Compact Upwind Research Articles

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.

A characteristic-based shock-capturing scheme for hyperbolic problems

In order to suppress numerical oscillations of linear compact schemes around discontinuities, a characteristic-based flux splitting limited method is introduced instead of ENO/WENO or other shock-capturing algorithms. This method begins with upwind schemes and flux vector splittings. The upwind schemes are projected along characteristic directions in a different way, and their amplitudes are carefully controlled by a special limiter in order to meet entropy condition and to prevent non-physical oscillations. A fifth-order linear compact upwind scheme is modified by this method for solving problems involving discontinuities. The properties of the numerical algorithm are checked on some benchmark problems in one, two and three space dimensions. Numerical results show that it is high-order accurate with high resolution and oscillation-free.

1604 超音速平面ジェットにおける斜めモードの組み合せに対する遷移構造に及ぼす影響(S35-1 圧縮性流れの基礎と応用(1),S35 圧縮性流れの基礎と応用)

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. Three cases for convective Mach number Mc=1.17 are presented. The fast case is the jet flow forced randomly. The second case is the jet forced by random disturbances and two pair of most unstable oblique modes with a subsonic phase speed. The third case is the jet forced by random disturbances, two pairs of most unstable oblique modes and a pair of subharmonic 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 jet flow structures. The numerical results show that the jet meandering is suppressed in the jet forced by oblique modes. Therefore, the intense sound radiation observed in the randomly forced jet can be reduced by forcing with a pair of oblique modes.

A class of compact upwind TVD difference schemes

A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD (CTVD) schemes were tested, one is third-order accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a two-dimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

Effects of Unstable Oblique Modes on the Acoustic Filed in a Supersonic Plane Turbulent 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. Two cases for convective Mach number Mc=1.17 are presented. The first case is the jet flow forced randomly. The second case is the jet forced by random disturbances and a pair of linear unstable oblique modes with a subsonic phase speed. The numerical results provide new physical insights into three-dimensional structures and acoustic wave generation mechanisms in a plane turbulent jet. Intense Mach waves radiating at low-frequencies are attributable to the growth of two-dimensional radiating mode with a supersonic phase speed in the jet shear layers. Upstream disturbance conditions play an important role for the evolution of the jet shear layers. The growth of a pair of oblique modes is responsible for the Λ structure in the jet shear layers, which yields sooner decay of the centerline velocity due to the shear layer rapid evolution. Therefore, the intense sound radiation observed in the randomly forced jet can be reduced by forcing with a pair of oblique modes.

908 超音速平面ジェットの音響場に対する斜めモードの影響(2)(OS9-2 高速流れと衝撃波現象の解明と制御,OS9高速流れと衝撃波現象の解明と制御)

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 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. Two cases for convective Mach number Mc=1.17 are presented. The first case is the jet flow forced randomly. The second case is the jet forced by random disturbances and a pair of linear unstable oblique modes with a subsonic phase speed. 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 jet shear layers. The growth of a pair of oblique modes is responsible for the Λ structure in the jet shear layers, which yields sooner decay of the centerline velocity due to the shear layer rapid evolution. Therefore, the intense sound radiation observed in the randomly forced jet can be reduced by forcing with a pair of oblique modes.

A preliminary study of 3D difference scheme with energy dynamic equilibrium

In this paper, a difference scheme with energy dynamic equilibrium (DS-EDE) is presented, which can be used for the simulation of long-term atmosphere and sea motion. Based on three dimensional nonlinear evolution equations for atmosphere and sea motion, a three dimensional compact upwind scheme (CUWS) is constructed, as the basis of the DS-EDE. The DS-EDE satisfies the following condition of energy dynamic equilibrium (EDE): the total work of external forces on the region boundary is equal to the sum of the total effective variation of the kinetic energy and the energy dissipation in the average flow motion and the effective variation of the potential energy per unit time within the region of interest. It really reflects the basic mechanism of the action of external forces and dissipation in atmosphere and sea movement. Therefore, the DS-EDE developed in this paper is a suitable model for simulating long-term atmosphere and sea movement with forcing and dissipation.

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.

Finite Volume Formulation of Compact Upwind and Central Schemes with Artificial Selective Damping

The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes. The different schemes are analyzed and tested on several numerical experiments. A new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes is presented. Results are shown for a 1D advection equation, a 2D rotating Gaussian pulse and a subsonic inviscid vortical flow on uniform and non-uniform meshes and for a non-linear acoustic pulse.

An AUSM‐based high‐order compact method for solving Navier‐Stokes equations

A high‐order compact upwind algorithm is developed for solving Navier‐Stokes equations in two‐space dimensions. The method is based on advection upstream splitting method and fourth‐order compact finite‐difference schemes. The convection flux terms of the Navier‐Stokes equations are discretized by a compact cell‐centered differencing scheme while the diffusion flux terms are discretized by a central fourth‐order compact scheme. The midpoint values of the flux functions required by the cell‐centered compact scheme are determined by a fourth‐order MUSCL approach. For steady‐state solutions; first‐order implicit time integration, with LU decomposition, is employed. Computed results for a laminar flow past a flat plate and the problem of shock‐wave boundary layer interaction are presented.

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.

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/

An explicit filtering method for large eddy simulation of compressible flows

A method for large eddy simulation (LES) is presented in which the sub-grid-scale modeling is achieved by filtering procedures alone. The procedure derives from a deconvolution model, and provides a mathematically consistent approximation of unresolved terms arising from any type of nonlinearity. The formal steps of primary filtering to obtain LES equations, approximate deconvolution to construct the subgrid model term and regularization are combined into an equivalent filter. This filter should be an almost perfect low pass filter below a cut-off wavenumber and then fall off smoothly. The procedure has been applied to a pressure-velocity-entropy formulation of the Navier–Stokes equations for compressible flow to perform LES of two fully developed, turbulent, supersonic channel flows and has been assessed by comparison against direct numerical simulation (DNS) data. Mach numbers are 1.5 and 3.0, and Reynolds numbers are 3000 and 6000, respectively. Effects of filter cut-off location, choice of differentiation scheme (a fifth-order compact upwind formula and a symmetric sixth-order compact formula were used), and grid refinement are examined. The effects are consistent with, and are readily understood by reference to, filtering characteristics of the differentiation and the LES filter. All simulations demonstrate a uniform convergence towards their respective DNS solutions.

Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction

In the present paper an efficient hybrid compact-WENO scheme is proposed to obtain high resolution in shock-turbulence interaction problems. The algorithm is based on a fifth-order compact upwind algorithm in conservation form to solve for the smooth part of the flow field, which is coupled with a high-resolution weighted essentially nonoscillatory (WENO) scheme to capture the discontinuities. The derivation of the compact scheme is discussed in detail, and a stability study of the full discretization is included. The performance of the numerical algorithm has been assessed by performing preliminary simulations on benchmark problems, such as the interaction of a shock wave with entropy and vortical disturbances. The algorithm here developed is proven to have better resolution properties than standard WENO schemes and hybrid compact-ENO schemes as well, at a lower computational cost. In addition, the application to more realistic shock-turbulence interaction problems is discussed.

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.

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.

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.

A characteristic-type formulation of the Navier–Stokes equations for high order upwind schemes

We propose a formulation of the three-dimensional Navier–Stokes equations, which expresses the inviscid part of the equations as a decomposition into several plane waves which are aligned with the numerical grid. The resulting equations are very well suited to numerical solution using compact high order upwind schemes. Boundary conditions and blockwise decomposition of the computational domain are particularly straightforward. Such advantages make the formulation attractive, even when using central or pseudospectral differencing methods. Numerical examples are presented.

Compact upwind algorithms for viscous compressible flow calculations

A family of high-order accurate compact upwind difference operators have been used together with the split fluxes of the KFVS (Kinetic Flux Vector Splitting) scheme to obtain high order semi-discretizations of nonlinear convective terms in 2-D Navier-Stokes equations governing viscous compressible flows. A TVD multi-stage Runge-Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic inviscid and viscous flow problems which indicate the higher accuracy and low diffusion realizable in such schemes.

On Performance of Methods with Third- and Fifth-Order Compact Upwind Differencing

The difference schemes for fluid dynamics type of equations based on third- and fifth-order Compact Upwind Differencing (CUD) are considered. To validate their properties following from a linear analysis, calculations were carried out using the inviscid and viscous Burgers' equation as well as the compressible Navier–Stokes equation written in the conservative form for curvilinear coordinates. In the latter case, transonic cascade flow was chosen as a representative example. The performance of the CUD methods was estimated by investigating mesh convergence of the solutions and comparing with the results of second-order schemes. It is demonstrated that the oscillation-free steep gradients solutions obtained without using smoothing techniques can provide considerable increase of accuracy even when exploiting coarse meshes.