Upwind Compact Scheme Research Articles

SUPERSONIC PLANE JET AT HIGHT CONVECTIVE MACH NUMBER(Plane Jet)

The three-dimensional time dependent compressible Navier-Stokes equations are numerically solved to study acoustic emission mechanism in a supersonic plane jet at high convective Mach numbers using high-order compact upwind schemes. High-order compact schemes of 5th order developed by Deng and Maekawa (1996, 1997) [1] [2] are used for spatial derivatives and a 4th order Runge-Kutta scheme is employed for time advancement. Navier-Stokes characteristic boundary conditions are used in the streamwise and vertical directions and periodic boundary conditions in the spanwise direction. Numerical results for the convective Mach number Mc=1.17 are presented (Mc is denned by eq. (16) in Section 2). Two cases were investigated. The first case is the jet flow forced randomly. The second case is the jet forced by random disturbamces and linear unstable oblique modes. The numerical results provide new physical insights into three-dimensional structures and acoustic wave generation mechanisms in a plane turbulent jet. Upstream disturbance conditions play an important role for the evolution of the downstream structure, such as development of shear layers in a jet. Growth of the oblique mode is responsible for the A structure in a plane jet, which yields sooner decay of the centerline velocity. The result indicates that the jet forced with the oblique modes can mix faster with the surounding fluid medium than the jet forced randomly. The intense sound radiation observed in the randomly forced jet can be reduced by forcing with a pair of oblique modes.

An upwind compact mpact approach with group velocity control for compressible flow fields

AbstractIn this paper we construct an upwind compact finite difference scheme with group velocity control for better simulation of compressible flow fields. Compared with traditional difference schemes, compact schemes have higher accuracy for the same stencil width. By means of the characteristic analysis of the operators, the group velocity of wave packets will be controlled to suppress the non‐physical oscillations in numerical solutions. In numerical simulation of the 3D compressible flow fields the third‐order accurate upwind compact operator is used to approximate the derivatives in the convection terms of the compressible N–S equations, the traditional finite difference scheme is used to approximate the viscous terms. Numerical solutions indicate that the method is satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.

Compact difference approximation with consistent boundary condition*

For simulating multi-scale complex flow fields it should be noted that all the physical quantities we are interested in must be simulated well. With limitation of the computer resources it is preferred to use high order accurate difference schemes. Because of their high accuracy and small stencil of grid points computational fluid dynamics (CFD) workers pay more attention to compact schemes recently. For simulating the complex flow fields the treatment of boundary conditions at the far field boundary points and near far field boundary points is very important. According to authors' experience and published results some aspects of boundary condition treatment for far field boundary are presented, and the emphasis is on treatment of boundary conditions for the upwind compact schemes. The consistent treatment of boundary conditions at the near boundary points is also discussed. At the end of the paper are given some numerical examples. The computed results with presented method are satisfactory.

Analysis of central and upwind compact schemes

Central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space, numerical stability and dispersion relation preservation. A von Neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously. This allows one to evaluate various boundary closures and their effects on the domain interior. The same method can be used for stability analysis performed for the semi-discrete initial boundary value problems (IBVP). This analysis tells one about the stability for every resolved length scale. Some well-known compact schemes that were found to be G-K-S and time stable are shown here to be unstable for selective length scales by this analysis. This is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming. To demonstrate the asymptotic stability of the resultant schemes, numerical solution of the wave equation is compared with analytical solution. Furthermore, some of these schemes are used to solve two-dimensional Navier–Stokes equation and a computational acoustic problem to check their ability to solve problems for long time. It is found that those schemes, that were found unstable for the wave equation, are unsuitable for solving incompressible Navier–Stokes equation. In contrast, the proposed compact schemes with improved boundary closure and an explicit higher-order upwind scheme produced correct results. The numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite DRP property.

LES에서 중심 및 상류 컴팩트 차분기법의 적합성에 관하여 (I) - 수치 실험 -

The suitability of high-order accurate, centered and upwind-biased compact difference schemes is evaluated for large eddy simulation of turbulent flow. Two turbulent flows are considered: turbulent channel flow at Re = 23000 and flow over a circular cylinder at Re = 3900. The effects of numerical dissipation on the finite differencing and aliasing errors and the subgrid-scale stress are investigated. It is shown through the simulations that compact upwind schemes are not suitable for LES, whereas the fourth order-compact centered scheme is a good candidate for LES provided that proper dealiasing of nonlinear terms is performed. The classical issue on the aliasing error and the treatment of nonlinear terms is revisited with compact difference schemes.br/

Start-Time of Magnetic Reconnection in Interplanetary Space

Start-time of magnetic reconnection under typical interplanetaryparameters has been numerically simulated by using the two-dimensionalcompressible magnetohydrodynamic equations with a third-order compactupwind scheme. Magnetic reconnection would occur near theinterplanetary current sheet impacted by a plasmoid. Its initiation isassociated with the interplanetary plasma parameter β and the momentumof the plasmoid. The higher the β value is, the faster the reconnectiontakes place. Meanwhile the reconnection occurs earlier with increasing the plasmoid momentum, and increasing driving velocityis more effective in initializing the reconnection than that of theplasma density when the other factors are kept to be the same. Theevolution of the reconnection with the heliocentric distance is alsoinvestigated.

CABARET FINITE‐DIFFERENCE SCHEMES FOR THE ONE‐DIMENSIONAL EULER EQUATIONS

In the present paper we consider second order compact upwind schemes with a space split time derivative (CABARET) applied to one‐dimensional compressible gas flows. As opposed to the conventional approach associated with incorporating adjacent space cells we use information from adjacent time layer to improve the solution accuracy. Taking the first order Roe scheme as the basis we develop a few higher (i.e. second within regions of smooth solutions) order accurate difference schemes. One of them (CABARET3) is formulated in a two‐time‐layer form, which makes it most simple and robust. Supersonic and subsonic shock‐tube tests are used to compare the new schemes with several well‐known second‐order TVD schemes. In particular, it is shown that CABARET3 is notably more accurate than the standard second‐order Roe scheme with MUSCL flux splitting.

Fourth order accurate compact scheme with group velocity control (GVC )

For solving complex flow field with multi-scale structure higher order accurate schemes are preferred. Among high order schemes the compact schemes have higher resolving efficiency. When the compact and upwind compact schemes are used to solve aerodynamic problems there are numerical oscillations near the shocks. The reason of oscillation production is because of non-uniform group velocity of wave packets in numerical solutions. For improvement of resolution of the shock a parameter function is introduced in compact scheme to control the group velocity. The newly developed method is simple. It has higher accuracy and less stencil of grid points.

Numerical study of magnetic reconnection process near interplanetary current sheet

The third order accurate upwind compact difference scheme has been applied to the numerical study of the magnetic reconnection process possibly occurring near the interplanetary current sheet, under the framework of the two-dimensional compressible magnetohydrodynamics (MHD). Our results here show that the driven reconnection near the current sheet can occur within 10–30 min for the interplanetary high magnetic Reynolds number,RM=2 000– 10 000, the stable magnetic reconnection structure can be formed in hour-order of magnitude, and there are some basic properties such as the multiple X-line reconnections, vortical velocity structures, filament current systems, splitting and collapse of the high-density plasma bulk. These results are helpful in understanding and identifying the magnetic reconnection phenomena near the interplanetary current sheets.

Turbulent supersonic channel flow

The effects of compressibility are studied in low Reynolds number turbulent supersonic channel flow via a direct numerical simulation. A pressure-velocity-entropy formulation of the compressible Navier-Stokes equations which is cast in a characteristic, non-conservative form and allows one to specify exact wall boundary conditions, consistent with the field equations, is integrated using a fifth-order compact upwind scheme for the Euler part, a fourth-order Padé scheme for the viscous terms and a third-order low-storage Runge-Kutta time integration method. Coleman et al fully developed supersonic channel flow at M?=?1.5 and Re?=?3000 is used to test the method. The nature of fluctuating variables is investigated in detail for the wall layer and the core region based on scatter plots. Fluctuations conditioned on sweeps and ejections in the wall layer are especially instructive, showing that positive temperature, entropy and total temperature fluctuations are mainly due to sweep events in this specific situation of wall cooling. The effect of compressibility on the turbulence structure is in many respects similar to that found in homogeneous shear turbulence and in mixing layers. The normal components of the Reynolds stress anisotropy tensor are increased due to compressibility, while the shear stress component is slightly reduced. Characteristic of the Reynolds stress transport is a suppression of the production of the longitudinal and the shear stress component, a suppression of all velocity-pressure-gradient correlations and most of the dissipation rates. Comparison with incompressible channel flow data reveals that compressibility effects manifest themselves in the wall layer only.

Simulation of turbulent magnetic reconnection in the small-scale solar wind

Some observational examples for the possible occurrence of the turbulent magnetic reconnection in the solar wind are found by analysing Helios spacecraft's high resolution data. The phenomena of turbulent magnetic reconnections in small scale solar wind are simulated by introducing a third order accuracy upwind compact difference scheme to the compressible two-dimensional MHD flow. Numerical results verify that the turbulent magnetic reconnection process could occur in small scale interplanetary solar wind, which is a basic feature characterizing the magnetic reconnection in high-magnetic Reynolds number ( R M = 2 000-10 000) solar wind. The configurations of the magnetic reconnection could evolve from a single X-line to a multiple X-line reconnection, exhibiting a complex picture of the formation, merging and evolution of magnetic islands, and finally the magnetic reconnection would evolve into a low-energy state. Its life-span of evolution is about one hour order of magnitude. Various magnetic and flow signatures are recorded in the numerical test for different evolution stages and along different crossing paths, which could in principle explain and confirm the observational samples from the Helios spacecraft. These results are helpful for revealing the basic physical processes in the solar wind turbulence.

Implicit, High-Resolution, Compact Schemes for Gas Dynamics and Aeroacoustics

Implicit, high-order schemes are developed for time-accurate numerical solutions of hyperbolic equation systems. High-order spatial accuracy for the implicit operators is obtained at no additional computing cost by performing compact differentiation. The resulting alternating direction implicit and unfactored algorithms yield improved dispersion characteristics compared to second-order accurate in space implicit schemes which makes them suitable for high-resolution numerical simulations in gas dynamics and computational aeroacoustics. First, a fourth-order accurate in space implicit, factorized scheme, which requires block-tridiagonal matrix inversion, is presented. Next, a class of implicit factorized schemes, which require scalar matrix inversions, is presented. Higher order of accuracy in space of the implicit operators is achieved at the expense of inverting scalar matrices with larger bandwidth. Finally, extensions to unfactored algorithms, which use upwind compact schemes, are obtained. The proposed high-order schemes can be implemented with little modification of existing second-order accurate in space, implicit CFD methods. The efficiency, accuracy, and convergence characteristics of the new, high-resolution implicit schemes are demonstrated by their implementation for test problems.

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.

Compact finite difference-Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Appendix B: Acronyms and abbreviations

High-accuracy, time-accurate compressible Navier–Stokes solvers have been developed for transonic flows. These solvers use optimized upwind compact schemes (OUCS) and four-stage, fourth order explicit Runge–Kutta (RK4) time integration scheme, details of which can be obtained in Sengupta [Sengupta TK. High Accuracy Computing Methods, fluid flows and wave phenomena. UK: Cambridge Univ. Press; 2013]. Although these compact schemes have been developed originally for direct simulation of incompressible flows, it is shown here that the same can be used for compressible flows, with shock-boundary layer interactions clearly captured for flow past NACA 0012 and NLF airfoils. Numerical higher order diffusion terms which are used for incompressible flows, have been replaced here by the pressure-based artificial diffusions proposed by Jameson et al. [Jameson A, Schmidt W, Turkel E. Numerical solution of the Euler equations by finite volume methods using Ruge–Kutta time stepping schemes. AIAA Paper 1981-1259. AIAA 14th fluid and plasma dynamics conference. Palo Alto, CA; 1981]. Such second and fourth order diffusion terms are used adaptively at selective points, located by the pressure switch. Developed computational methods used here are validated for cases with and without shocks, for which experimental results are available. Apart from surface pressure coefficient, contours of physical quantities are presented to explain the time-accurate results. Presented methods are robust and the results can be gainfully used to study shock formation, drag divergence and buffet onset of flow over airfoils.

Upwind compact finite difference schemes

It was shown by Ciment, Leventhal, and Weinberg ( J. Comput. Phys. 28 (1978), 135) that the standard compact finite difference scheme may break down in convection dominated problems. An upwinding of the method, which maintains the fourth order accuracy, is suggested and favorable numerical results are found for a number of test problems.