described and validated.

Numerical studies of a detonation analogue

Abstract As an aid to deciding a computational strategy for problems involving strong detonation waves, we have applied a variety of numerical techniques to a mathematical problem devised by Fickett, which exhibits many of the essential computational difficulties and possesses analytical solution corresponding to overdriven and underdriven reacting flows. Classical shock capturing methods such as those of MacCormack and Godunov, do not produce acceptable solutions. Better solutions can be provided by Flux Difference Methods (Roe's method) and by Random Choice Methods (still in one dimension), especially by using a new variant that yields second order accuracy. Adaptive gridding techniques for these methods are currently being investigated with encouraging preliminary results.

An upwind differencing scheme for the equations of ideal magnetohydrodynamics

Recently, upwind differencing schemes have become very popular for solving hyperbolic partial differential equations, especially when discontinuities exist in the solutions. Among many upwind schemes successfully applied to the problems in gas dynamics, Roe's method stands out for its relative simplicity and clarity of the underlying physical model. In this paper, an upwind differencing scheme of Roe-type for the MHD equations is constructed. In each computational cell, the problem is first linearized around some averaged state which preserves the flux differences. Then the solution is advanced in time by computing the wave contributions to the flux at the cell interfaces. One crucial task of the linearization procedure is the construction of a Roe matrix. For the special case γ = 2, a Roe matrix in the form of a mean value Jacobian is found, and for the general case, a simple averaging procedure is introduced. All other necessary ingredients of the construction, which include eigenvalues, and a complete set of right eigenvectors of the Roe matrix and decomposition coefficients are presented. As a numerical example, we chose a coplanar MHD Riemann problem. The problem is solved by the newly constructed second-order upwind scheme as well as by the Lax-Friedrichs, the Lax-Wendroff, and the flux-corrected transport schemes. The results demonstrate several advantages of the upwind scheme. In this paper, we also show that the MHD equations are nonconvex. This is a contrast to the general belief that the fast and slow waves are like sound waves in the Euler equations. As a consequence, the wave structure becomes more complicated; for example, compound waves consisting of a shock and attached to it a rarefaction wave of the same family can exist in MHD.

A MICROMODIFICATION OF THE SMITH AND ROE METHOD FOR THE DETERMINATION OF AMYLASE IN BODY FLUIDS

A MICROMODIFICATION OF THE SMITH AND ROE METHOD FOR THE DETERMINATION OF AMYLASE IN BODY FLUIDS

Studie, s on the Determination of Vitamin C in the Foods by 2, 4-Dinitrophyenylhydrazine

Some problems on the estimation of Vitamin C in foods by Roe's method were studied. The disturbing action of various kinds of pigments contained in vegetables and alkali-treated fructose solution produced by the heating of foods must be considered in this procedure. Disturbing action of pigments as flavon-series and antocyan-series can be diminished or eliminated through the diluting of test solutions. The available range of dilution of the test solutions for this method are 10 and 100 in cases of egg-plant and beefstak-plant respectively.Though alkali-treated fructose solution also presents somewhat disturbing action, the degree of this disturbance can be eliminated from the relation between the wave-length and their extinction coefficients applied to colorimetrie estimation.The Roe's method of Vitamin C estimation can be sufficiently applied to vegetables and heated foods after these considerations.Vitamin C contents in several kinds of foods were estimated by the Roe's method.

Determination of Inulin by Means of Resorcinol

SummaryA simplified method, based on Roe's resorcinol method and avoiding the use of yeast, is described for the determination of inulin in plasma and urine. The method has been applied under clinical conditions to clearance determinations in patients with renal disease, and shown to yield results in good agreement with the diphenylamine method.

Origin and Site of Formation of Fructose in the FŒtal Sheep

Bacon and Bell1 isolated and identified fructose in fœtal sheep's blood. This was the first conclusive proof of its existence prenatally. They also showed it disappears from the blood of the new-born lamb within three days of birth. During 1948 we performed experiments designed to throw some light on the mode of formation and site of origin of this fructose. Caesarean section was performed under spinal anaesthetic supplemented by pentothal (which we find to be without effect on the blood sugar). The ewe was immersed in saline at 38° C. into which the fœtus was delivered. Blood samples were taken from the maternal dorsalis pedis artery and the umbilical vein. The total reducing substances and the fructose were estimated by the Somogyi–Nelson2 method and the Cole3 modification of the Roe's method respectively. The difference was taken as an approximate measure of the glucose.