Flux-difference Splitting Research Articles

Numerical Simulation of Acoustic Waves Interacting with a Shock Wave in a Quasi-ID Convergent-Divergent Nozzle Using an Unstructured Finite Volume Algorithm

Numerical simulations of a very small amplitude acoustic wave interacting with a shock wave in a quasi-ID convergent-divergent nozzle is performed using an unstructured finite volume algorithm with piece-wise linear, least square reconstruction, Roe flux difference splitting, and second-order MacCormack time marching. First, the spatial accuracy of the algorithm is evaluated for steady flows with and without the normal shock by running the simulation with a sequence of successively finer meshes. Then the accuracy of the Roe flux difference splitting near the sonic transition point is examined for different reconstruction schemes. Finally, the unsteady numerical solutions with the acoustic perturbation are presented and compared with linear theory results.

An approximate Riemann solver for a two-phase flow model with numerically given slip relation

In this paper an approximate Riemann solver is constructed for solving one class of two-phase models. These models describe the gas–liquid flow in a long tube where the flow behaviour perpendicular to the tube axis is averaged, so that the model is essentially one-dimensional in the direction of the axis. The model consists of equations for the conservation of mass for each of the phases and the conservation of momentum of the mixture. In addition, an equation is supplied which relates the velocities of the two phases at any point, the slip relation, which may have a different shape for each of the flow regimes of interest and change in time. Generally, a slip relation will not be known entirely in algebraic form, but given partly in numerical form. Under certain restrictions the resulting system of conservation laws is hyperbolic and allows discontinuous solutions. The general idea is that the numerical algorithm for solving this model must be able to handle any valid slip relation. As the slip relation affects the Jacobian of the flux function to a large extent, this means that flux vector or flux difference splittings cannot be based on algebraic manipulation of the Jacobian. We propose to use a first-order upwind scheme of Roe-type, where the construction of the approximate Riemann solver is fully numerical. This basic scheme, however, has the same limitation as the original Roe method, namely, that for systems the positivity of the solution is not guaranteed. The modification of the basic scheme to ensure positivity of the solution, is based on the HLL Riemann solver.

A new total variation diminishing scheme for the calculation of one-dimensional flow in inlet and exhaust pipes of internal combustion engines

Total variation diminishing (TVD) schemes have been recently introduced for the calculation of the one-dimensional unsteady flow in the inlet and outlet pipes of internal combustion engines. This paper describes the flux difference splitting technique (with first- or second-order upwind fluxes) for the classic TVD schemes. To avoid problems at nodes with a section change, a new TVD scheme is developed. This paper further describes a method to impose the boundary condition at the pipe end, independent of the numerical scheme used. This is shown for a reservoir inlet of the pipe and a subsonic outlet flow. For two test cases (the shock-tube and the tapered-pipe calculation), the new TVD algorithm is compared with the classic TVD schemes. The evaluation shows that the new cell—vertex TVD scheme with superbee limiter in two stage form combines a high accuracy with an exact representation of the mass flow in each of the nodes.

Assessment of Artificial Dissipation Models for Three-Dimensional Incompressible Flow Solutions

Various approaches for constructing artificial dissipation terms for three-dimensional artificial compressibility algorithms are presented and evaluated. Two, second-order accurate, central-differencing schemes, with explicitly added scalar and matrix-valued fourth-difference artificial dissipation, respectively, and a third-order accurate flux-difference splitting upwind scheme are implemented in a multigrid time-stepping procedure and applied to calculate laminar flow through a strongly curved duct. Extensive grid-refinement studies are carried out to investigate the grid sensitivity of each discretization approach. The calculations indicate that even the finest mesh employed, consisting of over 700,000 grid nodes, is not sufficient to establish grid independent solutions. However, all three schemes appear to converge toward the same solution as the grid spacing approaches zero. The matrix-valued dissipation scheme introduces the least amount of artificial dissipation and should be expected to yield the most accurate solutions on a given mesh. The flux-difference splitting upwind scheme, on the other hand, is more dissipative and, thus, particularly sensitive to grid resolution, but exhibits the best overall convergence characteristics on grids with large aspect ratios.

Strongly-Coupled Multigrid Method for 3-D Incompressible Flows Using Near-Wall Turbulence Closures

An efficient artificial compressibility algorithm is developed for solving the three-dimensional Reynolds-averaged Navier-Stokes equations in conjunction with the low-Reynolds number k-ω turbulence model (Wilcox, 1994). Two second-order accurate central-differencing schemes, with scalar and matrix-valued artificial dissipation, respectively, and a third-order accurate flux-difference splitting upwind scheme are implemented for discretizing the convective terms. The discrete equations are integrated in time using a Runge-Kutta algorithm enhanced with local time stepping, implicit residual smoothing, and V-cycle multigrid acceleration with full- and semi-coarsening capabilities. Both loosely and strongly-coupled strategies for solving the turbulence closure equations are developed and their relative efficiency is evaluated. Calculations are carried out for turbulent flow through a strongly-curved 180 deg pipe bend discretized with fine, highly-stretched and skewed meshes. It is shown that the strongly-coupled multigrid algorithm, with semi-coarsening in the transverse plane, is an efficient approach for simulating flows of practical interest with advanced near-wall turbulence closures.

A Robust Low Diffusive Kinetic Scheme for the Navier–Stokes/Euler Equations

A new kinetic scheme based on the equilibrium flux method (EFM) and modified using Osher intermediate states is proposed. This new scheme called EFMO combines the robustness of the equilibrium flux method and the accuracy of flux-difference splitting schemes. The original EFM scheme is expressed in terms of simple wave decomposition in which only the linearly degenerate subpath is calculated from Osher numerical flux while nonlinear waves are still evaluated from the regular EFM splitting. Owing to its capability of withstanding intense nonlinear waves and yet exactly resolving contact discontinuities, EFMO is particularly well suited for the resolution of the Navier–Stokes equations as demonstrated by a series of severe test cases including the high-speed viscous flow around a cone, a shock-boundary layer interaction problem, a vacuum apparition problem, the hypersonic flow around a circular cylinder at Mach 100, and the forward-facing step at Mach 3.

Comparative Study of Flux splitting Method for Steady/Unsteady Flow.

When introducing the TVD method to the system equation, it is important to treat discontinuity near the shock waves. Flux splitting method handles this discontinuity by separating flux at the cell interface. The flux splitting method is divided into two categories : FDS (Flux Difference Splitting) and FVS (Flux Vector Splitting). However, they have some defects. For example, in the case of FDS, it becomes diffcult to maintain stability because of strong expansion wave. Also, excessive numerical diffusivity makes the results far from the reality in the case of FVS. Recently AUSM (Advection Upstream Splitting Method) has been proposed. In the present paper, a blunt body with a compressive corner are investigated using three flux splitting methods. We demonstrate the efficiency of AUSM, compared to the other flux splitting methods.

Approximate Riemann Solvers in FVM for 2D Hydraulic Shock Wave Modeling

This paper presents three approximate Riemann solver schemes, namely: the flux vector splitting (FVS), the flux difference splitting (FDS), and the Osher scheme. Originally used to solve the Euler equations in aerodynamic problems, these Riemann solvers based on the characteristic theory are used in the finite volume method (FVM) for solving the two-dimensional shallow water equations. The three solvers are compared in this paper according to theoretical development, difference schemes, practical applications to shock wave problems, and sensitivity analysis on the computational stability of the methods. The effects of changes in bed elevations on the solutions are also investigated. Comparison of numerical and analytical solutions indicates that very good agreement can be obtained by all three approximate Riemann solvers. Differences in accuracy, computer time, and numerical stability among the three schemes are not significant. For practical purposes, all of them can satisfactorily simulate the hydraulic phenomena in subcritical and supercritical flows as well as in smooth and discontinuous flows, especially shock wave modeling. These solvers are useful for studying levee failure or dam break due to extreme flood events, or the sudden opening or closing of sluice gates in a channel.

Numerical study of multidirectional-curvature waverider with finlets

A comprehensive numerical study is performed for the flowfields associated with multidirectional-curvature waveriders with finlets. This hypersonic vehicle, composed of an airframe and reflexed trailing edge, is developed by means of the waverider concept. The airframe and finlets can he constructed simultaneously by making an appropriate selection of a freestream trailing-edge curve/In this study, a three-dimensional grid system generated by elliptic differential equations is developed. Additionally, the Navier-Stokes equations with a flux-difference splitting numerical method are utilized to simulate the flowfields associated with the design and off-design flight conditions. At the design Mach number 4, the inviscid numerical results are compared with analytical estimates; in addition, the viscous effects for aerodynamic performance are evaluated. Also, an investigation is made of the variation in aerodynamic performance caused by altering the flight Mach number from 2 to 6. Moreover, the influence on aerodynamic performance and stability of the addition of finlets is discussed in detail.

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.

Cone-derived waverider flowfield simulation including turbulence and off-design conditions

Using the numerical scheme of Roe's flux-difference splitting, the Baldwin-Lomax algebraic turbulent model, and the time-averaged thin-layer Navier-Stokes equation, the flowfields for a selected Rasmussen's waverider forebody are investigated. Specific cases include turbulent flow on-design condition (M^ = 4, a = 0), turbulent flow off-design condition (Moo = 4, a = 5 deg), and laminar (and Euler) flow off-design condition (M^ = 5, a = 0). For comparison purposes, some results previously reported by the authors are presented. The off-design flowfields are more complicated than the on-design ones. For the M^ = 5 waverider, there is very strong stock-wave/boundarylayer interaction. For the a = 5 deg waverider, there is a detached shock, resulting in a small separation bubble and a vortex on the upper surface. The shock-wave/boundary-layer interaction and shock attachment pattern strongly affect the waverider pressure and local skin-friction coefficient distributions.

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.

A numerical scheme for steady. Supercritical flows with boundary fitted coordinates

An algorithm based on flux difference splitting is presented for the solution of two‐dimensional, steady, supercritical open channel flows. A transformation maps a non‐rectangular, physical domain into a rectangular one. The governing equations are then the shallow water equations, including terms of slope and friction, in a generalised coordinate system. A regular mesh on a rectangular computational domain can then be employed. The resulting scheme has good jump capturing properties and the advantage of using boundary/body‐fitted meshes. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.

Accurate Upwind Methods for the Euler Equations

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate and can be considered as extensions of Godunov’s scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer’s scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the “smooth” part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope-steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a “smooth” interface, for economy, the linearization employs the arithmetic average of the left and right states, and the flux is obtained by one of the four simple models: the primitive-variable splitting, a simplified flux splitting, the one-intermediate-state model, or the central difference with artificial viscosity. Near a discontinuity, Roe’s flux-difference splitting is used. The current presentation of Roe’s method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement and an efficient approximation to Osher’s approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions and yield high resolution at discontinuities.

Calculation of Reynolds stresses in turbulent supersonic flows

AbstractThe baseline numerical procedure of interest in this study combines flux vector splitting, flux difference splitting and an explicit treatment of the diffusion terms of the flow equations. The viscous terms are treated explicitly to preserve the wave propagation properties of the Euler fluxes and permit splitting. The experience with this scheme has been limited to laminar or, at best, ‘eddy viscosity’ flows. In this paper the applicability of the scheme is extended to include the calculation of turbulent Reynolds stresses in supersonic flows. The schemes and our implementation are discussed. Both laminar and turbulence subsets of the Reynolds/Favre‐averaged equations are tested, with a discussion of relative performance. The test problem for turbulence consists of a zero‐pressure‐gradient supersonic boundary layer as well as a supersonic boundary layer experiencing the combined effects of adverse pressure gradient, bulk compression and a concave streamline curvature. Excellent agreement with experimental measurements is observed for most of the quantities compared, which suggests that the numerical procedures presented in this paper are potentially very useful.

Supersonic flux-split procedure for second moments of turbulence

A finite volume procedure which combines flux-difference splitting and flux-vector splitting is presented in this paper. The diffusive fluxes are treated explicitly to preserve the upwinding structure of the split Euler fluxes. This procedure is extended in this paper from its essentially laminar or eddy viscosity form to include the equations for the six components of the Reynolds stress tensor and an additional equation for the solenoidal dissipation. The models used for the unclosed terms are presented as are the extensions of the numerical procedures to cover the turbulence equations. To validate the proposed procedures, a compressible turbulent flow over a flat plate at Mach number M∞ = 2.87 and Reynolds number per unit length Re/m of 6.5X10 7 was calculated and compared with experimental measurements. Also simulated was the case when the foregoing boundary-layer flow was made to pass over a ramp. A strong adverse pressure gradient case in which boundary-layer thickness-to-curvature ratio (δ/R c ) is 0.1 was considered. In this case, the supersonic turbulent boundary layer experiences the combined effects of an adverse pressure gradient, bulk compression, and concave streamline curvature.

Simulation of three-dimensional turbulent flows on unstructured meshes

A three-dimensional Navier-Stokes flow solver is developed on unstructured tetrahedral meshes. For a turbulence closure, a standard high-Reynolds-number k-e model with a wall function boundary condition is used. The seven equations of motion are discretized and integrated in a tightly coupled manner. The time integration is achieved using an explicit Runge-Kutta time-stepping scheme. The inviscid flux terms are discretized based on a cell-centered finite volume formulation with Roe's flux-difference splitting. The numerical method is applied for flows on a two-dimensional backward-facing step and a three-dimensional turbomachinery geometry. The results are compared with analytical and experimental data for validations.

Comparison of real gas simulations using different numerical methods

Two different numerical simulation codes (the SPLIT/NE code and the PoliTo code) are examined in this paper by comparing results for non-equilibrium flow cases using two different geometries, a hemisphere-cylinder-flare (axisymmetric) and a hyperbola-flare-cylinder configuration (axisymmetric and 2-D plane). The solution methods implemented in the codes are essentially different (finite difference, bow shock fitting, matrix splitting and Runge-Kutta high accuracy schemes for SPLIT/NE and finite volume, shock capturing, flux difference splitting and E.N.O. high accuracy schemes for PoliTo). The flow simulations are done employing physical models of different complexity (inviscid and viscous flows, chemically reacting flows in vibrational equilibrium and non-equilibrium, steady and unsteady flows). Comparing the solutions for the different codes, a very good agreement is obtained for both the hemisphere-cylinder-flare flow and the 2-D plane hyperbola-flare-cylinder flow. For relatively small flare angles, this is also true for the axisymmetric hyperbola-flare-cylinder flow; however, for larger flare angles, differences in the solutions can be detected which can be addressed to the formation of a strong recirculation bubble behind the detached flare shock.

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.

AN UNCOUPLED ALGORITHM FOR CALCULATING STEADY-STATE LAMINAR HEAT TRANSFER IN THREE-DIMENSIONAL INCOMPRESSIBLE FLOW

An uncoupled algorithm for the solution of the steady-state thermal energy equation in three-dimensional generalized curvilinear coordinates for an incompressible flow is described. The algorithm uses a higher-order flux-difference splitting technique for the convective terms and a second-order central difference scheme for the conductive terms. The equation is solved using a line-relaxation technique that allows the use of very large time steps, leading to rapid convergence. Preliminary results that demonstrate the code`s predictive capability are presented. The results agree well with test cases for which analytical solutions can be obtained and with classical boundary-layer theory.