Roe's Approximate Riemann Solver Research Articles

Two-dimensional scour simulations based on coupled model of shallow water equations and sediment transport on unstructured meshes

A two-dimensional scour model based on coupled system of shallow water equations (SWEs) and sediment transport on unstructured mesh is developed. The coupled system of hydrodynamic and morphodynamic equations is solved by finite volume method using Godunov scheme. Roe's approximate Riemann solver is used to calculate the inviscid fluxes. The use of unstructured mesh makes the model applicable to complex domains. However, it is difficult to evaluate the eigenvalues and eigenvectors of the Jacobian matrix in the global coordinate. The method proposed herein to deal with this difficulty is to transform the system into the local coordinate with one of the axes in the same direction as the interface outward normal vector. In the local coordinate system, the Jacobian matrix is simplified and the eigenvalues are analyzed using asymptotic method. Regular expansion breaks down when the flow is near critical. Uniformity of the expansion is achieved by changing the scales. Rotational invariance theorem is used to relate the interfacial fluxes in the global and local coordinate systems. Special treatment of the source term on unstructured grid makes the scheme stable and physically balanced (both mass and momentum). The method proposed in this paper for the eigen-system is very efficient comparing to iterative numerical methods. Results from the test cases show good agreement with the experiments.

A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains

AbstractThe propagation, runup and rundown of long surface waves are numerically investigated, initially in one dimension, using a well‐balanced high‐resolution finite volume scheme. A conservative form of the nonlinear shallow water equations with source terms is solved numerically using a high‐resolution Godunov‐type explicit scheme coupled with Roe's approximate Riemann solver. The scheme is also extended to handle two‐dimensional complex domains. The numerical difficulties related to the presence of the topography source terms in the model equations along with the appearance of the wet/dry fronts are properly treated and extended. The resulting numerical model accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuities. Numerical results show very good agreement with previously presented analytical or asymptotic solutions as well as with experimental benchmark data. Copyright © 2007 John Wiley & Sons, Ltd.

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.

Numerical simulations of three‐dimensional transitional compressible flows in turbomachinery cascades

PurposeThis paper aims to provide a validation of a state‐of‐the‐art methodology for computing three‐dimensional transitional flows in turbomachinery.Design/methodology/approachThe Reynolds‐averaged Navier‐Stokes equations for compressible flows are solved. Turbulence is modeled using an explicit algebraic stress model and k−ω turbulence closure. A numerical method has been developed, based on a cell‐centered finite volume approach with Roe's approximate Riemann solver and formally second‐order‐accurate MUSCL extrapolation. The method is validated versus two severe test cases, namely, the subsonic flow through a turbine cascade with separated‐flow transition; and the transonic flow through a compressor cascade with transitional boundary layers, shock‐induced separation and corner stall. For the first test case, the transition model of Mayle for separated flow has been employed, whereas, for the second one, the transition has been modeled employing the Abu‐Ghannam and Shaw correlation.FindingsThe comparison of numerical results with the experimental data available in the literature shows that, for such complex flow configurations, an improved numerical solution could be achieved by employing transition models. Unfortunately, the available models are case‐dependent, each of them being suitable for specific applications.Originality/valueA state‐of‐the‐art numerical methodology has been developed and applied to compute very complex flows in turbomachinery. Through an original analysis of the results, the merits and limits of the considered approach have been assessed. The paper points up the fundamental role of transition modeling for turbomachinery flow simulations.

Unstructured Grid Finite-Volume Algorithm for Shallow-Water Flow and Scalar Transport with Wetting and Drying

A high-resolution, unstructured grid, finite-volume algorithm is developed for unsteady, two-dimensional, shallow-water flow and scalar transport over arbitrary topography with wetting and drying. The algorithm uses a grid of triangular cells to facilitate grid generation and localized refinement when modeling natural waterways. The algorithm uses Roe's approximate Riemann solver to compute fluxes, a multidimensional limiter for second-order spatial accuracy, and predictor-corrector time stepping for second-order temporal accuracy. The novel aspect of the algorithm is a robust and efficient procedure to consistently track fluid volume and the free surface elevation in partially submerged cells. This leads to perfect conservation of both fluid and dissolved mass, preservation of stationarity, and near elimination of artificial concentration and dilution of scalars at stationary or moving wet/dry interfaces. Multi-dimensional slope limiters, variable reconstruction, and flux evaluation schemes are optimized in the algorithm on the basis of accuracy per computational effort.

A conservative 2D model of inundation flow with solute transport over dry bed

In this paper, a transient 2D coupled vertically averaged flow/transport model is presented. The model deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. The model is based on an upwind finite volume method, using Roe's approximate Riemann solver. A specific modification of the Riemann solver is proposed to overcome the generation of negative values of depth and concentration, that can appear as a consequence of existing wetting/drying and solute advance fronts over variable bed levels, or by the generation of new ones when dry areas appear. The numerical stability constraints of the explicit model are stated incorporating the influence of the flow velocity, the bed variations and the possible appearance of dry cells. Faced to the important restriction that this new stability condition can impose on the time step size, a different strategy to allow stability using a maximum time step, and in consequence a minimum computational cost is presented.

Discontinuous Galerkin Spectral Element Simulation of a Thermal-Hydraulic Problem in Superconducting Magnets

ABSTRACT In this article we simulate one-dimensional quench propagation in superconducting magnets using cable-in-conduit conductors (CICCs) by a discontinuous Galerkin (DG) spectral element method (SEM) and explicit Runge-Kutta time integration. The supercritical helium flow in the CICC channel can be described by the Euler equations with an additional friction force and coupled heat transfer among the helium, the conductor, and the conduit. Roe's approximate Riemann solver for a real gas/fluid is used to compute numerical flux, and a nonreflecting boundary condition is introduced for the approximation of the outflow boundary condition. The method used here is highly parallelizable. Some numerical results are given and compared with those obtained by a finite-element method and with experimental data. This method shows high robustness to simulate the thermal-hydraulic problem with large nonlinear source terms.

Discontinuous Galerkin Spectral Element Simulation of Quench Propagation in Superconducting Magnets

In this paper we simulate 1D quench propagation in superconducting magnets using cable-in-conduit conductors (CICC) by a discontinuous Galerkin (DG) spectral element method (SEM) and explicit Runge-Kutta time integration. The supercritical helium flow is considered in the modeling of quench propagation in CICC, which can be expressed by the Euler equations with additional friction and coupled heat transfer between helium and conductor and conduit. Roe's approximate Riemann solver for real gas/fluid is used to compute numerical flux and nonreflecting boundary condition is introduced in the algorithm. The method used here is highly parallelizable. Some numerical results are given and compared with those obtained by another simulation method and experimental data.

AN EFFICIENT IMPLICIT MESH-FREE METHOD TO SOLVE TWO-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS

Local radial basis function-based differential quadrature (RBF-DQ) method is a natural mesh-free approach, in which any derivative of a function at a point is approximated by a weighted linear sum of functional values at its surrounding scattered points. In this paper, the weighting coefficients in the spatial derivative approximation of the Euler equation are determined by using a weighted least-square procedure in the frame of RBFs, which enhances the flexibility of distributing points in the computational domain. An upwind method is further introduced to cope with discontinuities by using Roe's approximate Riemann solver for estimation of the inviscid flux on the virtual mid-point between the reference knot and its surrounding knot. The lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm, which is implemented in a matrix-free form like the one used in the finite-volume method, is introduced in the work to speed up the convergence. The proposed approach is validated by its application to simulate transonic flows over a NACA 0012 airfoil. It was found that the present mesh-free results agree very well with available data in the literature, and the implicit LU-SGS algorithm can greatly save the computational time as compared with explicit time marching methods.

THE NUMERICAL SIMULATION OF VARIABLE-PROPERTY REACTING-GAS DYNAMICS: NEW INSIGHTS AND VALIDATION

ABSTRACT A large number of numerical algorithms have been reported to solve the Euler equations of motion for a variety of mechanical and aerospace engineering applications. Based on a review of the past and current history of these solvers, a MUSCL-based solver that uses Hancock's predictor-corrector method incorporating Roe's approximate Riemann solver was found to be the most efficient second-order-accurate numerical method to solve the Euler equations. This method was subsequently extended to account for variable properties and then extensively validated for the effects of friction, heat transfer, variable area, and chemical kinetics.

Godunov‐type adaptive grid model of wave–current interaction at cuspate beaches

AbstractThis paper presents a second‐order accurate Godunov‐type numerical scheme for depth‐ and period‐averaged wave–current interaction. A flux Jacobian is derived for the wave conservation equations and its eigensystem determined, enabling Roe's approximate Riemann solver to be used to evaluate convective fluxes. Dynamically adaptive quadtree grids are used to focus on local hydrodynamic features, where sharp gradients occur in the flow variables. Adaptation criteria based on depth‐averaged vorticity, wave‐height gradient, wave steepness and the magnitude of velocity gradients are found to produce accurate solutions for nearshore circulation at a half‐sinusoidal beach. However, the simultaneous combination of two or more separate criteria produces numerical instability and interference unless all criteria are satisfied for mesh depletion. Simulations of wave–current interaction at a multi‐cusped beach match laboratory data from the United Kingdom Coastal Research Facility (UKCRF). A parameter study demonstrates the sensitivity of nearshore flow patterns to changes in relative cusp height, angle of wave incidence, bed roughness, offshore wave height and assumed turbulent eddy viscosity. Only a small deviation from normal wave incidence is required to initiate a meandering longshore current. Nearshore circulation patterns are highly dependent on the offshore wave height. Reduction of the assumed eddy viscosity parameter causes the primary circulation cells for normally incident waves to increase in strength whilst producing rip‐like currents cutting diagonally across the surf zone. Copyright © 2004 John Wiley & Sons, Ltd.

Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations

The goal of this work is to extend finite volume WENO and central WENO schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function. In particular, we apply proposed schemes to the shallow water and the open-channel flow equations where the source term depends on the channel geometry. For obtaining stable numerical schemes that are free of spurious oscillations, it becomes crucial to use the decomposed source term evaluation, which maintains the balancing between the flux gradient and the source term. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the source term decomposition, in combination with the Roe approximate Riemann solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property – the property of preserving the quiescent flow exactly. When the central WENO schemes are applied, additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes, as well as with some other first- and second-order numerical schemes.

Numerical Solution of the Dam-Break Problem with a Discontinuous Galerkin Method

A discontinuous Galerkin method for the solution of the dam-break problem is presented. The scheme solves the shallow water equations with spectral elements, utilizing an efficient Roe approximate Riemann solver in order to capture bore waves. The solution is enhanced by a projection limiter that eliminates spurious oscillations near discontinuities. The main advantage of the model is the flexibility in approximating smooth solutions with high-order polynomials and resolving at the same time discontinuous shock waves. Furthermore, the finite element discretization is capable of handling complex geometries and producing correct results near the boundaries. Both the h- and p-type extensions are investigated for the one-dimensional dam break, and the results are verified by comparison with analytical solutions. The application to a two-dimensional dam-break problem shows the efficiency and stability of the method.

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

AbstractA wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.

ターボ機械翼列を通る湿り空気・蒸気の乱流解析(OS23c 相変化・超臨界流体の計算力学)

In this study, transonic humid-air and wet-steam turbulent flows through a steam turbine cascade channel are calculated. Transonic condensate flows in turbomachinery are numerically predicted by using the high-resolution method developed by our group. Fundamental equations consist of conservation laws of mixed gas, water vapor, water liquid, and the number density of water droplets, coupled with the momentum equations and the energy equation. The classical condensation theory is employed for modeling homogeneous nucleation and non-equilibrium condensation. These equations are solved by a high-order high-resolution finite-difference method based on the fourth-order compact MUSCL TVD scheme, Roe's approximate Riemann solver, and the LU-SGS scheme. SST model is used for evaluating eddy-viscosity.

Iced airfoil simulation using generalized grids

A new strategy for simulating the flow around iced airfoils is presented in this paper. Two different approaches are used to generate quality grids over the iced airfoil. Both grids can be categorized as generalized grids since multiple element types are employed in each. In the first approach, a structured grid is generated near the body using a marching scheme and the rest of the domain is filled with an unstructured grid. In the second approach, the marching strategy is coupled with a node deletion/insertion algorithm to generate a quad-dominant grid. An edge based data structure is used to store the grid information to handle polygons with an arbitrary number of sides. A finite-volume, cell-centered scheme is used to solve the integral form of the full Navier–Stokes equations. The numerical fluxes crossing the cellfaces are calculated using Roe's approximate Riemann solver. The turbulent viscosity is estimated using Spalart–Allmaras one equation turbulence model. The results of computations are presented, together with comparison to NPARC results.

A numerical model for the flooding and drying of irregular domains

AbstractA numerical technique for the modelling of shallow water flow in one and two dimensions is presented in this work along with the results obtained in different applications involving unsteady flows in complex geometries. A cell‐centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the flow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. It is shown that this modification reproduces exactly steady state of still water in configurations with strong variations in bed slope and contour. The applications presented are mainly related with unsteady flow problems. The scheme is capable of handling complex flow domains as will be shown in the simulations corresponding to the test cases that are going to be presented. Comparisons of experimental and numerical results are shown for some of the tests. Copyright © 2002 John Wiley & Sons, Ltd.

Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography

A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe's approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.

Finite Volume Model for Nonlevel Basin Irrigation

A finite volume model for unsteady, two-dimensional, shallow water flow is developed and applied to simulate the advance and infiltration of an irrigation wave in two-dimensional basins of complex topography. The fluxes are computed with Roe's approximate Riemann solver and the monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to eliminate spurious oscillations and the model incorporates an efficient and robust scheme to capture the wetting and drying of the soil. Model predictions are compared with experimental data for one- and two-dimensional problems involving rough, impermeable, and permeable beds, including a poorly leveled basin.

Dissipation Improvement of MUSCL Scheme for Computational Aeroacoustics

ABSTRACTAn upwind finite-volume scheme is studied for solving the solutions of two dimensional Euler equations. It based on the MUSCL (Monotone Upstream Scheme for Conservation Laws) approach with the Roe approximate Riemann solver for the numerical flux evaluation. First, dissipation and dispersion relation, and group velocity of the scheme are derived to analyze the capability of the proposed scheme for capturing physical waves, such as acoustic, entropy, and vorticity waves. Then the scheme is greatly enhanced through a strategy on the numerical dissipation to effectively handle aeroacoustic computations. The numerical results indicate that the numerical dissipation strategy allows that the scheme simulates the continuous waves, such as sound and sine waves, at fourth-order accuracy and captures the discontinuous waves, such a shock wave, sharply as well as most of upwind schemes do. The tested problems include linear wave convection, propagation of a sine-wave packet, propagation of discontinuous and sine waves, shock and sine wave interaction, propagation of acoustic, vorticity, and density pulses in an uniform freestream, and two-dimensional traveling vortex in a low-speed freestream.