Roe Scheme Research Articles

Resolution of the bidimensional shallow water equations with horizontal temperature gradients by a well Balanced Roe scheme

This work presents a numerical resolution of the two-dimensional Ripa system using Roe scheme. A discretization of the source term satisfying the conservation property is presented. The mathematical model is based on the ordinary shallow water equations in merging the horizontal temperature gradient. The numerical approach employs unstructured grids and integrates Runge–Kutta method and the minmod limiter to achieve second order accuracy in both time and space domains. A strategy of mesh adaptation based on temperature gradient is implemented to refine the study domain and gain in computational cost. Various numerical tests are conducted to verify the effeciency of the numerical method.

Numerical simulating the blood flow model via nonhomogeneous Riemann solver scheme

This paper examines a one-dimensional (1D) model that appears in arterial blood flow. The mathematical model for blood flow via arteries is similar to that of unstable incompressible flows in thin-walled collapsible tubes. We present the Riemann invariants of the suggested model, which is one of the fundamental components of this work. For numerical modeling of blood flow model, we present a nonhomogeneous Riemann solver (NHRS) technique. Next, we demonstrate the simulation of how pressure, velocity, and cross section area waveforms propagate through arteries. Specifically, we present numerical test cases with various initial data sets. In addition, we compare the NHRS scheme to the classic Rusanov, Lax–Friedrichs, and Roe schemes. Theoretical models for thin-walled collapsible tubes are applicable to a wide range of physiological events and may be used to build clinical devices for actual biomedical science. The NHRS method’s accuracy and efficiency are demonstrated by the numerical tests.

Well-Balanced conservative central upwind scheme for solving the dam-break flow problem over erodible bed

This work deals with the numerical solution of dam-break flow over an erodible bed. The mathematical model is a combination of the shallow water, the transport diffusion and the bed morphology change equations. The system is solved by a well-Balanced central upwind scheme with conservative property. Several tests are illustrated in order to validate the accuracy and the performance of the model. A comparison of central upwind scheme and Roe scheme is presented.

A Comprehensive Numerical Overview of the Performance of Godunov Solutions Using Roe and Rusanov Schemes Applied to Dam-Break Flow

As open channel simulations are of great economic and human significance, many numerical approaches have been developed, with the Godunov schemes showing particular promise. To evaluate, confirm, and extend the simulation results of others, a variety of first- and second-order FVMs are available, with Rusanov and Roe schemes being used here to simulate the demanding case of 1D and 2D flows following a dam break. The virtual boundary cells approach is shown to achieve a monotonic solution for both interior and boundary cells, and while flux computation is employed at boundary cells, a refinement is only rarely used in existing models. A number of variations are explored, including the TVD MUSCL-Hancock (monotone upwind scheme for conservation laws) numerical scheme with several slope limiters in a quest to avoid spurious oscillations. The sensitivity of the results to both channel length and the ratio of downstream to initial upstream water depth is explored using 1D and 2D models. The Roe scheme with a Van Leer limiter as a slope limiter is shown to be both fast and slightly more accurate than other slope limiters for this problem, but the Rusanov scheme with different slope limiters works well for 1D simulations. Significantly, the selection of an appropriate slope limiter is shown to be best based on the ratio of the downstream to upstream water depth. However, this study focuses on the special case where the ratio of the initial depth downstream to upstream of the dam is equal to or less than 0.5, and these outcomes are compared to theoretical results. The 2D dam-break problem is used to further explore first- and second-order methods using different slope limiters, and the results show that the Superbee limiter can be problematic due to an observed large dispersion in depth contours. However, the most promising approaches from previous studies are confirmed to deserve the high regard given to them by many researchers.

Numerical Investigation on the Flow Field of Muzzle Decompression Device for the Barrel Recoil Gun

A structured dynamic overlapping grid and a user-defined function are used to study the projectile launching process, and the hybrid Roe scheme is used to solve the flow field with strong shock wave. The launching process of a projectile with a muzzle decompression device is numerically simulated by using a three-dimensional transient model, and the flow field inside the muzzle decompression device and the development process of the muzzle flow field in the projectile launching process are discussed in detail. Compared with no device, the muzzle decompression device can effectively reduce the peak pressure around the muzzle; the numerical results are in agreement with the corresponding experimental values. The numerical investigation in this paper is helpful to understand the mechanism of pressure reduction of the device. It also provides a new way to reduce the muzzle pressure of aircraft gun.

Extension of an all-Mach Roe scheme able to deal with low Mach acoustics to full Euler system

We propose to extend the fix of Roe’s approximate Riemann solver developed for the Barotropic Euler equations in [2] to the full Euler equations. This scheme is built mainly to handle low Mach acousticwaves. Moreover, compared to pressure-centered type schemes, this numerical fix has the advantage of improving the numerical solution in the sense that the oscillating modes are reduced. The theoretical study is based on a two-time scales asymptotic analysis. It is proved that the Euler system equipped with a general equation of state is consistent with a first-order wave system in a low Mach number regime. Similar analysis is performed at the discrete level on the Roe scheme to derive the new fix. Numerical tests confirm the results obtained for the Barotropic case about the ability of this fix to deal with both steady and low Mach acoustic computations also in the case of full Euler equations.

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

The article describes initial steps toward the development of our nonlinear magnetohydrodynamics code, MUSES. Unstructured grids and general coordinates are employed so that MUSES code is expected to have a capability to represent realistic geometries of fusion devices. Discontinuous Galerkin method is chosen as a code framework because of its high spatial accuracy and numerical stability on unstructured grids. The governing equation of the ideal magnetohydrodynamics in this work is Powell 8-wave model which can mitigate the divergence-free issue. Roe scheme is used as a numerical flux since the Powell 8-wave model is a nonconservative system. A cross code comparison is carried out with a linear magnetohydrodynamics code, MINERVA. Owing to discontinuous Galerkin method, a linear theory of an ideal internal kink mode is reproduced quantitatively although the upwind method is employed for numerical stability.

Mechanism of width effects on cavity-induced transitions at hypersonic speed using direct numerical simulation

Cavities on the surfaces of hypersonic vehicles cannot be avoided easily. Moreover, they can trigger boundary layer transition under certain conditions. However, little progress has been reported on boundary layer transitions induced by a three-dimensional (3-D) shallow cavity. In this study, transitions induced by six 3-D shallow cavities with the same length–depth ratios of 20 and different widths of 0.25, 0.5, 0.75, 1.0, and 1.5 times the baseline width (W), as well as infinite width, were investigated. A direct numerical simulation was conducted using the Roe scheme with 4th-order minimum dispersion and controllable dissipation, and weighted essentially non-oscillatory reconstruction, based on our in-house code: Unsteady NavIer–STokes equations solver. Cavity width was observed to have non-monotonic influences on transition. Both the 0.25 and 1.0 W cavities could induce transition constantly. Moreover, flow was maintained as laminar past the 1.5 W and InfW cavities. For the 0.5 and 0.75 W cavities, intermittent transition was observed with different intermittency factors. The intermittent transition phenomenon was determined to be caused by the periodic increase and decrease in the adverse pressure gradient (APG) in the front part of cavity. Notably, the recirculation with a synchronic size change was the origin of the APG oscillation.

A shock-stable numerical scheme accurate for contact discontinuities: Applications to 3D compressible flows

The low-dissipation Roe scheme is a popular shock capturing scheme but the shock instability, such as the infamous carbuncle phenomenon, seriously affects its application in high Mach number flows. Although considerable research has been undertaken in two dimensions, mechanism analysis of the three-dimensional shock instability is relatively scarce. Numerical simulations of the Sedov blast wave indicate that the shock instability is more prominent in three dimensions, so it has theoretical and practical significance to analyze and heal the numerical instability. The mechanism for three-dimensional shock instability has been thoroughly studied by means of the linearized stability analysis and the dissipation-controlling approach which introduces the required transverse dissipation in the numerical shock layer and is adopted to enhance the Roe scheme’s shock stability. Furthermore, the unphysical expansion shock resulting from the violation of entropy condition is eliminated by a simple modification of numerical signal speed. For the improvement of the resolution for contact discontinuities and shear waves, an algebraic method which combines the THINC reconstruction scheme and the BVD algorithm is employed to minimize the density difference in the numerical diffusion term. A series of benchmark numerical experiments fully demonstrate that the proposed scheme is endowed with excellent robustness against the shock instability and high accuracy for contact discontinuities and shear waves.

Numerical Study of Benchmark Experiments of Supersonic Compressor Cascades

Benchmark cases of turbulent supersonic flow through compressor cascades were computed using a modern upwind scheme (Roe with MUSCL pre-processing), with the k-ω turbulence model, and the classical MacCormack method. The test cases derive from experiments through ARL SL 19 blade cascades at Detroit Diesel Allison (DDA) at the design Mach number, and at an off-design condition from ONERA. A verification of the codes was obtained by computing an inviscid flow through a wedge cascade at similar conditions. Predictions are in very good agreement for the DDA test case for the blade surface pressure distributions. Cleaner solutions were obtained using the Roe scheme. The MacCormack solution exhibits oscillations corresponding to oscillations of the shock-induced separating boundary layer on the pressure surface near the trailing edge. Wake flow Mach number distributions were very well predicted, but not flow angle distributions. Predictions are less accurate for the ONERA test case which was at a slightly higher Mach number and which moves shock locations sufficiently to cause a more sensitive response from the suction surface boundary layer.

A Comparison of Numerical Schemes for Simulating Reflected Wave on Dry and Enclosed Domains

This paper is to investigate the capability of six numerical schemes to simulate reflected wave over a dry and closed domain with and without building, namely: (a) two proposed 2D numerical models solving the conservation form of 2D Shallow Water Equations (2D-SWEs) by Finite Volume Method (FVM) with Roe and HLLC schemes are invoked to approximate Reimann solver; (b) three options of shallow models in the commercial software Flow 3D based on a non-conservation form of 2D-SWEs and (c) the Flow 3D with turbulence modules. By analyzing flooding maps, the area of the reflected wave, and water level profiles on a dry and closed domain, two proposed models give reasonable solutions, while three options of the shallow module of Flow 3D originate result less accurately when initial wave celerity (c0) is small. The accuracy level will be increased if c0 value increases. The 3D model presented the best performance of the complex flow pattern in the dry and enclosed domain in both cases without and with building.

Numerical modelling of dense snow avalanches with a well-balanced scheme based on the 2D shallow water equations

Abstract A common technique for simulating non–Newtonian fluid dynamics, such as snow avalanches, is to solve the Shallow Water Equations (SWE), together with a rheological model describing the momentum dissipation by shear stresses. Friction and cohesion terms are commonly modelled using the Voellmy friction model and, recently, the Bartelt cohesion model. Here, an adaptation of the Roe scheme that ensures the balance between the flux and pressure gradients and the friction source term is presented. An upwind scheme was used for the discretisation of the SWE numerical fluxes and the non–velocity-dependent terms of the friction–cohesion model, whereas a centred scheme was used for the velocity-dependent source terms. The model was tested in analytically solvable settings, laboratory experiments and real cases. In all cases, the model performed well, avoiding numerical instabilities and achieving stable and consistent solution even for an avalanche stopping on a sloping terrain.

Enhancing the Resolution of Blade Tip Vortices in Hover with High-Order WENO Scheme and Hybrid RANS–LES Methods

The accurate prediction of blade tip vortices continues to be challenging for the investigation of rotor aerodynamics. In the current work, the blade tip vortex system of a Caradonna–Tung rotor in the hovering state is simulated on account of the framework of a high-order WENO scheme and hybrid RANS/LES method progressively. With the RANS method based on a fifth-order WENO–Roe scheme, the spatial resolution of wake age and capture the accuracy of tip vortices are improved significantly. However, the unsteadiness of the vortex system fails to be distinguished. Then, the DDES and IDDES methods, coupled with a fifth-order WENO–Roe scheme, are implemented to further enhance the resolution of the spatial turbulence. Compared with RANS, the improvement of spatial resolution is reflected in both the statistical and transient results with DDES. The identifiable vortex core distributions can be predicted at older wake ages. The asymmetrical characteristic of blade tip vortices is revealed along with the release of fluctuation while the predicted distribution of velocity profiles in the vortex core is enhanced. With the application of IDDES, the secondary turbulent structures are captured around the primary vortex core of blade tip vortices while the complex behaviors of the vortices are observed. However, the statistics of the averaged flow field are similar to DDES.

Influence of different interface component distributions on Richtmyer-Meshkov instability

In this paper, the Richtmyer-Meshkov instability is studied numerically by using the high-resolution Roe scheme based on the two-dimensional unsteady Euler equation, which is caused by the interaction between shock wave and the helium circular light gas cylinder with different component distributions. The numerical results are used to further discuss the deformation process of the gas cylinder and the wave structure of the flow field, and also to quantitatively analyze the characteristic dimensions (length, height and central axial width) of the gas cylinder, the time-dependent volume compression ratio of the cylinder. In addition, the flow mechanism of shock-driven interface gas mixing is analyzed from multiple perspectives by combining the flow field pressure, velocity, circulation and gas mixing rate. Then the effects of different initial component distribution conditions on interface instability are investigated. The results show that when the diffusion interface transforms into the sharp interface, the reflection coefficient gradually increases on both sides of interface. When the incident shock wave interacts with the cylinder, the transmission of the shock wave will transform from conventional transmission into unconventional transmission. At the same time, the reflected shock wave is gradually strengthened and the transmitted shock wave is gradually weakened, which leads the Richtmyer-Meshkov instability to be strengthened. Moreover, the Atwood numbers on both sides of the interface also increase as the diffusion interface transforms into the sharp interface, which leads the Rayleigh-Taylor instability and the Kelvin-Helmholtz instability to be strengthened. Therefore, the increase of instability will cause the circulation to increase, resulting in the increase of the growth rate of gas mixing rate.

Using a Combination of Roe and Rusanov Schemes for the Numerical Solution of the Equations of Magnetohydrodynamics in the Problems of Cosmic Plasma

Using a Combination of Roe and Rusanov Schemes for the Numerical Solution of the Equations of Magnetohydrodynamics in the Problems of Cosmic Plasma

Development of a carbuncle-free and low-dissipation Roe-type scheme: Applications to multidimensional Euler flows

The Roe scheme is known to have high resolution but to be afflicted by the notorious shock anomalies (such as the carbuncle phenomena) and the unphysical expansion shocks. This paper aims to develop a new Roe-type scheme which is robust for strong shock waves and low dissipative for contact waves and shear layers. Based on the mechanism analysis of shock instability, a dissipation-controlling strategy is adopted to adjust the entropy wave viscosity and shear wave viscosity of the transverse numerical flux in the numerical shock layer and the robustness of the Roe scheme is thus enhanced. The violation of entropy condition and the problem of expansion shocks are also fixed to broaden the scheme’s application scope. In addition, the THINC reconstruction procedure and a BVD algorithm are implemented to further improve the resolution for contact waves and shear layers. Numerical results of a series of classical one-, two- and three-dimensional test problems fully demonstrate that the proposed new Roe-type scheme has not only better robustness for strong shock waves but also higher resolution for contact waves and shear layers than the original Roe scheme.

A class of structurally complete approximate Riemann solvers for trans- and supercritical flows with large gradients

The wave structure of approximate Riemann solvers has a significant impact on the accuracy and computational requirements of finite volume codes. We propose a class of structurally complete approximate Riemann solvers (StARS) and provide an efficient means for analytically restoring the expansion wave to pre-existing three-wave solvers. The method analytically restores the expansion, is valid for arbitrary thermodynamics, and has comparable complexity to the popular Harten-Hyman entropy fix. The StARS modification is applied to a Roe scheme, resulting in Roe-StARS with noticeable improvements in unsteady transcritical and supercritical conditions with large flow gradients. A novel scaling analysis is performed on the flow conditions that cause rarefaction fluxes and the magnitude of errors if the rarefaction is omitted. Four test cases are examined: a transcritical shock tube, a shock tube with periodic bounds resulting in interfering shocks and rarefactions, a two-dimensional Riemann problem, and a “gradient” Riemann problem—a variant on the traditional Riemann problem featuring an initial gradient of varying slope rather than an initial step function. The results highlight the complex causes and effects of entropy violations, and encourage further study of StARS-type solvers for modern flow problems in which high flow speeds, large gradients, and non-ideal thermodynamics are increasingly common.

Depth effects on the cavity induced transition at hypersonic speed by DNS

Cavity-induced transition (CIT) in a hypersonic boundary layer was simulated by employing direct numerical simulation (DNS). The upwind Roe scheme was discretized by high-order minimum dispersion and controllable dissipation (MDCD) reconstruction based on high-quality multi-block structured grids. The grid scale and time step are determined after evaluating the grid scale and nondimensional time step and comparing with the available measurements. Four depths, 1.0, 1.5, 2.0, and 2.5, multiplied by the baseline depth (D), are evaluated to determine their effects on the CIT on the wall after the cavity. The baseline is a typically “closed” cavity and the width of turbulent region at one cavity length downstream cavity (x = 2L) is 67 mm, caused by the strong adverse pressure gradient and horseshoe vortex instability. Typically, the other three are “open” cavities with no transition for the 1.5D cavity, a turbulent region width of 26 mm for the 2.0D cavity, and 45 mm for the 2.5D cavity at x = 2L were obtained after the cavity. Moreover, the relationship between the flow structures and a typical high heat flux region is explored. Reattachment supplies heat to the wall, and separation removes heat from the wall.

Decoupled solution of the sediment transport and 2D shallow water equations using the finite volume method

In this paper was successfully implemented a numerical solution of the sediment transport and 2D shallow water equations using the decoupled approach. Different discharge formulas to approximate the solid discharge of the sediment were used. The flow field was calculated by solving the shallow water equations and was discretized using the finite volume method; the Godunov's method along with the Roe's Riemann solver was used to approximate the numerical fluxes. The sediment transport was modeled with the Exner equation which was discretized using the Roe's scheme and the finite volume method. In addition, it was proposed an algorithm to guarantee the mass balance in the whole domain for water and sediment at all time steps. The results were compared with experimental measurements reported by other authors to validate the implemented solution.

Development of a Shock-Stable and Contact-Preserving Scheme for Multidimensional Euler Equations

The shock instability, notably the disastrous carbuncle phenomenon, has plagued many low-dissipation Godunov-type numerical schemes in calculations of high Mach flows. In the current work, two popular stability analysis tools and corresponding numerical experiments are used to investigate the root cause of shock instability of the low-dissipation Harten, Lax, and van Leer with contact (HLLC) scheme. The local linear stability analysis reveals that the HLLC flux could attenuate all perturbations in the streamwise direction but not perturbations of density and shear velocity in the transverse direction. Numerical results also indicate that the shock instability of the HLLC scheme is only related to the transverse flux. The viscosity terms of entropy wave and shear wave are introduced into the transverse flux, and the stability analysis shows that with the help of additional viscosities the shock instability of the HLLC scheme is cured. To preserve the capability of resolving contact discontinuities and shear waves, a pressure-based switching function is incorporated into the expressions of viscosity terms so that the additional viscosities are activated only when calculating the transverse flux in the shock layer. A series of numerical experiments give evidence of the robustness and accuracy of the proposed scheme, and the strategy adopted here can be easily applied to cure shock instabilities of other low-dissipation numerical schemes, e.g., the Roe scheme and the AUSM+ scheme.