Upwind TVD Scheme Research Articles

超軌道速度再突入物体前面における対流・ふく射熱伝達の推算

The convective and radiative heat transfer rates are calculated over the super-orbital reentering body surface like the MUSES-C capsule. Thermally and chemically nonequilibrium 11-species (O2, N2, O, N, NO, O+2, N+2, O+, N+, NO+ and e-) air is considered using Park's two temperature model. Chemical species number densities and temperature distributions are obtained by solving the Navier-Stokes equations and the vibrational-electronic energy conservation equation, using a Harten-Yee-type upwind TVD scheme. For the isothermal wall condition the minimum grid interval is set to 1μm in order to maintain the cell Reynolds number of O(1). The radiative heat transfer rate is calculated by integrating the radiative heat transfer equation using emission and absorption coefficients that are determined by NEQAIR (Nonequilibrium Air Radiation) and SPRADIAN (Structured Package for Radiation Analysis). Those results are compared with convective heat transfer rate.

On the Accuracy of Upwind and Symmetric TVD Schemes in Simulating Low Mach Number Flow

The accuracy of MUSCL upwind and Yee-Roe-Davis symmetric TVD schemes for simulating low Mach number flow is studied through a numerical experiment of the 2-D lid driven cavity problem. The steady slate solution is reached by using a marching approach based on the pseudocompressibilty method in conjunction with implicit approximate factorization. A finite volume discretization of the conservation equations is used with a four level multigrid method to accelerate the convergence. The tests performed which were in the range of 100 ≤ Re ≤ 5000, show that the Yee-Roe-Davis symmetric scheme generates results in very good agreement with the benchmark results over this range of Re. The MUSCL upwind scheme accuracy deteriorates with the increasing Re.

超軌道速度再突入カプセル前方衝撃波層の数値解析

The calculation code originally developed for a hypersonic nonequilibrium flow is applied to the flowfield in front of a super orbital-velocity reentry capsule. The re-entry speed is assumed to be 12km/sec. The governing equations for a chemically and thermally nonequilibrium flow consist of the full Navier-Stokes equations, including the 11 chemical species mass conservation equations and the vibrational-electronic energy conservation equation. The Park's two-temperature model is used for the chemical reaction rates. A Harten-Yee type upwind TVD scheme is used to solve the governing equations with a fractional step method. A semi-implicit scheme is introduced because of the stiffness of chemical source terms. Under a super orbital-velocity re-entry conditions, the results of numerical calculations indicate that the vibrational excitation behind a strong bow shock is prevented by nitrogen molecule dissociation. The obtained distributions of vibrational and translational temperatures and species mass fractions are essentially sufficient to yield not only the convective but also the radiative heat transfer to the capsule surface using the existing NEQAIR code.

High-order upwind method for hyperbolic grid generation

A new hyperbolic grid generation method based on a high-order accurate upwind approximation is presented and developed for two- and three-dimensional spaces in this paper. The method uses a system of hyperbolic equations for grid generation. Due to a hyperbolic property of the grid generation equations, there exist characteristic directions and high-order TVD upwind scheme can be applied to the directions. The method employs a three-point backward difference scheme for descritization of a derivative in a marching direction, while a high-order accurate upwind scheme using MUSCL interpolation with minmod limiter is employed according to the plus or minus sign of eigenvalues of coefficient matrices to descritize derivatives in the other directions. Making use of the upwind scheme the present hyperbolic grid generation method needs no artificial damping term, which a conventional hyperbolic grid generation method requires to prevent oscillations. The method incorporates a subiteration procedure to solve nonlinear equations at each marching step in order to obtain both sufficient robustness and satisfactory grid orthogonality realized by the high-order upwind scheme. For two dimensions, the Newton subiteration is performed at each marching step before advancing to next grid line. For three dimensions, a new pseudo-time subiteration approach is introduced and incorporated in the method to achieve further robustness. It is demonstrated that in the hyperbolic grid generation the first-order accurate upwind scheme is too dissipative to generate an orthogonal grid, although the higher-order accurate upwind scheme can generate the orthogonal grid. The generated grids around bodies of complex geometry with sharp edges, deep concave and high convex show excellent orthogonality and smoothness, and demonstrate promising properties of the method.

The flow of wet steam in a one-dimensional nozzle

The formation of water droplets in a low-pressure steam turbine, seriously degrades the efficiency of the generator. A model has been developed which includes the nucleation and subsequent growth of the droplets as the extra equations to the usual Euler equations for dry steam. A feature of this work is that all the equations are cast in Eulerian form compared to much of the previous work which considered the droplets in Lagrangian form. The ensuing equations are solved using a second-order upwind TVD scheme which can cope with the steep gradients which occur in the solution. The results for a 1-D nozzle are presented and compared with experimental results. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical Analysis of Passive Control of Pseudo-Shock.

Oscillation of the pseudo shock in the rectangular duct with the porous wall was investigated numerically. In the simulation, 2nd order accurate, Harten Yee's upwind TVD scheme was used for high resolution and numerical stability, and Chien's κ-e two equation model was used as the turbulence model. To simulate the effect of the blowing and suction on the porous wall, Darcy's law was used on the porous wall boundary. The boundary layer thickness under the first shock wave oscillates due to the shock oscillation, which changes the strength of the first shock wave changes, so that it suppresses the pressure fluctuation downstream of the first shock wave. The effects of some parameters of the passive control on the pseudo-shock oscillation were investigated.

Pseudo-Shock in Straight Duct with Porous Wall and Bypass System.

As the basic study of the passive control of the shock oscillation, steady pseudo shock in the rectangular ducts with the porous wall and the bypass system was investigated experimentally and numerically. The expriments were carried out in a blow-down supersonic wind tunnel at a free stream Mach number of 2.0. Numerical simulations were made with the 2-D compressible Navier Stokes code. In sloving the equations, 2nd-order accurate Harten-Yee's upwind TVD scheme and κ-ε turbulence model were used. It is shown that the boundary layer thickness under the first shock wave of the pseudo-shock is controlled by the passive control method without noticeable changes of the pseudo-shock strcuture. But the total pressure loss downstream of the pseudo-shock is reduced by this method. And the effects of some parameters influencing the passive control method are studied numerically.

Oscillation of Circular Pseudo-Shock in Radial and Spiral Supersonic Flow.

Oscillation of the circular pseudo shock in the radial and spiral supersonic flow between two parallel discs was investigated both numerically and experimentally. In the simulation, 2 nd-order accurate, Harten-Yee's upwind TVD scheme was used for high resolution and numerical stability, and Baldwin-Lomax's algebraic model was used as the turbulence model. To simulate shock wave oscillation, the back pressure was fluctuated at the outlet boundary. The X-type pseudo shock, which appears at higher Mach number, is more stable than the A type one, which appears at lower Mach number. Oscillation of mode 0 is predominant when the swirl is absent, while it becomes weaker and higher modes appear with the increase in the swirl.

Numerical Simulation of Peseudo-Shock Oscillation in Straight Channel.

Numerical simulation of the oscillation of a pseudo-shock in a straight channel was made, using the explicit, second-order accurate Harten-Yee's upwind TVD scheme and the Reynolds averaged Navier Stokes equations. The basic frequency of the shock wave is the same as that of the back pressure fluctuation. The oscillation of the shock wave is similar to the forced oscillation of the simple one freedom vibration system with damping. At low Mach number, the damping effect is high, while at high Mach number it is low, and the strong resonance occurs.

Modeling of Axisymmetric Two-phase Dilute Flows

This paper deals with the numerical treatment of Eulerian approach for dilute two-phase compressible flows (gas-particles mixtures) in axisymmetric configurations. For dilute flows, two classes of models depending on the dispersed phase volumetric fraction can be found. The volume occupied by the particles may be considered, that yields a model in which the gas phase and the dispersed phase equations are coupled through the void fraction and the source terms (Delhaye model). The void fraction effects can be neglected, that means the gas phase is a carrier phase for the particles (Ishii model). The mathematical nature of the two models is demonstrated from analysis of characteristic directions. For the Delhaye's model, a centered scheme is used to solve the system of partial differential equations, while an upwind TVD scheme is used for the Ishii's one. Then, it is shown that the problem of symmetry boundary conditions does not depend on the physical approach, as long as the flow remains dilute. However, a classical treatment for symmetry boundary conditions at the geometrical axis leads to large errors. A particular treatment for this boundary is presented: a new class of particles, described by a supplementary system of equations, is required.

Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock–spherical fast/slow bubble interactions

Collapsing shock-bounded cavities in fast/slow (F/S) spherical and near-spherical configurations give rise to expelled jets and vortex rings. In this paper, we simulate with the Euler equations planar shocks interacting with an R12 axisymmetric spherical bubble. We visualize and quantify results that show evolving upstream and downstream complex wave patterns and emphasize the appearance of vortex rings. We examine how the magnitude of these structures scales with Mach number. The collapsing shock cavity within the bubble causes secondary shock refractions on the interface and an expelled weak jet at low Mach number. At higher Mach numbers (e.g. M=2.5) ‘vortical projectiles’ (VP) appear on the downstream side of the bubble. The primary VP arises from the delayed conical vortex layer generated at the Mach disk which forms as a result of the interaction of the curved incoming shock waves that collide on the downstream side of the bubble. These rings grow in a self-similar manner and their circulation is a function of the incoming shock Mach number. At M=5.0, it is of the same order of magnitude as the primary negative circulation deposited on the bubble interface. Also at M=2.5 and 5.0 a double vortex layer arises near the apex of the bubble and moves off the interface. It evolves into a VP, an asymmetric diffuse double ring, and moves radially beyond the apex of the bubble. Our simulations of the Euler equations were done with a second-order-accurate Harten–Yee-type upwind TVD scheme with an approximate Riemann Solver on mesh resolution of 803×123 with a bubble of radius 55 zones.

Investigation of Compression Wave Propagating in Slab Track Tunnel of High-Speed Railway. 1st Report. Field Test and One-Dimensional Numerical Analysis.

An impulsive pressure which is called a micro-pressure wave or a sonic boom radiating from a tunnel exit is one of the important environmental problems in high-speed railways. The strength of the impulsive depends on the waveform of the compression at the tunnel exit. In this study, the distortion of the compression during its propagation through the tunnel is investigated by field measurement and numerical analysis. The field measurement is conducted in the concrete slab (ballastless) track tunnel of the Shinkansen. The numerical analysis is also carried out on one-dimensional compressible flow using upwind TVD scheme. It takes account of steady and unsteady wall friction and of heat transfer to the tunnel wall. Our original numerical analysis method is based on Galilei transformation of the coordinate system moving with the compression wave. The results show that the agreement of the numerical analysis and the field measurement is good.

平板に衝突する超音速噴流に関する数値解析及び実験

An impingement of two-dimensional axisymmetric supersonic jet on a plate is simulated numerically to evaluate the jet flow configuration on the plate with a real gas assumption. An explicit TVD-Upwind scheme is used for the Euler equations to evaluate interaction between shock waves. An experimental study of the underexpanded axisymmetric supersonic N2 jets impinging on a plate is also discussed. Pressure measurements on the plate surface and Schlieren photometries are performed to investigate the impinging jets. Numerical results with the Euler equations have a good agreement with the experimental results at the shock structure and pressure distribution on the plate. The results show that three types for complicated shock-shock interactions are revealed and that a stagnation bubble is formed due to triple point near the plate.

Propagation of strongly nonlinear plane N-waves

Formation and evolution of N (-like) waves is studied without the restriction of low amplitude, namely weak nonlinearity. To this end, the classical piston problem of gasdynamics is investigated, in which the wave is radiated by a piston executing a single cycle of harmonic oscillation into an inviscid perfect gas. The method of analysis is based on the simple-wave theory up to the shock formation time, and beyond that time on the numerical calculation by a high-resolution TVD upwind scheme. The initial sinusoid-like wave profile is rapidly distorted as the wave propagates, and this leads to the formation of head and tail shocks. The main effects of strong nonlinearity may be listed as follows: (i) entropy production at shock fronts, (ii) the existence of waves reflected from shocks, (iii) an asymmetric wave profile stemming from the boundary condition at the source of the strongly nonlinear problem. As the result, the strongly nonlinear wave possesses the following remarkable distinctive features, in contrast to its counterpart in the weakly nonlinear regime. The tail shock is not formed at the tail of the wave, and the expansion wave behind the head shock has non-uniform intensity. The N (-like) wave propagates with some excess mass. Thereby a region with low density, associated with the entropy production, appears in the vicinity of the source.

Thermochemical Nonequilibrium in Rapidly Expanding Flows of High-Temperature Air

Abstract • This paper describes a computational study of the thermal and chemical nonequilibrium occuring in a rapidly expanding flow of high-temperature air transported as a free jet from an orifice into low-density stationary air. Translational, rotational, vibrational and electron temperatures are treated separately, and in particular the vibrational temperatures are individually treated; a multi-vibrational temperature model is adopted. The governing equations are axisymmetric Navier-Stokes equations coupled with species vibrational energy, electron energy and species mass conservation equations. These equations are numerically solved, using the second order upwind TVD scheme of the Harten-Yee type. The calculations were carried out for two different orifice temperatures and also two different orifice diameters to investigate the effects of such parameters on the structure of a nonequilibrium free jet.

A Preconditioned Flux-Differencing Scheme for Chemically Reacting Flows at all Mach Numbers

A new computation procedure is developed for solving time-accurate, chemically reacting flows over a wide range of Mach numbers. The algorithm is based on scaling the pressure terms in the momentum equations and preconditioning the conservation equations to circumvent numerical difficulties at low Mach numbers. The resulting equations are solved using the lower-upper (LU) factorization method in a fully-coupled manner, with incorporation of a flux-differencing upwind TVD scheme to achieve high-order spatial accuracy. The transient behavior of the modeled system is preserved through implementation of the dual time-stepping integration technique. The capabilities of the scheme are illustrated by applying it to selected problems, including one-dimensional nozzle flows, two-dimensional channel flows, rocket-motor internal flows, and acoustic waves in a porous chamber with surface transpiration.

自由ピストン２段膜衝撃波管の特性

An experimental and numerical investigation of a free-piston double-diaphragm shock tube is conducted. In shock tube flow sequence, shock speeds up to 12.7km/s are achieved in the low pressure tube for initial pressure 13.3Pa by using helium driver gas. The present paper is described especially on characteristics of the buffer tube. In the numerical simulation, the flow is modeled as unsteady, inviscid and compressible. The numerical method employs the Harten-Yee second order upwind TVD scheme. The influence of the primary and secondary diaphragm rupture process on flow characteristics is investigated in experiment and numerical simulation.

Mass, momentum and total excess energy transported by a weak planeN wave

A weakly nonlinear plane acoustic wave is emitted into an ideal gas of semi-infinite extent from an infinite plate by its sinusoidal motion of single period. The wave develops into anN wave in the far field, as long as the energy dissipation is negligible everywhere except for discontinuous shock fronts. The third-order effects at shock fronts are evaluated, i.e., the generation of reflected acoustic wave as a result of the interaction of shock and expansion wave and the production of entropy by the energy dissipation at shock fronts. Consideration of these effects enables one to estimate the whole mass, momentum and total excess energy (sum of the kinetic energy and excess of internal energy over an initial undisturbed value) transported by theN wave to the accuracy of third order of wave amplitude. It is shown that the mass and total excess energy transported by theN wave increase and the momentum decreases to asymptotic limits as the wave propagates. The result shows good agreement with a numerical result obtained by solving the Euler equations with a high-resolution TVD upwind scheme.

An improved multistage time stepping for second-order upwind TVD schemes

An improved multistage time stepping for second-order upwind TVD schemes to compute the Euler equations is suggested by splitting the residual into the first-order flux and the antidiffusive terms. The flows passing through the cascade built of two circular arc blades and through the NACA0012 airfoil are selected to test this method. Present time stepping shows excellent results in view of the convergence acceleration ability for supersonic, transonic, and subsonic flows compared with the Wiedermann and Iwamoto's time stepping.

FULLY DISCRETE HIGH-RESOLUTION SCHEMES FOR HYPERBOLIC CONSERVATION LAWS

The present paper is a sequel to two previous papers in which rigorous, up to fourth-order, fully discrete (FD) upwind TVD schemes have been presented. In this paper we discuss in detail the extension of these schemes to solutions of non-linear hyperbolic systems. The performance of the schemes is assessed by solving test problems for the time-dependent Euler equations of gas dynamics in one and two space dimensions. We use exact solutions and experimental data to validate the results.