Flux-difference Splitting Method Research Articles

Investigate Impact Force of Dam-Break Flow against Structures by Both 2D and 3D Numerical Simulations

The aim of this paper was to investigate the ability of some 2D and 3D numerical models to simulate flood waves in the presence of an isolated building or building array in an inundated area. Firstly, the proposed 2D numerical model was based on the finite-volume method (FVM) to solve 2D shallow-water equations (2D-SWEs) on structured mesh. The flux-difference splitting method (FDS) was utilized to obtain an exact mass balance while the Roe scheme was invoked to approximate Riemann problems. Secondly, the 3D commercially available CFD software package was selected, which contained a Flow 3D model with two turbulent models: Reynolds-averaged Navier-Stokes (RANs) with a renormalized group (RNG) and a large-eddy simulation (LES). The numerical results of an impact force on an obstruction due to a dam-break flow showed that a 3D solution was much better than a 2D one. By comparing the 3D numerical force results of an impact force acting on building arrays with the existence experimental data, the influence of velocity-induced force on a dynamic force was quantified by a function of the Froude number and the water depth of the incident wave. Furthermore, we investigated the effect of the initial water stage and dam-break width on the 3D-computed results of the peak value of force intensity.

Numerical experiments of flux difference splitting methods with high resolution scheme for supersonic flows

In this work, we have carried out the assessment of a high resolution scheme for unsteady compressible flow. For high order spatial accuracy, we have used fifth order weighted essentially non oscillatory (WENO) scheme. This scheme is applied to four flux difference splitting (FDS) methods: Harten-Lax-van-Leer (HLL), Roe solver, Harten-Lax-van Leer-Contact (HLLC), and Rusanov methods. We have compared results of these flux schemes with each other. WENO scheme is used for the reconstruction of left and right state variable across the cell interface for high resolution. The reconstruction procedure is performed in terms of primitive variables instead of conservative variable, in order to avoid spurious oscillation. We have considered two test cases: shock wave reflection and supersonic viscous flow over a flat plate, to access the performance of FDS schemes. An explicit third order TVD Runge-Kutta method is used for advancement of solution in time. The present results are compared with available numerical solutions. WENO-HLLC has good shock capturing capabilities as compare to WENO-Roe, WENO-HLL and WENO-Rusanov methods. It also provides best results inside and outside the boundary layer.

A flux splitting method for the Euler equations

The solution of the Euler and Navier–Stokes equations was the focus of this research. As with methods such as the AUSM-family, this work further attempted to construct an explicit upwind method that combines both simplicity and accuracy. In particular, with an appropriate choice of working variables, the aim was to develop a simple method that is based on the advection of the characteristic variables. The end result is a computationally efficient method valid for all homogeneous fluids with accuracy comparable to that of expensive flux-difference splitting methods. Good performance was extensively verified with Riemann test problems and results were on par with Roe's scheme. Furthermore, at least one formulation of the method passed all test problems satisfactory without requiring manual intervention. The method also illuminates some similarities between flux-difference splitting and the AUSM-family of methods.

Simulation of dam-break problem by a robust flux-vector splitting approach in Cartesian grid

A robust and effective flux-vector splitting method is utilized to simulate dam-break problems based on a finite-volume method on a Cartesian grid. The method combines the effectiveness of Flux-Difference Splitting (FDS) methods and the robustness of Flux-Vector Splitting (FVS) methods to precisely estimate the numerical flux at each cell interface. A nonlinear second-order term, which is evaluated in local characteristics field, is added to decomposed fluxes to enhance the accuracy of the scheme. Several classical dam-break problems including 1D dam-break problem, hypothetical 2D dam-break problem with non-symmetric breach, circular dam-break problem and 2D dam-break problem with a baffle in the floodplain have been tested by the model. The results show very satisfactory agreement with analytical solutions and other numerical methods. The paper shows the accuracy, effectiveness and robustness of the hybrid FVS and FDS methods, both over wet and dry bed.

On the Development of Turbomachine Blade Aerodynamic Design System

A turbomachine blade aerodynamic design process is proposed to design turbomachine blades. The design system, including a global optimization of through flow for whole machine and a local optimization of the airfoil design and airfoil section, stacks up. The airfoil generator code employs Bezier polynomial curves to produce smooth airfoil shapes. A meanline program combined with an optimizer was used to perform the global optimization. In the airfoil section design, for fast calculations of the airfoil pressure distributions a discrete vortex method is developed. A novel Navier-Stokes (N-S) solver is developed for further examination of the airfoil performance for final section design. The N-S code is used to obtain the blade-to-blade quasi-three-dimensional, turbulent, and viscous flow characteristics. The time-dependent N-S equations are discretized and integrated in a coupled manner based on a finite-volume formulation, as well as a flux-difference splitting. The flux-difference splitting method enables us to compute with a rapid convergence. The present N-S code can handle computations of both subsonic and supersonic flows and can be connected to an external optimizer code with the airfoil generator. After optimized airfoil was designed, a three-dimensional code was used for final tuning of the design.

Experimental and numerical study of combusting solid propellant in an axisymmetric flow field with two moving boundaries

Simulation of compressible gases produced by combustion of solid propellant inside a cylinder with a moving boundary experimentally and numerically is considered. One-dimensional combustion model is used for solid propellant while, axisymmetrical Euler Equation can be used to simulate combusting gases. A finite volume upwind scheme based on the Roe's flux-difference splitting method is employed. VOF technique is applied to capture moving boundary. Test technique applied to this case measures chamber pressure vs. time with piezoelectric transducer and final velocity with two light screens. The numerical results compared well with experimental results.

High-Order Unstructured Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Aerodynamic Flows

In the present work, essentially nonoscillatory schemes and weighted essentially nonoscillatory schemes are implemented in a cell centered finite volume context on unstructured meshes. The two-dimensional Euler equations are considered to represent the flows of interest. The essentially nonoscillatory and weighted essentially nonoscillatory schemes have been developed with the purpose of accurately capturing discontinuities appearing in problems governed by hyperbolic conservation laws. In the aerodynamic studies of interest in the paper, these discontinuities are mainly represented by shock waves. The entire reconstruction process of essentially nonoscillatory and weighted essentially nonoscillatory schemes is described in detail for any order of accuracy with an emphasis on the implementation of second-order and third-order schemes. A flux-difference splitting method and a flux-vector splitting method are tested and compared. The code developed also features an agglomeration multigrid procedure for convergence acceleration, and an adaptive mesh refinement tool. Applications for aerodynamic flows are performed to assess the capability implemented against data available in the literature.

Developing High-Order Weighted Compact Nonlinear Schemes

The weighted technique is introduced in the compact high-order nonlinear schemes (CNS) and three fourth- and fifth-order weighted compact nonlinear schemes (WCNS) are developed in this paper. By Fourier analysis, the dissipative and dispersive features of WCNS are discussed. In view of the modified wave number, the WCNS are equivalent to fifth-order upwind biased explicit schemes in smooth regions and the interpolations at cell-edges dominate the properties of WCNS. Both flux difference splitting and flux vector splitting methods can be applied in WCNS, though they are finite difference schemes. Boundary and near boundary schemes are developed and the asymptotic stability of WCNS is analyzed. Several numerical results are given which show the good performances of WCNS for discontinuity capture high accuracy for boundary layer calculation, and good convergent rate. We also compare WCNS with MUSCL scheme and spectral solutions. WCNS are more accurate than MUSCL, as expected, especially for heat transfer calculations.

Modelling of freeway merging and diverging flow dynamics

In this paper, a new high-order continuum model for a freeway with merging and diverging geometries is presented. Three formulations describing traffic friction at freeway merging or diverging areas are reviewed. A new formulation for traffic friction is proposed and compared with the existing formulations based on tests against field data. In addition, a numerical modification for avoiding unjustified upstream flow propagation that has been observed in high-order models is developed. For this purpose, a Riemann problem-based method, namely the Harten-Lax-van Leer (HLLE) flux-difference splitting method, is employed. Intuitive tests demonstrate that such an approach can be very successful in preventing the flow from moving at negative speed. Field data tests for merging and diverging freeway geometries show that implementation of the proposed high-order model with the HLLE method yields low error levels and is more accurate than the simple continuum model at congested flows. Furthermore the proposed formulation for traffic friction seems to yield a smaller error in volume than the existing formulations. In addition the proposed high-order model is less sensitive to the choice of the speed-density relationship.

Influence of Reactive Particles on The Formation of a One-Dimensional Detonation Wave

ABSTRACT A model is proposed which demonstrates the influence of reactive particles on the formation of a detonation wave in a combustible two-phase system. Detailed kinetics was used to describe the gas phase processes whereas two different models for the description of the heterogeneous gas-solid reactions were introduced to describe the combustion. The governing equations, formulated for two interacting continua, were solved numerically using state of the art flux-difference splitting methods and implicit time integration. The shock and detonation waves were resolved correctly by employing a dynamically moving grid. Results obtained indicate the formation of detonation wave with a double-head structure.

Dissipation Additions to Flux-Difference Splitting

Although the flux-difference splitting methods for solving the Euler equations are generally very robust and no explicit dissipation is required. There are situations where explicit dissipation is needed. Two cases, a slowly moving shock problem and a blunt body calculation, are discussed in this paper. The slowly moving shock problem is tested extensively by Roe's Riemann solver and a cure for Roe's Riemann solver is proposed. For the second-order scheme it is found necessary to reduce the second-order accuracy to first-order accuracy inside the shock layer. For the supersonic blunt body calculation adding dissipation in the linear waves in Roe's Riemann solver can prevent numerical instability in the subsonic pocket. The drawback of Yee's formula to cure the instability when used on viscous flow calculation is demonstrated. A better solution based on the pressure gradient is proposed.

Two-dimensional unsteady leading-edge separation on a pitching airfoil

The initial stages of two-dimension al unsteady leading-edge boundary-layer separation of laminar subsonic flow over a pitching NACA-0012 airfoil have been studied numerically at Reynolds number (based on airfoil chord length) Rec = 104, Mach number Mx = 0.2, and nondimensional pitch rate H£ = 0.2. Computations have been performed using two separate algorithms for the compressible laminar Navier-Stokes equations. The first method, denoted the structured grid algorithm, utilizes a structured, boundary-fitted C grid and employs the implicit approximate-factorization algorithm of Beam and Warming. The second method, denoted the unstructured grid algorithm, utilizes an unstructured grid of triangles and employs the flux-difference splitting method of Roe and a discrete representation of Gauss' theorem for the in viscid and viscous terms, respectively. Both algorithms are second-order accurate in space and time and have been extensively validated through comparison with analytical and previous numerical results for a variety of problems. The results show the emergence of a primary clockwise-rotating recirculating region near the leading edge which can be traced to a pair of critical points (a center and a saddle) that appear within the flowfield, followed by a secondary counterclockwise-rotating recirculating region and a tertiary clockwise-rotating recirculating region. The primary and secondary recirculating regions interact with each other to give rise to the unsteady separation (breakaway) of the boundary layer.

Computation of Incompressible Viscous Flow around a Marine Propeller

This paper describes an application of CFD (Computational Fluid Dynamics) to the computation of incompressible viscous flow around a marine propeller. High Reynolds-number flow around a marine propeller operating in a uniform flow is simulated using a newly developed CFD scheme, which is based on cell-centered, finite-volume method with global conservation property for mass and momentum flux.Governing equations are full Navier-Stokes equations and continuity equation with pseudo-compressibility, which are written in boundary-fitted curvilinear coordinate system fixed on a propeller blade. For discretization in space, 3rd-order upwind scheme derived from flux-difference splitting method is adopted for convection terms and 2nd-order central difference scheme for viscous terms, respectively. 1st-order Euler implicit scheme is adopted for the time integration and the resulted discretized equations are solved by Implicit Approximate Factorization (IAF) scheme with high efficiency.A series of high Reynolds-number flow computations around a SEIUNMARU conventional propeller are made to show validity and availability of the present scheme as a practical design tool. Through quantitative comparison of the computed results with experimental data for surface pressure distributions, thrust and torque coefficients, it is clearly shown that present scheme can predict a viscous flow around a marine propeller qualitatively well and that several problems, such as open boundary conditions and excessive grid skewness, should be solved in order to improve accuracy of the present scheme.By flow visualization technique using the computed results, some important characteristics of a rotating propeller flow such as the development of boundary layers and separation on a propeller blade, tip vortex generation and a structure of trailing vortices, are found to be well simulated qualitatively by the present scheme, which may be useful tool for the development of the inviscid-flow calculation method.

Multigrid formulation of polynomial flux-difference splitting for steady Euler equations

A flux-difference splitting method based on the polynomial character of the flux vectors is applied to steady Euler equations. The discretization is done with a vertex-centered finite volume method. In first-order form, a discrete set of equations is obtained which is both conservative and positive. The flux-difference splitting is done in an algebraically exact way, so that shocks are represented without wiggles. Due to the positivity, the set of equations can be solved by collective relaxation methods. A full multigrid method based on symmetric successive relaxation, full weighting, bilinear interpolation and W-cycle is presented. In firstorder form, typical full multigrid efficiency is achieved. This is demonstrated on the GAMM transonic bump test-case. The second-order formulation is based on the Roe-Chakravarthy minmod-limiter. The discrete system is solved using a multigrid defect-correction formulation. The second-order formulation is demonstrated on Harten's shock reflection problem.

Flux-difference splitting for inviscid, real gases with non-equilibrium chemistry

A flux-difference splitting method is presented for the inviscid terms of the compressible flow equations for chemical non-equilibrium gases.

A Cell-Centered, Finite-Volume Upwind Scheme with Global Conservation

A finite-volume, conservative upwind scheme has been developed, based on the flux-difference splitting method. Pseudo-compressibility is introduced to the continuity equation. The cell-centered is adopted, i. e., nodes for flow variables were placed at the center of each grid cell. With this combination of the scheme and the node-cell layout, the global conservation property has been derived in a straightforward manner.The scheme was applied to two types of flows. First the flow past a circular cylinder was computed using the O-grid at the Reynolds number Re =40. The integrated momentum and mass fluxes at inner and outer boundaries agreed up to more than 9 significant figures after 1, 000 time steps. Thus the global conservation property was confirmed. The computed drag coefficient value agreed well with other computed values. The same flow was computed using the H-grid. The drag coefficient value thus obtained differed very little from the O-grid value.The flow past a flat plate with a point of mapping singularity was computed at an attack angle of 30 degrees. It was confirmed that the global conservation property of the present scheme is not affected by the presence of mapping singularities.

Multigrid acceleration of a flux-difference splitting method for steady Euler equations

A flux-difference splitting based on the polynomial character of the flux-vectors is introduced for steady Euler equations. This splitting is applied to finite volumes centered around the vertices of the computational grid. A discrete set of equations is obtained which is both conservative and positive. The flux-difference splitting is done in an algebraically exact way. As a consequence, shocks are represented as sharp discontinuities, without wiggles. Due to the positivity, the set of equations can be solved by collective relaxation methods. A full multigrid method based on symmetric successive relaxation, full weighting, bilinear interpolation and W-cycle is presented. Typical full multigrid efficiency is achieved for the GAMM transonic bump test case since after the starting cycle and one multigrid cycle, the solution cannot be distinguished anymore from the fully converged solution.

A flux-difference splitting method for steady euler equations

The polynomial flux-difference splitting method is applied to steady Euler equations. A discrete set of equations which is both conservative and positive is obtained. The set of equations is solved by the relaxation technique.