Flux-difference Splitting Research Articles

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.

Inviscid flux-splitting algorithms for real gases with non-equilibrium chemistry

Several flux-splitting methods for the inviscid terms of the compressible-flow equations are derived for gases that are not in chemical equilibrium. Formulas are presented for the extension to chemical nonequilibrium of the Steger-Warming and Van Leer flux-vector splittings, and the Roe flux-difference splitting. The splittings are incorporated in a TVD algorithm and applied to one-dimensional shock-tube and nozzle flows of dissociating air, including five species and 11 reaction steps for the chemistry.

Computation of vortex-dominated flow for a delta wing undergoing pitching oscillation

The conservative, unsteady Euler equations for the flow relative to a moving frame of reference are used to solve for the three-dimensional steady and unsteady flows around a sharp-edged delta wing. The resulting equations are solved by using an implicit, approximately factored, finite-volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in crossflow planes at the grid-chord stations. The computational applications cover a steady flow around a delta wing, whose results serve as the initial conditions for the unsteady flow around a pitching delta wing at a large mean angle of attack. The steady results are compared with the experimental data, and the unsteady results are compared with results of a flux-difference splitting scheme.

Upwind parabolized Navier-Stokes code for chemically reacting flows

A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic, viscous, chemically reacting flow around two-dimensional or axisymmetric bodies. The new code is an extension of the upwind (perfect gas) PNS code of Lawrence et al. (1986). The upwind algorithm is based on Roe's flux-difference splitting scheme which has been modified to account for real gas effects. The algorithm solves the gas dynamic and species continuity equations in a 'loosely' coupled manner. The new code has been validated by computing the laminar flow (at free stream Mach number 25) of chemically reacting air over a wedge and a cone. The results of these computations are compared with the results from a centrally-differenced, fully coupled, nonequilibrium PNS code. The agreement is excellent, except in the vicinity of the shock wave where the present code exhibits superior shock capturing capabilities.

Three-dimensional upwind parabolized Navier-Stokes code for real gasflows

A real gas, upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional hypersonic flow of equilibrium air around various body shapes. The new code is an extension of the upwind (perfect gas) PNS code of Lawrence et al. (1986). The upwind algorithm is based on Roe's (1981) flux-difference splitting scheme which has been modified to account for real gas effects using the nearly exact approach of Vinokur and Liu (1988). Simplified curve fits are employed to obtain the thermodynamic and transport properties of equilibrium air. The new code has been validated by computing the M-infinity = 25 laminar flow of air over cones at various angles of attack. The results of these computations are compared with the results from a conventional centrally-differenced, real gas PNS code and the previous axisymmetric, upwind, real gas code. The agreement is excellent in all cases.

Splitting of inviscid fluxes for real gases

Flux-vector and flux-difference splittings for the inviscid terms of the compressible flow equations are derived under the assumption of a general equation of state for a real gas in equilibrium. No unnecessary assumptions, approximations, or auxiliary quantities are introduced. The formulas derived include several particular cases known for ideal gases and readily apply to curvilinear coordinates. Applications of the formulas in a TVD algorithm to one-dimensional shock-tube and nozzle problems show their quality and robustness.

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.

Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme, based on flux-difference splitting, is used to compute the convective terms. The upwind differencing is biased, based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. Both third-order and fifth-order differencing schemes are used on the convective fluxes throughout the grid's interior. The equations are solved using an implicit line relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudo-time, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, and good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at —90° angle of attack is computed.

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.

Unsteady transonic airfoil computation using implicit Euler scheme on body-fixed grid

The unsteady Euler equations have been derived for the flow relative motion with respect to a frame of reference that is rigidly attached to the moving airfoil. The resulting equations preserve the conservation form. The grid is generated once by an elliptic solver without any need for dynamic grid computation. An implicit approximately factored finite-volume scheme has been developed and implemented through a fully vectorized computer program. The scheme is based on the spatial approximate factorization of Beam and Warming. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The scheme is applied to steady and unsteady transonic rigid-airfoil flows. For forced harmonic airfoil motions, periodic solutions are achieved within the third cycle of oscillation. The results are in good agreement with the experimental data. sonic flows past a NACA 0015 airfoil at a constant pitch rate.18 The latter case is the same application considered in Ref. 16. The numerical solution has been obtained by using the flux-vector splitting and the flux-difference splitting meth- ods extended for dynamic meshes.19 In Ref. 18, it is noted that different turbulence models give different normal-force and pitching-moment coefficients. Numerical solutions of the unsteady Euler equations are less expensive than those of the unsteady Navier-Stokes equations. For the unsteady transonic flow, the unsteady Euler equations adequately model most of the real flow features, with the exception of viscous effects whenever they are substantial. The Euler equations model shock waves and their motion, entropy increase across shocks and entropy gradient, and vorticity production and convection behind shocks, as can be seen from Crocco's theorem and the inviscid vorticity transport equa- tion. Recently, successful time-accurate solutions of the un- steady Euler equations have been presented for pitching air- foils19 and wings in transonic flows,20 and for the rolling oscillation of delta wings—a vortex-dominated flow problem in a locally conical supersonic flows.21 To the best of bur knowledge, Ref. 21 is the first work done for unsteady vortex- dominated flows with shock waves, which is directly applica- ble to maneuvering delta wings. In this paper, we present an implicit approximately-factored finite-volume scheme for the time-accurate numerical solution of the unsteady Euler equations of the flow relative motion with respect to an airfoil-fixed frame of reference. The scheme is applied to steady and unsteady transonic flows around a NACA 0012 airfoil. For the unsteady flows, the airfoil is in pitching oscillation about a small and moderate mean angle of attack at a large amplitude. The computational results are compared with the experimental data.

Flux difference splitting for the Euler equations in generalised coordinates using a local parameterisation of the equation of state

An efficient algorithm based on flux difference splitting is presented for the solution of the three-dimensional Euler equations of gas dynamics in a generalised coordinate system with a general equation of state. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The algorithm uses a local parameterisation of the equation of state and as a consequence requires only one function evaluation in each computational cell. The scheme has good shock capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for Mach 8 flow of “equilibrium air” past a circular cylinder.

Transonic flow about a circular cylinder

Transonic flows governed by the Euler equations, about half and full circular cylinders, are investigated numerically. The tool to carry out this study is a finite-difference method based upon an upwind hybrid formulation, a blend of the “lambda” and the “flux-difference splitting” formulations. In the case of the half cylinder, a separation occurs behind the shock, with the generation of a circulating bubble. It is worthwhile mentioning that for M ∞ = 0.5 we have not been able to reach a steady configuration, instead a barely noticeable and perfectly periodic unsteadiness develops. The phenomenon is much more evident at M ∞ = 0.60 . For the full cylinder case, we have found that the symmetrical configuration with respect to the longitudinal geometrical symmetry plane is not stable. Rather, asymmetric, unsteady, periodic flows are predicted with the shedding of eddies behind the cylinder, which trap the vorticity generated by the shocks. Numerical experiments with different computational parameters and grid sizes seem to remove any doubt about the reliability of the present numerical results to represent solutions of the Euler equations in these particular problems.

Flux-split algorithms for the multi-dimensional Euler equations with real gases

Upwind algorithms are developed for the numerical solution of the multi-dimensional Euler equations for real gases. Flux splitting methods are derived which account for a general equation of state. Approximations to the state equation based on physical arguments result in simplified algorithms which may be implemented into existing perfect-gas codes. Details of this method with the Steger-Warming and Van Leer flux-vector splittings and the Roe flux-difference splitting for 3-D curvilinear coordinates are given. Applications of the method to several high Mach number, high temperature flows are presented for two and three space dimensions.

High-resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations

Based on flux-difference splitting, implicit high resolution schemes are constructed for efficient computations of steady-state solutions to the three-dimensional, incompressible Navier-Stokes equations in curvilinear coordinates. These schemes use first-order accurate Euler backward-time differencing and second-order central differencing for the viscous shear fluxes. Up to third-order accurate upwind differencing is achieved through a reconstruction of the solution from its cell averages. The reconstruction is accomplished by linear interpolation, where the node stencils are selected such that in regions of smooth solution the flow is highly resolved while spurious oscillations in regions of rapid changes in gradient are still suppressed. Fairly rapid convergence to steady-state solutions is attained with a completely vectorizable hybrid time-marching method. Flows around a sharp-edged delta wing are computed with the maximum accuracy of the upwind-differencing restricted to first-, second-, and third-order, to illustrate the effect of accuracy on the global and on the local vortical flow fields. The results are validated with experimental data.

A shock‐capturing scheme for body‐fitted meshes

AbstractA finite difference scheme based on flux difference splitting is presented for the solution of the two‐dimensional Euler equations of gas dynamics in a generalized co‐ordinate system. The scheme is based on numerical characteristic decomposition and solves locally linearized Riemann problems using upwind differencing. The decomposition is for a generalized co‐ordinate system and a convex equation of state. This ensures good shock‐capturing properties when incorporated with operator splitting and the advantage of using body‐fitted co‐ordinates. The resulting scheme is applied to supersonic flow of real air' past a circular cylinder.

Approximate Riemann solutions of the shallow water equations

A finite difference scheme based on flux difference splitting is presented for the solution of the onedimensional shallow water equations of ideal fluid flow. A linearised problem, analogous to tha...

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.

Quasi-conservative lambda formulation

The numerical simulation of inviscid transonic flows by means of a lambda takes in- to account the shock transition from supersonic to subsonic flow conditions, thus allowing the coupling of the supersonic region with the shocked subsonic one and, as a consequence, the upstream movement of the shock. The present methodology is applied to one- and two-dimensional transonic flows. Although the two-dimensional flow calculations involve Cartesian coordinates and are limited to the thin-airfoil approximation, the method can be generalized to arbitrary two- and three-dimensional flow cases. In all of the computed cases, the shock is found to have appropriate strength and position for steady flow conditions and to move upstream properly when a change in the downstream pressure warrants it. A MONO the several numerical methods developed for the JTjLnumerical simulation of transonic inviscid flows, the so- called lambda formulation, recently developed and employed by Moretti 1 and Zannetti and Colasurdo,2 has several very desirable features. The time-dependent, compressible Euler equations are recast in terms of compatibility conditions for characteristic (Riemann) variables along characteristic lines and discretized by means of upwind differences which cor- rectly take into account the direction of wave propagation. In this way a numerical technique is obtained which com- bines the coding simplicity of finite-differ ence methods with the intrinsic accuracy and physical soundness of the method of characteristi cs. Since its first appearance, the lambda formulation has undergone several improvements. In particular, various im- plicit integration schemes3'5 and relaxation methods6'7 have been developed in order to enhance the efficiency of the lambda methodology by removing the Courant-Friedrichs- Lewy (CFL) stability limitation of the explicit schemes previously employed. Also, a significant improvement in ac- curacy is achieved by employing a perturbative formula- tion8'9 such that only the compressibility effects with respect to a suitable incompressible flow solution need to be computed. The lambda formulation has its drawbacks; in particular, numerical experiments have shown that for one-dimensional transonic flows the captured shock cannot move upstream in the supersonic flow region and that this problem persists in some cases for two-dimensional transonic flows, even if the supersonic bubble is embedded in a subsonic region, which would be expected to allow for upstream propagation. This drawback is due to the decoupling of the supersonic region from the subsonic one once the shock wave is established. Recently, the splitting,101 3 which uses governing equations in conservation form and splits the fluxes according to the wave nature of the flow, has been proposed to solve compressible inviscid flows. A comparison of this methodology with the lambda formulation applied to a system of linear hyperbolic equations14 shows that they both lead to identical conclusions, but an analysis of the flux-difference splitting applied to the one-dimensional Euler equations demonstrates that it allows a coupling of the supersonic region with the shocked subsonic one by splitting the waves containing a sonic point, i.e., a vertical characteristic in the (x,t) plane. Such a situation occurs in those mesh intervals where the shock is numerically cap- tured. On the contrary, this basic mechanism is neglected by the classical lambda formulation, but it can be taken into ac- count in order to obtain a modified lambda formulation characterized by a supersonic region coupled with the shocked subsonic one, as done in the present paper. The method will be first described for one-dimensional flows of a test gas with a specific heat ratio equal to unity; the method will then be extended to classical one-dimen- sional nonisentropic flows; and finally to the case of nonhomentropic two-dimensional flows within the frame- work of thin-airfoil theory. The validity and usefulness of the modified lambda methodology will be demonstrated by means of a few example calculations.