Roe Scheme Research Articles

An investigation of the unstable phenomena of natural convection in parallel square plates by multi-GPU implementation

ABSTRACTUnstable phenomena induced by natural convection of parallel square plates are investigated numerically. The geometry belongs to the open boundary problem, and the compressibility of fluids is taken into consideration. The absorbing boundary condition and modified local one-dimensional inviscid relation method are adopted for open boundaries. Numerical methods of the Roe scheme, preconditioning, and dual time stepping matching the data-parallel lower-upper relaxation method are coordinated with multi-GPU implementation. Unstable phenomena of natural convection caused by the buoyancy force can be observed. Present results have good agreement with the experimental results in the overlap range of the existing study.

Large Time Step Roe scheme for a common 1D two-fluid model

We present the Large Time Step (LTS) extension of the Roe scheme and apply it to a standard two-fluid model. Herein, LTS denotes a class of explicit methods that are not limited by the CFL (Courant–Friedrichs–Lewy) condition, allowing us to use very large time steps compared to standard explicit methods. The LTS method was originally developed in the nineteen eighties (LeVeque, 1985), where the Godunov scheme was extended to the LTS Godunov scheme. In the present work, the relaxation of the CFL condition is achieved by increasing the domain of dependence. This might lead to difficulties when it comes to boundary and source terms treatment. We address and discuss these difficulties and propose different ways to treat them. For a shock tube test case, where there are neither source terms nor difficulties associated with the boundaries, the method increases both accuracy and efficiency. For a water faucet test case that includes a source term, the method increases the efficiency, while the accuracy strongly depends on the appropriate treatment of boundary conditions and source terms.

Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

Effects of Reducing Convective Acceleration Terms in Modelling Supercritical and Transcritical Flow Conditions

Modelling floods and flood-related disasters has become priority for many researchers and practitioners. Currently, there are several options that can be used for modelling floods in urban areas and the present work attempts to investigate effectiveness of different model formulations in modelling supercritical and transcritical flow conditions. In our work, we use the following three methods for modelling one-dimensional (1D) flows: the MIKE 11 flow model, Kutija’s method, and the Roe scheme. We use two methods for modelling two-dimensional (2D) flows: the MIKE21 flow model and a non-inertia 2D model. Apart from the MIKE11 and MIKE21 models, the code for all other models was developed and used for the purposes of the present work. The performance of the models was evaluated using hypothetical case studies with the intention of representing some configurations that can be found in urban floodplains. The present work does not go into the assessment of these models in modelling various topographical features that may be found on urban floodplains, but rather focuses on how they perform in simulating supercritical and transcritical flows. The overall findings are that the simplified models which ignore convective acceleration terms (CATs) in the momentum equations may be effectively used to model urban flood plains without a significant loss of accuracy.

Role of the momentum interpolation mechanism of the Roe scheme in shock instability

SummaryThe shock instability phenomenon is a well‐known problem for hypersonic flow computation by the shock‐capturing Roe scheme. The pressure checkerboard is another well‐known problem for low‐Mach‐number flow computation. The momentum interpolation method (MIM) is necessary for low‐Mach‐number flows to suppress the pressure checkerboard problem, and the pressure‐difference‐driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high‐Mach‐number flows or low‐Mach‐number cells in numerical shock. Therefore, MIM should be retained for low‐Mach‐number flows and be completely removed for high‐Mach‐number flows and low‐Mach‐number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.

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.

Characteristic methods and Roe’s method for the incompressible two-fluid model for stratified pipe flow

This article examines the use of characteristic methods in stratified two-phase pipe flow simulations for obtaining non-dissipative flow predictions. A Roe scheme and several methods based on the principle of characteristics are presented for the two-fluid model. The main focus is finding numerically efficient ways of capturing wave dynamics and flow regime transitions through direct simulation. Capturing flow regime transitions both effciently and precisely necessitates schemes which remain stable in a numerical sense, yet predicts wave growth once the flow conditions turn unstable. Characteristic methods offer the possibility of simulating hyperbolic systems without numerical dissipation. These methods do however lack certain fundamental conservation properties. Challenges related to information scattering and clustering in space and time, particularly around shocks, are also an issue in some method variants. Hybridisations with the finite volume method are proposed which overcome these shortcomings. All methods are compared, evaluating predictions on the onset of linear wave growth and simulations of non-linear, discontinuous roll-waves. The following observations are made: 1. Characteristic methods are excellent at predicting the onset of linear hydrodynamic instability, even with a small number of computational nodes. 2. Dissipative errors in finite volume methods and characteristic hybrids will be closely linked to the CFL number. The Roe scheme and the characteristic hybrids give very little dissipation error as the CFL number approaches unity. This then becomes a question of numerical stability. 3. Adapting dynamic grid cells, moving along with the characteristics information drift, greatly improves simulation efficiency by allowing for longer time steps. Dynamic grid cells are also useful for alleviating the need for interpolation in characteristic methods.On the whole, the performance of the characteristic hybridisations are similar to that of the Roe scheme at large CFL numbers. Characteristic hybridisations perform somewhat better in predicting linear instability while the Roe scheme is better suited for the shock fronts found in the roll-wave flow regime. These method easily out-perform the more basic upwind and Lax–Friedrich schemes.

A Jacobian-free Edge-based Galerkin Formulation for Compressible Flows

A parallel formulation of a Jacobian-free all Mach numbers solver on unstructured hybrid meshes is proposed. The Finite Element formulation is edge-based with flow stabilization obtained with either AUSM+ -up or Roe scheme. The linear system is solved via a Jacobian-Free Newton-Krylov (JFNK) method with Lower-Upper Symmetric GaussSeidel (LU-SGS) used as matrix-free preconditioner. The traditional formulation of LU-SGS is enriched by including the contributions from viscous fluxes and boundary conditions. The accuracy and efficiency of the proposed approach are demonstrated over cases ranging from low to high Mach numbers: subsonic flow over the Trap Wing, transonic flow over the ONERA M6 wing, supersonic flow over a sphere, supersonic flow over a waverider and finally hypersonic flow over a sphere.

Rescaling of the Roe scheme in low Mach-number flow regions

A rescaled matrix-valued dissipation is reformulated for the Roe scheme in low Mach-number flow regions from a well known family of local low-speed preconditioners popularized by Turkel. The rescaling is obtained explicitly by suppressing the pre-multiplication of the preconditioner with the time derivative and by deriving the full set of eigenspaces of the Roe–Turkel matrix dissipation. This formulation preserves the time consistency and does not require to reformulate the boundary conditions based on the characteristic theory. The dissipation matrix achieves by construction the proper scaling in low-speed flow regions and returns the original Roe scheme at the sonic line. We find that all eigenvalues are nonnegative in the subsonic regime. However, it becomes necessary to formulate a stringent stability condition to the explicit scheme in the low-speed flow regions based on the spectral radius of the rescaled matrix dissipation. With the large disparity of the eigenvalues in the dissipation matrix, this formulation raises a two-timescale problem for the acoustic waves, which is circumvented for a steady-state iterative procedure by the development of a robust implicit characteristic matrix time-stepping scheme. The behaviour of the modified eigenvalues in the incompressible limit and at the sonic line also suggests applying the entropy correction carefully, especially for complex non-linear flows.

A new method of adjustment of inlet boundary for improving heat transfer of mixed convection in a vertical channel

This article is dedicated to a new method for improving heat transfer of mixed convection in a vertical channel from the numerical perspective. An adjustable inlet boundary is proposed in the new method to substitute an inlet that is completely occupied by the amount of fluid provided by forced convection included in mixed convection. According to this new method, the flow reversal occurring in the large Richardson number situation can be eliminated significantly, which results in an improvement in the heat transfer rate of mixed convection. In order to investigate a compressible flow problem, this study focuses on four methods, i.e., Roe scheme, preconditioning, dual time stepping, and the LUSGS (Lower–Upper Symmetric-Gauss–Seidel), to solve the governing equations. Except for the aperture occupied by the mount of fluid provided by forced convection, this study simulates the non-reflecting boundary conditions in other open boundaries. Based on our previous study which has successfully explored a phenomenon of flow reversal, this study intends to enhance the heat transfer rate under this circumstance, and the result has shown a significant improvement, by 29 percentage, comparing with the previous study. Also, a criterion of estimation of the necessity of forced convection included in mixed convection is proposed for enhancing heat transfer rate. Present results and existing results have good agreement.

A two-fluid four-equation model with instantaneous thermodynamical equilibrium

We consider an equilibrium version of a common two-fluid model for pipe flow, containing one mixture mass equation and one mixture energy equation. This model can be derived from a five-equation model with instantaneous thermal equilibrium, to which additional phase relaxation terms are added. An original contribution of this paper is a quasilinear formulation of the instantaneous phase relaxation limit. From this, the mixture sound speed intrinsic to the model can be extracted. This allows us to directly prove some subcharacteristic conditions with respect to a previously established model hierarchy of different relaxation processes. These subcharacteristic conditions reveal the fundamental insight of this paper; in the hierarchy, thermodynamic versus velocity relaxation both reduce the mixture sound velocity with a factor that is independent of whether the other type of relaxation has been performed.

Evaluation of rotated upwind schemes for contact discontinuity and strong shock

Rotated upwind schemes have been evaluated as carbuncle-free shock-capturing methods. In order to reveal the relation between the rotated upwind-differencing direction and the performance of rotated upwind schemes, theoretical and numerical investigations are made. Three commonly used upwind schemes, including Roe scheme, AUSM+ scheme and Van Leer scheme are investigated. According to the theoretical analysis, the dissipative terms of each upwind scheme are changed by the rotation procedure. Especially, the dissipative density term in the numerical flux function of rotated upwind schemes introduces extra dissipation. The numerical cases include simulations of laminar boundary layer, steady twin vortices and strong shock. The numerical results show that the carbuncle phenomenon can be eliminated by rotated upwind schemes, and the contact resolving capability of rotated upwind schemes is deteriorated by the rotation procedure. Monotonicity of the dissipation with rotation angle is also shown in the numerical results. Therefore, a novel rotation strategy that defines the rotation angle by a pressure weight function is introduced for tuning the dissipative effects adaptively. This strategy shows not only carbuncle-free result in hypersonic inviscid flow simulations, but also accurate results in viscous flow simulations.

Framework for simulation of natural convection in practical applications

Natural convection is commonly used as a means of heat transfer in many practical products because it is highly cost-effective. The development of simulation tools for this type of application is generally accompanied by several critical issues, including high-temperature differences, rapid turnaround demand, and complex geometries. Under conditions of natural convection with high-temperature differences, the density of the medium is variable but the flow speed is low. Therefore, a compressible solver, i.e., Roe scheme developed by P.L. Roe in 1981, must be combined with a preconditioning method that can make the Roe scheme available at low speeds to allow the above situation to be addressed. The building cube method is adopted to make our method suitable for massive parallelization systems, which can reduce the calculation and turnaround times immensely. An immersed boundary method for compressible flows combined with a fast, easy to implement, and robust interpolation method is developed to handle flows with complex immersed geometries. The results show that the program described here is suitable for application to product design and analysis because of its wide applicability to natural convection with high-temperature differences, its capacity to handle complex geometries, and its feasibility for use in massive parallelization systems.

A Roe-like numerical method for weakly hyperbolic systems of equations in conservation and non-conservation form

We describe a scheme for the numerical approximation of uniformly or locally weakly hyperbolic systems of equations in conservation and non-conservation forms. The strategy focuses on generalizations of the Roe scheme for non-conservation systems that can be considered as upstream differencing discretizations. The present work explicitly treats an advection matrix that is not diagonalizable by extending previous ideas that introduce a perturbation into the weakly hyperbolic problem to convert it to a nearby strongly hyperbolic problem. The perturbative expansion allows one to identify and remove the terms that would otherwise result in an apparent singularity and isolate the regular part of the numerical method. The performance of the method is showcased for uniformly and locally weakly hyperbolic systems in one and two dimensions. Generalization to higher dimensional systems can be performed by following the usual practice of dimension-by-dimension splitting.

Development of a discrete gas-kinetic scheme for simulation of two-dimensional viscous incompressible and compressible flows.

In this work, a discrete gas-kinetic scheme (DGKS) is presented for simulation of two-dimensional viscous incompressible and compressible flows. This scheme is developed from the circular function-based GKS, which was recently proposed by Shu and his co-workers [L. M. Yang, C. Shu, and J. Wu, J. Comput. Phys. 274, 611 (2014)]. For the circular function-based GKS, the integrals for conservation forms of moments in the infinity domain for the Maxwellian function-based GKS are simplified to those integrals along the circle. As a result, the explicit formulations of conservative variables and fluxes are derived. However, these explicit formulations of circular function-based GKS for viscous flows are still complicated, which may not be easy for the application by new users. By using certain discrete points to represent the circle in the phase velocity space, the complicated formulations can be replaced by a simple solution process. The basic requirement is that the conservation forms of moments for the circular function-based GKS can be accurately satisfied by weighted summation of distribution functions at discrete points. In this work, it is shown that integral quadrature by four discrete points on the circle, which forms the D2Q4 discrete velocity model, can exactly match the integrals. Numerical results showed that the present scheme can provide accurate numerical results for incompressible and compressible viscous flows with roughly the same computational cost as that needed by the Roe scheme.

Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall

Phenomena of the flow downward penetration in vertical parallel plates natural convection with an asymmetrically heated wall are investigated numerically. For clarifying the occurrence of flow downward penetration, the width of parallel plates is further broadened and broader than that investigated in the previous study in which the flow reversal was studied in detail. In order to simulate realistically, the compressibility and the viscosity of the working fluid are simultaneously considered, and the non-reflecting boundary is adopted at apertures of plates. The methods of the Roe scheme, preconditioning, and dual time stepping matching LUSGS are coordinated to solve governing equations of a low-speed compressible flow problem. The flow downward penetration, in which fluids are sucked from surroundings outside the outlet to flow into the parallel plates and through the inlet flow out of parallel plates, is composed of a few fluids and firstly observed. Also, two flow reversal caused by the buoyancy force appear in the upper left region, and one flow reversal induced by the flow downward penetration appears in the low right region. The usage of the existing results and present study, an equivalent ratio of the width to the length of parallel plates, of which the heat transfer rate is the same as the single vertical heated plate, can be expediently obtained. In this study, the ratio is about 0.6.

Analytical and numerical solutions of the Local Inertial Equations

Neglecting the convective terms in the Saint-Venant Equations (SVE) in flood hydrodynamic modelling can be done without a loss in accuracy of the simulation results. In this case the Local Inertial Equations (LInE) are obtained. Herein we present two analytical solutions for the Local Inertial Equations. The first is the classical instantaneous Dam-Break Problem and the second a steady state solution over a bump. These solutions are compared with two numerical schemes, namely the first order Roe scheme and the second order MacCormack scheme. Comparison between analytical and numerical results shows that the numerical schemes and the analytical solution converge to a unique solution. Furthermore, by neglecting the convective terms the original numerical schemes remain stable without the need for adding entropy correction, artificial viscosity or special initial conditions, as in the case of the full SVE.

All-speed Roe scheme for the large eddy simulation of homogeneous decaying turbulence

ABSTRACTAs a type of shock-capturing scheme, the traditional Roe scheme fails in large eddy simulation (LES) because it cannot reproduce important turbulent characteristics, such as the famous k−5/3 spectral law, as a consequence of the large numerical dissipation. In this work, the Roe scheme is divided into five parts, namely, ξ, δUp, δpp, δUu, and δpu, which denote basic upwind dissipation, pressure difference-driven modification of interface fluxes, pressure difference-driven modification of pressure, velocity difference-driven modification of interface fluxes, and velocity difference-driven modification of pressure, respectively. Then, the role of each part in the LES of homogeneous decaying turbulence with a low Mach number is investigated. Results show that the parts δUu, δpp, and δUp have little effect on LES. Such minimal effect is integral to computational stability, especially for δUp. The large numerical dissipation is due to ξ and δpu, each of which features a larger dissipation than the sub-grid scale model. On the basis of these conditions, an improved all-speed Roe scheme for LES is proposed. This scheme can provide satisfactory LES results even for coarse grid resolutions with usually adopted second-order reconstructions for the finite volume method.

Numerical investigation of transonic flow over deformable airfoil with plunging motion

In this article, the transonic inviscid flow over a deformable airfoil with plunging motion is studied numerically. A finite volume method based on the Roe scheme developed in a generalized coordinate is used along with an arbitrary Lagrangian-Eulerian method and a dynamic mesh algorithm to track the instantaneous position of the airfoil. The effects of different governing parameters such as the phase angle, the deformation amplitude, the initial angle of attack, the flapping frequency, and the Mach number on the unsteady flow field and aerodynamic coefficients are investigated in detail. The results show that maneuverability of the airfoil under various flow conditions is improved by the deformation. In addition, as the oscillation frequency of the airfoil increases, its aerodynamic performance is significantly improved.

A path-conservative finite volume scheme for compressible multi-phase flows with surface tension

The accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). The basic difficulties are the capturing of the correct interface dynamics between the two fluids as well as the computation of the interface curvature. In this paper, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer–Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. In the CSF method, the surface tension effect is replaced by a volume force, which is usually integrated as a classical volume source term. However, since this source term contains the gradient of a color function that is convected with the flow velocity, we propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path-conservative finite volume schemes put forward by Castro and Parés.For that purpose, we use the new generalized Osher-type Riemann solver (DOT), recently proposed by Dumbser and Toro and compare it with a path-conservative Roe and Rusanov scheme. Via numerical evidence we can show that if the curvature computation is exact, our new scheme is well-balanced for a steady circular bubble in equilibrium according to the Young–Laplace law. This means that the pressure jump term across the interface in the momentum equation is exactly balanced with the surface tension force.We apply our scheme to several one- and two-dimensional test problems, including 1D Riemann problems, an oscillating droplet, as well as a deforming droplet initially attached to a wall and which is subject to gravity and surface tension forces. Finally, we also check the rising of a gas bubble due to buoyancy forces. We compare our numerical results with those published previously in the literature.