Roe Scheme Research Articles

A Numerical Study of Shock and Heating With Rarefaction for Hypersonic Flow Over a Cylinder

Abstract The numerical computation of hypersonic flows over blunt bodies is challenging due to the difficulty in robust and accurate wall heat flux prediction and proper capturing of shock waves free from the “carbuncle” phenomenon and other shock anomalies. It is important to understand how this behavior is affected due to rarefaction, which in turn will help to improve the study of aerospace vehicles flowing in rarefied and hypersonic regime. Recently, the SLAU2 convective scheme was shown to suppress the shock anomalies found in capturing strong shocks, however, it still showed a wavy pattern of heating. We have proposed a modification to the SLAU2 convective scheme to improve the accuracy of flow predictions in the presence of strong shocks. We then perform the numerical simulation of hypersonic viscous flow over a cylinder at Mach 8 and 16.34 at different Knudsen numbers. We carry out the study using the modified SLAU2 and the classical Roe schemes. We study how the shock anomalies found in the continuum hypersonic flows behave with the degree of rarefaction. It is found that the modified SLAU2 captures the shock free from the shock anomalies at all Kn, while the Roe scheme lacks robustness for Kn≲10−3. The variation of different flow properties such as heat flux, wall shear stress, and the Mach number is investigated. The peak heating value was observed to decrease with the degree of rarefaction.

ОБТЕКАНИЕ ЦИЛИНДРА И АЭРОДИНАМИЧЕСКОГО ПРОФИЛЯ С УЧЕТОМ ЛАМИНАРНО-ТУРБУЛЕНТНОГО ПЕРЕХОДА

A simulation of the flow around a cylinder and a NACA 4412 airfoil was carried out using the γγ-Reθ laminar-turbulent transition model and without it. Numerical simulation was performed on the basis of the Reynolds-averaged Navier-Stokes equations (URANS), using the Spalart-Allmaras differential one-parameter turbulence model. The system of initial equations was written with respect to an arbitrary curvilinear coordinate system. The coordination of the pressure and velocity fields was carried out using the method of artificial compressibility modified to calculate non-stationary problems. The integration of the system of initial equations was carried out numerically using the control volume method. For convective flows, a countercurrent Rogers-Kwak approximation was used based on the third-order Roe scheme. In turbulence models, the TVD scheme with a third-order ISNAS flow limiter was used to approximate convective terms. Comparison of the calculation results for a flow past a cylinder is carried out using the laminar-turbulent transition model and without it. It is shown that at low Reynolds numbers, when the flow around a cylinder is laminar, and the track isturbulent use of the Spalart-Allmaras turbulence model leads to the development of a turbulent boundary layer on the cylinder and, consequently, to a change in the position of the separation point. The incorrect position of the separation point affects the pressure distribution in the bottom of the cylinder and, as a result, the integral aerodynamic characteristics. The use of the transition model allows one to adequately reconstruct the laminar separation near the leading edge of the airfoil with its subsequent reattachment. The use of the Spalart-Allmaras model alone leads to excessive generation of turbulent viscosity. It is shown that the use of the γ-Reθ laminar-turbulent transition model the qualitatively and quantitatively improves the results of numerical simulation. The results of numerical simulation of the flow around a circular cylinder and the NACA 4412 aerodynamic airfoil agree well with experimental data in a wide range of Reynolds numbers.

Shock-Stable Roe Scheme Combining Entropy Fix and Rotated Riemann Solver

The Roe scheme is an important shock-capturing scheme that is known for its good performance. However, the classical Roe scheme suffers from disastrous problems, such as shock instability for hypersonic flows. The entropy fix and the rotated Riemann solver are common methods used to cure the shock instability. However, these methods may introduce a large and unnecessary numerical dissipation. To cure the shock instability with a minimally increasing numerical dissipation, this study investigates the mechanisms of entropy fix and rotated Roe scheme and combines their complementary advantages. Thus, this paper proposes the Roe-ER scheme, which uses the rotated method to identify where the entropy fix is necessary. Therefore, the possible large dissipation of the rotated Roe scheme is replaced by the relatively small value of entropy fix, and areas that adopt the entropy fix are limited in regions where the shock instability may occur. Numerical cases validate the Roe-ER scheme, which is shock stable, features a low additional numerical dissipation, and can use high-order reconstruction to further reduce areas that activate the entropy fix.

Aerodynamics of the turbulent flow around a multi­element airfoil in cruse configuration and in takeoff and landing configuration

Numerical modeling of multi-element airfoil's aerodynamics employs the Reynolds averaged Navier-Stokes equations of incompressible environment, which are related via a single-parametric differential turbulence model by Spalart-Allmaras. The system of initial equations has been recorded with respect to an arbitrary curvilinear coordinate system. The pressure and velocity fields have been aligned by using an artificial compressibility method modified for calculating the nonstationary problems. The system of initial equations has been integrated numerically by applying a control volume method. The counter-flow Rogers-Kwak approximation has been used for convection flows, based on the Roe scheme of third-order accuracy. The turbulence models, in order to approximate the convection components, employed a TVD scheme with the third-order flow limiter ISNAS. The paper reports results from calculating a turbulent flow around a multi-element airfoil in a wide range of the angles of attack. The result of the current research is the performed analysis of the flow field around a multi-element airfoil, pressure coefficients, the lifting force, as well as the drag force. Physical features in a flow structure at flowing around the multi-element airfoil 30P30N have been identified. In the investigated range of the angles of attack, flowing around a airfoil in the takeoff and landing configuration is stationary in nature except for the regions where the flow is detached from sharp edges, such as the slat's inside part and a region in the tail part of the main profile. There are recirculation currents within these regions. With an increase in the angle of attack, dimensions of the detachable zone at the slat's inner surface decrease while remaining almost unchanged in the tail part of the main profile. At the top surface of the main profile there forms a jet of air, due to the acceleration of the flow between the slat and the leading edge of the main profile. The existence of a gap between the main profile and the flap leads to the interference of jet currents at the upper surface of the slat. It has been shown that the takeoff and landing configuration demonstrates the higher values of the lifting force coefficient than the cruise configuration, especially at large angles of attack. The calculation results agree well with the data by other authors.

2D numerical modelling of dam break- applying for Namchien dam in Vietnam

The paper is dedicated to study a numerical model simulating dam-break based on two dimensional nonlinear shallow water equations (2D-NSWE). Finite Volume Method-Godunov type is applied to discretize this equation. Roe scheme is utilized to approximate Riemann problem, meanwhile method of flux difference splitting is implemented to construct numerical solvers of SWE. Besides, the semi implicit scheme is also invoked to solve friction term in case of high roughness coefficient. The proposed model is verified through a comparison between computed results and empirical data of two reference tests. A dam break flow over floodable area with different roughness coefficients is also researched. A total collapsed dam scenario of an arch dam-Nam Chien in Vietnam is simulated by the proposed model. Several hydraulic characteristics such as flood extent, arrival time and time histories of water depth at different gauges are estimated with different grid sizes.

Dominating Unsteadiness Flow Structures in Corner Separation Under High Mach Number

Corner separation is a key factor affecting the aerodynamic performance of compressors, whereas corner separation under a high-Mach-number condition is a scientific problem with practical engineering significance. Using a self-developed high-precision (Roe scheme for convective flux calculation and fifth-order weighted essentially non-oscillatory for reconstruction) delay detached-eddy simulation (DDES) code, a compressor cascade with massive corner separation under a high Mach number is simulated. The primary aim of the simulations is to obtain the dominating fluctuation vortex structures in and out of the blade passage, as well as to analyze the reason for the formation of these structures. The vortex structures are analyzed after verifying the accuracy of the code. An advanced data mining technology (that is, dynamic mode decomposition) was performed to extract the dominating fluctuation flowfield simulated by DDES for the first time. The results prove the credibility of the self-developed code. It is found that the passage vortex and the hairpin vortex are the primary sources of unsteady fluctuations in and out of the passage, respectively. The former is formed by endwall separation, and the position where it is generated is relatively stable; thus, the endwall suction technology is suggested for flow control. The latter is formed by the intermixing of the passage vortex, the trailing-edge shedding vortex, and the concentrated shedding vortex. Owing to the different frequencies of those three kinds of vortices, different hairpin scales are formed. The research results can provide a basis for flow control design.

Formulation of exactly balanced solvers for blood flow in elastic vessels and their application to collapsed states

In this work, numerical solvers based on extensions of the Roe and HLL schemes are adapted to deal with test cases involving extreme collapsing conditions in elastic vessels. To achieve this goal, the system is transformed to provide a conservation–law form, allowing to define Rankine–Hugoniot conditions. The approximate solvers allow to describe the inner states of the solution. Therefore, source term fixes can be used to prevent unphysical values of vessel area and, at the same time, the eigenvalues of the system control stability. Numerical solvers of different order are tested using a wide variety of Riemann problems, including extreme vessel collapse and blockage. In all cases, the robustness of the approximate solvers presented here is checked using first and third order methods in time and space, using the WENO reconstruction scheme in combination with the TVDRK3 method.

Comparison of Shallow Water Solvers: Applications for Dam-Break and Tsunami Cases with Reordering Strategy for Efficient Vectorization on Modern Hardware

We investigate in this paper the behaviors of the Riemann solvers (Roe and Harten-Lax-van Leer-Contact (HLLC) schemes) and the Riemann-solver-free method (central-upwind scheme) regarding their accuracy and efficiency for solving the 2D shallow water equations. Our model was devised to be spatially second-order accurate with the Monotonic Upwind Scheme for Conservation Laws (MUSCL) reconstruction for a cell-centered finite volume scheme—and be temporally fourth-order accurate using the Runge–Kutta fourth-order method. Four benchmark cases of dam-break and tsunami events dealing with highly-discontinuous flows and wet–dry problems were simulated. To this end, we applied a reordering strategy for the data structures in our code supporting efficient vectorization and memory access alignment for boosting the performance. Two main features are pointed out here. Firstly, the reordering strategy employed has enabled highly-efficient vectorization for the three solvers investigated on three modern hardware (AVX, AVX2, and AVX-512), where speed-ups of 4.5–6.5× were obtained on the AVX/AVX2 machines for eight data per vector while on the AVX-512 machine we achieved a speed-up of up to 16.7× for 16 data per vector, all with singe-core computation; with parallel simulations, speed-ups of up to 75.7–121.8× and 928.9× were obtained on AVX/AVX2 and AVX-512 machines, respectively. Secondly, we observed that the central-upwind scheme was able to outperform the HLLC and Roe schemes 1.4× and 1.25×, respectively, by exhibiting similar accuracies. This study would be useful for modelers who are interested in developing shallow water codes.

A simple extension of Roe's scheme for multi-component real gas flows

This paper proposes a numerical algorithm to model multi-component real gas flows in local thermodynamic equilibrium (LTE). The approach is based on the solution of the Euler equations for the mixtures which is coupled with the mass conservation equation for each component of fluid. This algorithm is based on the extension of Roe's scheme for real gases introduced in [21], which is robust, efficient and simpler than the other extensions. The Euler equations for the mixture are solved whereas fluid components are updated using a classical upwind approximation. As the solution evolves, the developed method preserves the positivity of the mass fractions of all components. The proposed technique does not directly involve the thermodynamics derivatives at the average state, which avoids assumptions and approximations in the calculation of these derivatives. The pressure oscillation problem appearing in the case of flow fields including material contact has been also discussed. The numerical experiments show the accuracy and robustness of the method in various cases.

A Riemann Problem Based Coupling Method for Predicting the Combustion of Propellant in a Gun Launching Process

AbstractMethods for improving the accuracy and guaranteeing the fidelity of numerical predictions are of great importance for physical problems accompanied by strong nonlinear multi‐physic interactions and extreme working conditions. The prediction of propellant combustion in a gun launching process is a typical example. A one‐dimensional two‐fluid model was utilized to govern the fluid field induced by the combustion. In order to guarantee the accuracy of the predictions, we utilized a Roe's scheme to solve the partial differential equations. To guarantee the accuracy of the moving boundaries in the expanding chamber, the coupling between the updating of the fluid field and the approximation of the mechanical interactions between the projectile and the barrel is realized based on the commercial software ABAQUS. The separate treatment of the advection and source terms in the split approach was verified through the agreement between the maximum value of the predicted average pressure in a closed bomb and the theoretical value. The coupled method dealing with the moving boundaries was validated by comparing the numerical results with the analytical results of a pressure‐driven piston case. Finally, the numerical method was applied to a large‐caliber gun. The predicted maximum pressure and the muzzle velocity agree well with experiments.

A Robust Rotated-Hybrid Riemann Scheme for Multidimensional Inviscid Compressible Flows

A robust algorithm is investigated for multidimensional inviscid compressible flows, based on rotated Riemann solver framework. Indeed, the proposed method combines the TV-HLL and Roe schemes. The upwind direction is imposed by the velocity-difference vector. Then, the TV-HLL solver is applied in the direction perpendicular to shocks in order to suppress carbuncle. In addition, the Roe solver is applied across shear layers to minimize the amount of dissipation. To assess the capabilities of the present method, numerous test problems (2D and 3D) are simulated.

Comparison of the AUSM+-up and other advection schemes for turbomachinery applications

In this paper, the AUSM+-up scheme is compared to other numerical flux schemes in the framework of a RANS/URANS code for turbomachinery applications. The considered advection schemes include central discretizations with artificial dissipation and the Roe upwind scheme. The comparison is carried out by studying a low-aspect-ratio turbine cascade over a wide range of expansion ratios that encompasses almost incompressible up to supersonic flow conditions. It is found that the dissipation scaling associated with the AUSM+-up scheme was effective over the whole range of analysed flow conditions. A detailed assessment with the aid of the available measurements will be exploited to show how the AUSM+-up is capable of a detailed and faithful reproduction of secondary flow features, with accuracy comparable to that of the Roe scheme and superior to that of central schemes.

The Study on the Influence of Boundary Condition Applied to the Case of Flow Past through Bump Model.

The present work focused on solving two dimensional compressible Euler equation based on cell-cantered finite volume discretization technique with the convective terms appear in the Euler equation were evaluated by using Roe scheme [4]. These methods used for solving the flow past through bump with different setting on their boundary conditions. Six different setting of boundary conditions had been applied; three of them are able to provide a convergent solution, while three others tent to diverge. It is clear the result shows that boundary condition plays an important role in determining the flow solution.

A low-dissipation DG method for the under-resolved simulation of low Mach number turbulent flows

In recent years the use of high-order Discontinuous Galerkin (DG) methods for the under-resolved direct numerical simulations (uDNS) of turbulent flows has received special attention. The suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme: while the dispersion errors are small over a broad range of frequencies, a relevant dissipation error mainly acts at the smallest under-resolved scales, resembling a high frequency filter. Nevertheless, it was recognized (Flad and Gassner, 2017; Mengaldo et al., 2018) that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the uDNS approach. In this regard, the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided, in particular when dealing with low-speed flows, since this behaviour is exacerbated approaching the incompressibility limit. Fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods, see for example Weiss and Smith (1995). In this work a simple modification of the dissipation term of the low Mach preconditioned Roe scheme proposed by Weiss and Smith is considered. The aim is to reduce further the amount of numerical dissipation with the intent of improving the results of uDNS. A spatial DG discretization coupled with a linearly-implicit Rosenbrock-type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions. Results on canonical turbulent flow problems as the Taylor–Green vortex and the flow in a straight sided channel are presented. The improved accuracy of the proposed flux function is demonstrated. The new low-dissipation flux can be useful also in the context of standard, lower order, finite volume methods.

Numerical investigation of laminar compressible natural convection flow in asymmetrically and isothermally heated open-ended inclined channel

This study presents a numerical investigation of laminar compressible natural convection flow induced under high-temperature difference in an open-ended inclined parallel-walled channel heated isothermally and asymmetrically at the top wall (hot plate facing downwards) while the lower plate is considered adiabatic. The investigation is carried out with air (Pr=0.72) as the working fluid for a temperature difference of 110 °C from the ambient, over of range of modified Rayleigh number 6.2×101 to 1.6×104 and inclination of channel varying between 10 and 90° with respect to horizontal position. For the simulation, compressible governing equations without Boussinesq approximation is solved by employing new modified all-speed Roe scheme with matching preconditioning method, dual time-stepping technique and utilizing the modified Local One-dimensional Inviscid (LODI) relations suitable for compressible natural convection flow as non-reflecting boundary conditions at the inlet and outlet of the channel. Average Nusselt number based on inter-plate spacing increases with the increase of modified Rayleigh number and also with the increase of inclination from the horizontal position. This increase in average Nusselt number can be described by the variation of the level of thermal saturation inside the channel. Significant reduction in average Nusselt number is observed in the combined zone of lower angle of inclination and lower modified Rayleigh number. Visualization of fluid flow (temperature contour and velocity magnitude contour) indicates the existence of both fully developed flow and developing flow regime within the span of investigated modified Rayleigh number. Based on the investigation, a composite correlation between average Nusselt number and product of modified Rayleigh number and the sine of the angle of inclination considering as a single parameter is presented which can be conveniently used for designing of engineering application dealing with natural convection heat transfer under high-temperature difference.

Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

A new implementation of the Roe scheme for solving two-layer shallow-water equations is presented in this paper. The proposed A-Roe scheme is based on the analytical solution to the characteristic quartic of the flux matrix, which is an efficient alternative to a numerical eigensolver. Additionally, an accurate method for maintaining the hyperbolic character of the governing system is proposed. The efficiency of the quartic closed-form solver is examined and compared to numerical eigensolvers. Furthermore, the accuracy and computational speed of the A-Roe scheme is compared to the Roe, Lax-Friedrichs, GFORCE, PVM, and IFCP schemes. Finally, numerical tests are presented to evaluate the efficiency of the iterative treatment for the hyperbolicity loss. The proposed A-Roe scheme is as accurate as the Roe scheme, but much faster, with computational speeds closer to the GFORCE and IFCP scheme.

Investigation on optimization of the thermal performance for compressible laminar natural convection flow in open-ended vertical channel

The present paper focuses on the optimization of the thermal performance for compressible laminar natural convection flow induced under high-temperature difference in an open-ended vertical channel by optimizing the channel inter-plate spacing using numerical simulation. The present investigation is conducted for a wide range of Rayleigh number (Ra) 104 to 107 in channel heated asymmetrically by uniform surface temperature with air (Pr0.72) as working fluid. Several values of channel gap between plates, new modified preconditioned all-speed Roe scheme along with dual time stepping technique and modified Local One-dimensional Inviscid (LODI) relations as channel inlet and outlet boundary conditions suitable for compressible laminar natural convection is employed for the current simulation. Heat transfer rate in terms of average Nusselt number is obtained for all Rayleigh number and channel aspect ratio is obtained. Variation of thermal and velocity profiles, the variation of average Nusselt number and mass flow rate into the channel for the combination of each Rayleigh number and channel aspect ratio is reported. From the results obtained, a correlation for optimum aspect ratio with Rayleigh number which maximizes the heat transfer within the channel is presented.

Numerical investigation of buoyancy-driven compressible laminar flow using new method preconditioned all-speed roe scheme

In this study, numerical simulation is performed with a focus on the application of new modified preconditioned all-speed Roe scheme to simulate natural convection flows. The new modified preconditioned all-speed Roe Scheme primarily utilizes the local flow parameters for calculation of the coefficients of numerator in dissipation term of the scheme instead of using the global cut-off Mach number strategy (which is advantageous for natural convection flows) and has been shown to have better accuracy than preconditioned Roe Scheme for Low Mach number flows. For the present simulation, the compressible governing equation in conservation form, new modified preconditioned all-speed Roe scheme and dual time stepping are employed. The validation of numerical algorithm is divided into two investigations a) natural convection flow within differentially heated enclosed square cavity and b) open-ended vertical channel asymmetrically heated for a wide range of Rayleigh number with air (Pr 0.72) as working fluid. Visualization of fluid flow dynamics conducted for both classes of geometries for all range of Rayleigh number show similar phenomena in accordance with previous literatures and compared data also show very good agreement with previous literatures. All results indicate that the new modified preconditioned all-speed Roe scheme is very much competent and accurate for simulation of buoyancy-induced compressible convection flows without relying on the correct prediction of global cut-off Mach number.

Time integration of unsteady nonhydrostatic equations with dual time stepping and multigrid methods

A time integration method based on dual time stepping coupled with a multigrid and implicit iterative solver is discussed for overcoming the temporal stiffness associated with modeling unsteady atmospheric flows. Adopting the dual time stepping allows solving an unsteady flow problem as a steady-state one by introducing a pseudo-time derivative into the governing equations. Consequently, the model uses a large time step and leads to a reduction in computational time. In the framework of the dual time stepping, a full approximation storage multigrid and lower–upper symmetric Gauss–Seidel implicit solver are employed to accelerate convergence in a structured, quadrilateral grid. The discretization of a physical-time derivative is based on a second-order backward differentiation formula or a family of singly diagonally implicit Runge–Kutta methods. The spatial terms of nonhydrostatic equations are discretized with a second-order Roe scheme in the finite volume method. To assess the performance of the model in terms of accuracy and efficiency, which are indicated by the normalized error norms and wall-clock time, respectively, three idealized tests are performed with the unsteady flow problems of isentropic vortex, rising thermal bubble and inertia-gravity waves. The results of numerical experiments consistently demonstrate that the dual time stepping coupled with explicit-first, singly diagonally implicit Runge–Kutta and multigrid methods is more efficient than the three-stage, third-order explicit Runge–Kutta method with comparable accuracy. The numerical experiment also indicates that the dual time stepping is more efficient when a stretched grid is employed. The efficiency in the framework of the dual time stepping is dependent on various numerical factors, including a multigrid level and cycle, Butcher coefficients, an implicit solver and a physical-time step.

Numerical study on lateral jet interaction in supersonic crossflows

This paper presents numerical analyses of lateral jets in supersonic cross flows on a flat plate and on a generic missile. The freestream Mach number is 4 for the flat plate and 3 for the missile, and the jets are sonic for both cases. The numerical results are validated with wind tunnel data such as Schlieren images and surface pressure distributions. The flow structure due to the jet interaction with the supersonic free-stream is examined in terms of the vortex structure. A 3-dimensional compressible RANS solver is used for the study. To describe the effects of high temperature, a thermally perfect gas is assumed. When high temperature is applied, the shock structure changes, which affects the separation region and recirculation zone. Next, the effects of turbulence models on the jet interaction flow are investigated. The Spalart–Allmaras, Menter's shear-stress transport k–ω, Huang and Coakley's k–ε, and Coakley's q–ω models are used to analyze the flows. The differences in pressure distribution among the turbulence models are larger in the case of the flat plate than the missile. In addition, several numerical flux functions are compared to investigate their effects on the jet interaction: the Roe scheme with Sanders's H-correction, RoeM scheme, and the HLLE scheme. Although the HLLE scheme shows a little difference at the indent of the barrel shock, the three numerical flux schemes give similar CP distributions on the wall.