Roe's Flux Difference Research Articles

An accurate, robust and efficient convection-pressure flux splitting scheme for compressible Euler flows

The popular Roe's flux difference splitting scheme has high resolution for various types of discontinuities but it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock waves, such as the post-shock oscillation of the slowly moving shock waves and the carbuncle phenomenon of the multidimensional strong shock waves. In the present work, a simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed. With application of the Toro-Vázquez splitting procedure, the flux of the 3D Euler equations is split into the convection part and the pressure part. Due to having a complete set of linearly independent eigenvectors, the pressure subsystem can be solved by the conventional flux difference splitting method. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and thus is solved by the flux difference splitting scheme based on the generalized eigenvectors [1]. In addition, a strategy that combines the THINC reconstruction with the BVD algorithm is employed on the numerical dissipation term of the convection flux to further improve the resolution for contact discontinuities. A comprehensive range of numerical simulations for classical 1D, 2D and 3D benchmark test problems demonstrate the good performance of the proposed scheme.

High-order simulation solving Navier-Stokes equations with Spalart–Allmaras turbulence model

The goal of this paper is to implement an accurate and robust solver for compressible Navier-Stokes equations coupled with the Spalart–Allmaras model, which possesses the capability of shock-capturing and predication of boundary layer and separated flow. In a given stencil width, a WENO-Z scheme equipped with Roe flux difference split method is used to calculate the inviscid flux, and central differencing scheme for the viscous terms are employed. The explicit Runge-Kutta is adopted for the temporal discretization. The simulation results of selected cases are given to verify the validation of the solver.

A simple extension of Roe's scheme for real gases

The purpose of this paper is to develop a highly accurate numerical algorithm to model real gas flows in local thermodynamic equilibrium (LTE). The Euler equations are solved using a finite volume method based on Roe's flux difference splitting scheme including real gas effects. A novel algorithm is proposed to calculate the Jacobian matrix which satisfies the flux difference splitting exactly in the average state for a general equation of state. This algorithm increases the robustness and accuracy of the method, especially around the contact discontinuities and shock waves where the gas properties jump appreciably. The results are compared with an exact solution of the Riemann problem for the shock tube which considers the real gas effects. In addition, the method is applied to a blunt cone to illustrate the capability of the proposed extension in solving two dimensional flows.

Numerical Study of Transonic Flows Using Various Turbulence Models

In this paper, validation and assessment of various turbulence models are performed in transonic flows, including an algebraic Baldwin Lomax model, Spalart Allmaras one equation, and two equation (k-e, k-ω and SST k-ω) turbulence models. The NACA 0012 airfoil has been chosen for the turbulence model validation studies. The three test cases selected here included both attached and separated transonic flows. This study shows that the five turbulence models provide satisfactory results for transonic attached flows. However, all models, fail to predict the location of the shock correctly when a strong shock and a shock induced flows separation are present. Compared to other models, it has been shown that the SST k-ω model is the most robust model in the prediction of lift coefficient for all cases. Computed results are performed with the CFD-FASTRAN code by using the fully implicit scheme for time integration, and the upwind Roe flux difference splitting scheme for space discretization augmented by a high order Min-Mod limiter to damp spurious oscillations and to allow for sharp shock resolution

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

SummaryIn this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high‐order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high‐order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock‐capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high‐order weighted compact nonlinear scheme is not only robust for shock‐capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.

NUMERICAL INVESTIGATION OF THE UNSTART SUPPRESSION IN A SUPERSONIC AIR INTAKE

The starting behavior of a supersonic axisymmetric mixed compression air intake was numerically investigated. The code solves Reynolds-averaged Navier–Stokes equations using an explicit finite volume method in a structured grid by the Roe flux difference splitting scheme. Further, it uses the Baldwin–Lomax algebraic model to compute the turbulent viscosity coefficient. The correct method of surveying the intake starting problem and effects of several geometrical parameters such as: intake throat area, cowl lip roundness and spike surface curvature upstream of the throat on the starting and performance of the intake were studied. Results showed that correct combinations of the mentioned parameters can suppress the intake unstart problem and in addition can prevent the reduction of the intake efficiency.

Multidimensional Flux Difference Splitting Schemes

A novel alteration to the Cauchy–Kowalevski procedure is here presented to obtain essentially monotonic solutions for multidimensional flows. It is argued that this can be accomplished by splitting the cross-derivative terms among the several dimensions, such that the coefficient of the cross derivatives remains small compared to the coefficient of the normal derivatives. The approach naturally lends itself to extending the Roe flux difference splitting scheme to multiple dimensions and is advantaged over previous Cauchy–Kowalevski-based methods by yielding a solution free of spurious oscillations in the vicinity of oblique shock waves. Several test cases ranging from low-speed subsonic flows in channels to hypersonic flows over ramp injectors indicate that the proposed genuinely multidimensional method generally achieves a twofold or more increase in resolution along each dimension over the dimensionally split Roe scheme while retaining its appealing attributes: the scheme has a compact three-node-bandwidth stencil, is a finite volume flux function, yields essentially monotonic solutions, introduces minimal dissipation within viscous layers, and is written in general matrix form. Although the method proposed is first-order accurate, it offers a resolution as high or higher than the dimensionally split second-order total-variation-diminishing schemes for many problems of interest and is expected to surpass significantly the latter when extended to second-order accuracy.

A flux vector splitting scheme for low Mach number flows in preconditioning method

The stiffness of Euler equations system increases, with the decreasing of Mach number, which results in poor convergence when solving steady flow problems numerically. In order to accelerate the convergence, a preconditioning method is proposed. A flux vector splitting scheme is derived from the classic Steger–Warming scheme and the Roe flux difference splitting scheme, and preconditioned systems developed for the two schemes, separately. Numerical results are presented and it is shown that flux vector splitting scheme can be preconditioned as successfully as flux difference splitting, and exhibits similar convergence acceleration and accurate solution.

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.

Positivity-preserving flux difference splitting schemes

A positivity-preserving variant of the Roe flux difference splitting method is here proposed. Positivity-preservation is attained by modifying the Roe scheme such that the coefficients of the discretization equation become positive, with a coefficient considered positive if all its eigenvalues are positive and if its eigenvectors correspond to those of the flux Jacobian. Because the modification does not alter the wave speeds at the interface, the appealing attributes of the Roe flux difference splitting schemes are retained, such as high-resolution capture of discontinuous waves, low amount of artificial dissipation within viscous layers, and ease of convergence to steady-state. The proposed flux function is advantaged over previous positivity-preserving variants of the Roe method by being written in general matrix form and hence by being readily deployable to arbitrary systems of conservation laws. The stencils are extended to second-order accuracy through a newly-derived positivity-preserving total-variation-diminishing limiting process that is applied to the characteristic variables and that yields positive coefficients. Also derived is a positivity-preserving restriction on the time step for flux difference splitting schemes that is shown to depart significantly from the CFL condition in regions with high property gradients.

An unstructured grid implementation of high-order spectral finite volume schemes

The present work implements the spectral finite volume scheme in a cell centered finite volume context for unstructured meshes. The 2-D Euler equations are considered to represent the flows of interest. The spatial discretization scheme is developed to achieve high resolution and computational efficiency for flow problems governed by hyperbolic conservation laws, including flow discontinuities. Such discontinuities are mainly shock waves in the aerodynamic studies of interest in the present paper. The entire reconstruction process is described in detail for the 2nd to 4th order schemes. Roe's flux difference splitting method is used as the numerical Riemann solver. Several applications are performed in order to assess the method capability compared to data available in the literature. The results obtained with the present method are also compared to those of essentially nonoscillatory and weighted essentially non-oscillatory high-order schemes. There is a good agreement with the comparison data and efficiency improvements have been observed.

Implicit High-Order Spectral Finite Volume Method for Inviscid Compressible Flows

The present work has the objective of demonstrating the capabilities of a spectral finite volume scheme implemented in a cell-centered finite volume context for unstructured meshes. The two-dimensional Euler equations are considered to represent the flows of interest. The spatial discretization scheme is developed to achieve high resolution for flow problems governed by hyperbolic conservation laws. Roe's flux difference splitting method is used as the numerical approximate Riemann solver. Several applications are performed in order to assess the method capability compared to data available in the literature and also compared to an weighted essentially nonoscillatory scheme. There is good agreement with the comparison data, and efficiency improvements over the weighted essentially nonoscillatory method are observed. The features of the present methodology include an implicit time-marching algorithm; second-, third-, and fourth-order spatial resolution; exact high-order domain boundary representation; and a hierarchical moment limiter to treat flow solution discontinuities.

Multidimensional Dissipation Technique for Roe's Flux-Difference Splitting Scheme on Triangular Grids

A multidimensional dissipation technique is implemented on the Roe's flux difference splitting scheme to avoid the numerical shock instability that may occur in compressible flow solutions. The damping characteristic of the proposed technique is presented through a linear perturbation analysis on the problem of a moving shock along an odd-even grid perturbation. The performance of the proposed technique is studied using well-known problems that exhibit the numerical shock instability. The technique is further extended to achieve higher-order solution accuracy and evaluated by several benchmark test cases.

Numerical Study of Heat Transfer Around the Small Scale Airfoil Using Various Turbulence Models

A numerical investigation of low-Reynolds number flows with thermal effect around the MAV airfoils using various turbulence models, including an algebraic Baldwin-Lomax model, Spalart-Allmaras one equation, and two equation (k-ω and SST-k-ω) turbulence models, is presented. First, the thermal effect on the aerodynamic efficiency is studied for flow around a rectangular MAV wing, based on the NACA0012 airfoil section at low-aspect ratio (AR = 2) and an angle of attack equal to 0°. Second, details of the thermal effect are limited to the two-dimensional NACA0012 airfoil with chord length of 3.81 cm. This study shows that the improvement of aerodynamic efficiency (increase lift and reduce drag) is achieved by the generation of a temperature difference between extrados and intrados of the airfoil (by cooling the upper surface and heating the lower surface). The numerical results obtained with various turbulence models are in good agreement with experiment data, except the k-ω turbulence model. These results are performed with the CFD-FASTRAN code, using the fully implicit scheme for time integration and the upwind Roe flux difference splitting scheme for space discretization augmented by a high order Osher-Chakravarthy limiter.

High-speed digital visualization and high-frequency automated shock tracking in supersonic flows

A low-cost, robust, versatile digital shadowgraph visualization system is presented that provides a fast nonintrusive diagnostic for unsteady high-speed flows. The technique is particularly designed for real-time automated tracking of shock positions, enabling high-speed active shock control. The visualization system is based on a high-intensity white LED light source combined with a CMOS-imaging sensor, providing the system with three modes of operation: (1) high-resolution overall instantaneous visualization; (2) high-resolution visualization showing spatial- temporal variations in the flow field, allowing direct identification of areas where changes occur; (3) adjustable windowed visualization at reduced resolution at high frame rate (currently up to 980 Hz) . Experimental results are presented together with numerical simulations based on the high-accuracy NTS Navier-Stokes solver and Roe's flux difference splitting method. The flow studied is an adjustable underexpanded jet flow coming from a nozzle that is placed in a counterflowing Mach-2 flow. The interaction of the two flows results in a complex shock and expansion pattern, providing a challenging configuration for the numerical flow solver. By modulating the jet, high-frequency changes are induced in the interaction pattern, allowing simulation of shock movement in a supersonic inlet. Good correspondence between measured and numerical shock position and angle isfound. (C) 2008 Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/1.2992621]

Assessment of Rotor Hover Performance Using a Node-based Flow Solver

A three-dimensional viscous flow solver has been developed for the prediction of the aerodynamic performance of hovering helicopter rotor blades using unstructured hybrid meshes. The flow solver utilized a vertex-centered finite-volume scheme that is based on the Roe's flux-difference splitting with an implicit Jacobi/Gauss-Seidel time integration. The eddy viscosity are estimated by the Spalart-Allmaras one-equation turbulence model. Calculations were performed at three operating conditions with varying tip Mach number and collective pitch setting for the Caradonna-Tung rotor in hover. Additional computations are made for the UH-60A rotor in hover. Reasonable agreements were obtained between the present results and the experiment in both blade loading and overall rotor performance. It was demonstrated that the present vertex-centered flow solver is an efficient and accurate tool for the assessment of rotor performance in hover.

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.

동압 계수의 불연속성을 이용한 내면파의 수치해석

Internal waves are computed using a ghost fluid method on an unstructured grid. Discontinuities in density and dynamic pressure are captured in one cell without smearing or oscillations along a multimaterial interface. A time-accurate incompressible Navier-Stokes/Euler solver is developed based on a three-point backward difference formula for the physical time marching. Artificial compressibility is introduced with respect to pseudotime and an implicit method is used for the pseudotime iteration. To track evolution of an interface, a level set function is coupled with the governing equations. Roe's flux difference splitting method is used to calculate numerical fluxes of the coupled equations. To get higher order accuracy, dependent variables are reconstructed based on gradients which are calculated using Gauss theorem. For each edge crossing an interface, dynamic pressure is assigned for a ghost node to enforce the continuity of total pressure along the interface. Solitary internal waves are computed and the results are compared with other computational and experimental results.

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.