Steger-Warming Flux Vector Splitting Research Articles

Numerical Investigation of Supersonic Flow over a Wedge by Solving 2D Euler Equations Utilizing the Steger–Warming Flux Vector Splitting (FVS) Scheme

Supersonic flow over a half-angle wedge (θ = 15°) with an upstream Mach number of 2.0 was investigated using 2D Euler equations where sea level conditions were considered. The investigation employed the Steger–Warming flux vector splitting (FVS) method executed in MATLAB 9.13.0 (R2022b) software. The study involved a meticulous comparison between theoretical calculations and numerical results. Particularly, the research emphasized the angle of oblique shock and downstream flow properties. A substantial iteration count of 2000 iteratively refined the outcomes, underscoring the role of advanced computational resources. Validation and comparative assessment were conducted to elucidate the superiority of the Steger–Warming flux vector splitting (FVS) scheme over existing methodologies. This research serves as a link between theoretical rigor and practical applications in high-speed aerospace design, enhancing the efficiency of aircraft components.

A 3-D Meshless Method for Solving Maxwell’s Equations by Using the Steger-Warming Flux Vector Splitting Approach

In order to solve Maxwell’s equations so as to study the electromagnetic stealth characteristics of targets, a 3-D meshless method is developed. According to the meshless method, the spatial derivatives on clouds of points are computed by using the weighted least square approach. After that, the Steger-Warming flux vector splitting approach is used to calculate the physical flux of Maxwell’s equations. By using the developed method, the calculated bistatic radar cross sections (RCS) of a 3-D sphere are obtained, which are agree with the series solutions. Finally, the electromagnetic scattering characteristics for a 3-D stealth aircraft model with different situations is given, which shows the developed method has the ability to accommodate complicated 3-D configurations with multi-element.

A 2-D meshless time-domain algorithm for solving Maxwell’s equations based on Steger-Warming flux vector splitting approach

A 2-D meshless time-domain algorithm is proposed for solving Maxwell’s equations. The spatial derivatives related to the algorithm are approximated by using an expanded Taylor series and the least square technique in each cloud of points, and then a Steger-Warming flux vector splitting approach is introduced for computing the physical flux of the governing equations. After that, an explicit four-stage Runge-Kutta scheme is employed in time-marching. Combined with solving Maxwell’s equations, the implementations of the present algorithm are described in details. Based on the proposed algorithm, bistatic radar cross sections for typical 2-D objects in different frequency regions are given, which show a good agreement with those of the series solutions. The paper ends with the presentation of the electromagnetic scattering fields for a 2-D aircraft model, which display the characteristics of the algorithm presented in dealing with time-domain problems to a certain extent.

A 3-D simulation tool for hydrogen detonation during severe accident and its application

A 3-D simulation tool (DEST, Detonation Simulation tool) for hydrogen detonation during severe accident has been developed. The compressible three-dimensional Euler equations in Cartesian coordinate system are solved by a second-order additive semi-implicit Runge–Kutta method with 5th weighted essentially non-oscillatory scheme (WENO) and Steger-warming flux vector splitting approach handling the convection flux. The multi-blocking patching method is used to handle the flow problem of complex geometry. Afterward, the hydrodynamics solver is tested by applying DEST to the classical shock tube problems, and the results show that hydrodynamics solver can capture the shock wave correctly and give a detailed description of shock wave even if the gradient of pressure, density or velocity is large. Then two detonation experimental tests of RUT facility are simulated by DEST, and the comparisons between the predicted results and experimental data show a reasonably good agreement. Furthermore, DEST is applied to a hypothetical detonation in two connected compartments of containment. The predicted results show that the maximum pressure and temperature always occur at the corner of the compartment. The influence of ignition position is analyzed and the results show that the pressure tended to be larger when the mixture is ignited at a farther location. The DEST can be used to analyze the pressure and temperature load during detonation process under severe accident and has guidance on the strategy of hydrogen mitigation.

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities, II

SummaryProblems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.

Comparative study of numerical schemes for strong shock simulation using the Euler equations

A numerical study of extremely strong shocks was presented. Various types of numerical schemes with first-order accuracy and higherorder accuracy with adaptive stencils were implemented to solve the one and twodimensional Euler equations based on the explicit finite difference method, including Roe’s first-order upwind, Steger-Warming Flux Vector splitting (FVS), Sweby’s flux-limited and Essentially Non-oscillatory (ENO) scheme. The result comparisons were carried out to discuss which scheme is the most suitable for strong shock problem. The dissipative nature of the firstorder scheme can be easily seen from the numerical solutions. High order ENO scheme had the best resolution for the case having weak discontinuity, but it over- predicted the shock wave location for the case of strong discontinuity.

Numerical solutions of coupling hydraulic transient and axial structural vibration in liquid-filled pipelines by flux vector splitting scheme

Method of characteristics is used widely to solve classical hydraulic transient problems. However, when it is applied to solve coupling hydraulic transient problems, excessive interpolation errors may be introduced into results because of unavoidable multi–wave interpolated calculations. To overcome the disadvantage, a novel finite difference scheme based on Steger–Warming flux vector splitting is proposed. Firstly, flux vector splitting scheme for coupling hydraulic transient model of liquid–filled pipelines is established. Then flux subvectors are discretised by Lax–Wendroff central difference scheme and Warming–Beam upwind difference scheme possessing second order precision in time and space. Finally, according to Rankine–Hugoniot conditions and corresponding boundary conditions, an effective method is developed to solve those points located at the boundaries, which can avoid the problem beyond calculation region induced by using second order discrete technology directly. The numerical simulation shows that the proposed computation scheme has several desirable advantages, such as high precision, excellent capability of capturing the shock waves, without false numerical oscillation, low sensitivity to Courant number and good stability.

Steger-Warming flux vector splitting method for special relativistic hydrodynamics

This paper discusses the properties of the rotational invariance and hyperbolicity in time of the governing equations of the ideal special relativistic hydrodynamics and proves for the first time that the ideal relativistic hydrodynamical equations satisfy the homogeneity property, which is the footstone of the Steger–Warming flux vector splitting method [J. L. Steger and R. F. Warming, J. Comput. Phys., 40(1981), 263–293]. On the basis of this remarkable property, the Steger–Warming flux vector splitting (SW-FVS) is given. Two high-resolution SW-FVS schemes are also given on the basis of the initial reconstructions of the solutions and the fluxes, respectively. Several numerical experiments are conducted to validate the performance of the SW-FVS method. Copyright © 2013 John Wiley & Sons, Ltd.

Solution of the Euler Equations Based on Flux Vector Splitting

The basic properties of the Euler equations are learned by studying hyperbolic PDEs and the Steger-Warming flux vector splitting approach and the NND scheme are given. The NND scheme based on improved Steger-Warming flux vector splitting is used for the solution of 1-D and shown that the numerical oscillation is restrained .The finite difference method has higher order of accuracy and better efficiency.

Numerical experiments using high-resolution schemes for unsteady, inviscid, compressible flows

The performance of seven high-resolution schemes is investigated in various unsteady, inviscid, compressible flows. We employ the Roe, HLL (Harten, Lax and van Leer), and HLLC (Toro et al.) Riemann solvers, two variants of the van Leer and Steger–Warming flux vector splitting (FVS) schemes, Rusanov's scheme, and a hybrid total variation diminishing (TVD) scheme that combines a high-order Riemann solver with a flux vector splitting scheme. The above schemes have been implemented in conjunction with an implicit-unfactored method which is based on Newton-type sub-iterations and Gauss–Seidel relaxation. The performance of the schemes has been assessed in six unsteady flow problems: two one-dimensional shock tube problems, shock-wave reflection from a wedge, shock-wave diffraction around a cylinder, blast-wave propagation in an enclosure, and interaction of a shock wave with a gas bubble. More dissipative schemes do not necessarily provide faster convergence per time step and also suppress instabilities that occur in certain unsteady flow problems. The efficiency of the solution depends strongly on the advective (high-resolution) scheme. The results reveal that the Roe, HLLC and hybrid TVD schemes provide similar and overall the best results. For the unsteady problems considered here, the computations show that an explicit implementation based on a TVD, fourth-order Runge–Kutta method results in longer computing times than the implicit-unfactored method.

A Triangle-Based Unstructured Finite-Volume Method for Chemically Reactive Hypersonic Flows

A triangle-based unstructured finite-volume method is developed for chemically reactive hypersonic calculations. The method is based on a Steger–Warming flux-vector splitting approach generalized to mixtures of thermally perfect gases. Second-order-in-space and time accuracy is provided by limited flux blending and an implicit multi-stage time marching scheme. The final stiff non-linear problem resulting from discretization presents a very peculiar block diagonal structure. This allows a decoupling of the species and gas dynamic equations in smaller subproblems. A linear algebra argument based on M-matrix theory makes it possible also to show that the method guarantees positivity of species mass densities and vibrational energies under a reasonable CFL-like constraint. Finally, a set of 2-D numerical test cases illustrates the performance of the method.

Focusing shock waves generated by an accelerating projectile

Unsteady flow fields past a projectile either in parabolic motion or in accelerated motion are investigated numerically, with special attention to shock wave focusing phenomena induced by the motions. The two-dimensional Euler equations are solved by a moving-cell, MUSCL TVD finite-volume method with Steger-Warming flux-vector splitting. The pattern of shock wave focusing induced by the projectile is found to be similar to that of shock waves reflected from a concave wall observed in a laboratory experiment using a shock tube. Effects of projectile shapes and trajectories on focusing are also examined.

Numerical simulation of nonequilibrium condensation in a hypersonic wind tunnel

A computational method for calculating supersonic two-dimensional nozzle flows with nonequilibrium phase change is developed. The code uses a fully implicit, finite volume formulation with Steger-Warming flux vector splitting and Jacobi point relaxation. Source terms for finite rate condensation and evaporation are constructed from the classical theories of nucleation and droplet growth. Calculations of the NASA Langley Research Center 8-ft High Temperature Tunnel nozzle are performed and compared with previously reported quasi-one-dimensional calculations and experimental data.

Zonal-local solution method for the turbulent Navier-Stokes equations

AZONAL-LOCAL solution method is developed for the solution of the compressible Navier-Stokes equations. The main feature of the method is coupling of the NavierStokes equations with the Euler equations and the local mesh solution procedure. Mesh sequencing (multilevel) technique is also used for the improvement of the efficiency of the algorithm. The methodology is applied to turbulent flowfields past an airfoil. A flux vector splitting method with an upwind scheme up to fourth-order accuracy is used for the discretization of the inviscid fluxes. The system of the equations is solved by an unfactored implicit method using Gauss-Seidel relaxation. Contents Techological improvements in supercomputer speed and memory size provided the means to solve the full compressible Navier-Stokes equations for turbulent flowfields and complex geometries. However, large amounts of computer time are required for the solution of the equations especially for problems in design practice. To reduce the computational time, a zonal-local solution method is presented for the solution of the two-dimensional Navier-Stokes equations in high Reynolds number flows. The equations are solved in time-dependent form and for a curvilinear body-fitted coordinate system JUt ^ + (Gvis)r] (1) where /is the Jacobian of the transformation of the Cartesian coordinates (x, z) to curvilinear coordinates (£, f), the subscripts inv, vis indicate the inviscid and viscous fluxes, respectively, and U is the conservative variables matrix. The equations are solved by a finite volume Navier-Stokes code1 which uses a modified 1 Steger-Warming flux vector splitting (FVS) method for the discretization of the inviscid fluxes. The FVS method split the fluxes into two parts, positive and negative, in accordance with the sign of the eigenvalues (2)