Hisaaki Daiguji

Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations

Abstract A fifth-order compact upwind TVD scheme and a fourth-order compact MUSCL TVD scheme are proposed for solving the compressible Euler and Navier-Stokes equations. The fundamental form of the present schemes is based on the second(third)-order-accurate upwind scheme. One of the distinctive points using the present MUSCL TVD scheme is the ability to capture the discontinuities, such as slip lines or contact surfaces as well as shocks, more sharply than the existing TVD scheme with a simpler algorithm than the so-called ENO scheme. The algorithms are relatively simple and the formulas are quite compact. They can be applied easily to the existing Euler and Navier-Stokes solvers based on the second(third)-order upwind scheme. Finally, we show some numerical results of steady and unsteady flows, including shocks, weak discontinuities and vortices, and the superiority of the present scheme is confirmed by comparison with the results of the ordinary numerical scheme.

An efficient CFD approach for simulating unsteady hypersonic shock-shock interference flows

Abstract An efficient CFD approach based on higher order accurate numerical schemes has been developed for simulating unsteady hypersonic viscous flows strongly associated with shock–shock interference flow phenomena. The principal concept of the present approach is to employ the higher-resolution finite difference schemes based on the fourth-order accurate compact MUSCL TVD scheme coupled with the AUSM-based scheme in space discretization and based on the maximum second-order accurate LU-SGS scheme modified using the Newton iteration in time integration. The hypersonic shock–shock interference flow which is well known as the Type IV is numerically investigated. The obtained numerical results are showing extremely complicated flow structures, such as an unsteady supersonic jet, a jet-bow-shock, and unsteady supersonic shear layers. The reliability of the calculated results is checked by comparison with the existing experimental and numerical results.

Application of an implicit time-marching scheme to a three-dimensional incompressible flow problem in curvilinear coordinate systems☆

Abstract An implicit finite-difference scheme based on the SMAC method for solving steady three-dimensional incompressible viscous flows is proposed. The three-dimensional incompressible Navier-Stokes equations in general curvilinear coordinates, in which the contravariant velocities and the pressure are used as the unknown variables, have been derived by the authors. The momentum equations for the contravariant velocity components and the elliptic equation for the pressure are solved directly in the transformed space by applying the delta-form approximate-factorization scheme and the Tschebyscheff SLOR method, respectively. The present implicit scheme is stable under correctly imposed boundary conditions, since the spurious error and the numerical instabilities can be suppressed by satisfying the continuity condition identically, and by employing the staggered grid and the TVD upwind scheme. Some numerical results for three-dimensional flow over a backward-facing step are shown to demonstrate the reliability of the present scheme and to clarify the three-dimensional effects of such complex flows.

A specially combined lower-upper factored implicit scheme for three-dimensional compressible Navier-Stokes equations

Abstract A specially combined lower–upper factored implicit scheme based on the lower–upper symmetric-Gauss–Seidel (LU-SGS) implicit scheme in conjunction with diagonalization and Gaussian elimination (GE) is proposed for solving the compressible, three-dimensional Navier–Stokes equations and the two-equation q – ω turbulence model. The present scheme, LU-SGS-GE, contains all features of the LU-SGS scheme. Because the similarity transforms are used to construct upstream Jacobian matrices of flux vectors, it leads to much faster convergence and better stability without improper numerical dissipation and free-parameters; besides, the block-diagonal matrix inversions are still eliminated partly, and the implicit operator can also be vectorized completely. The new implicit scheme can be used for solving the unsteady three-dimensional flows.

Stabilization of higher‐order high resolution schemes for the compressible Navier‐Stokes equations

Proposes a measure to stabilize the fourth(fifth)‐order high resolution schemes for the compressible Navier‐Stokes equations. Solves the N‐S equations of the volume fluxes and the low‐Reynolds number k‐e turbulence model in general curvilinear co‐ordinates by the delta‐form implicit finite difference methods. Notes that, in order to simulate the flow containing weak discontinuities accurately, it is very effective to use some higher‐order TVD upstream‐difference schemes in the right‐hand side of the equations of these methods; however, the higher‐order correction terms of such schemes in general amplify the numerical disturbances. Therefore, restricts these terms here by operating the minmod functions to the curvatures so as to suppress the occurrence of new inflection points. Computes an unsteady transonic turbine cascade flow where vortex streets occur from the trailing edge of blades and interact with shock waves. Finds that the stabilization measure improves not only the computational results but also the convergency for such a complicated flow problem.

A numerical method for the transonic cascade flow problem

Abstract An implicit time-marching finite-difference method for solving the three-dimensional compressible Navier-Stokes equations for the relative flow of a turbomachine impeller in general curvilinear coordinates is presented. The fundamental equations of the method have the distinctive feature that the momentums of the contravariant velocities are employed as the dependent variables. The use of the momentum equations of the contravariant velocities makes possible correct and simple treatments of some boundary conditions. In order to obtain the stable solution for high Reynolds number turbulent flow, the Navier-Stokes equations and the k - ϵ turbulence model equations are solved simultaneously, and a high-resolution TVD upwind scheme is introduced. The calculated results of some two-dimensional turbulent flows agreed well with the experimental data. The calculated results of an axial-flow transonic compressor rotor flow showed that the leakage vortex from the tip clearance as well as the shock waves can be captured vividly, in spite of the relatively coarse grid.

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.

An implicit time-marching method for solving the 3-D compressible Navier-Stokes equations

The implicit time-marching finite<difference scheme for the steady threedimensional compressible Euler equations developed by the authors[l] is extended to the Navier-Stokes equations taking account of the diffusion terms. The distinctive features of the previous scheme are to make use of the momentum equations of contravariant velocities instead of the physical velocities, and to be able to treat the solid wall boundary conditions exactly and the periodic boundary condition for the turbomachine impeller flow easily.

An implicit-explicit flux vector splitting scheme for hypersonic thermochemical nonequilibrium flows

An implicit-explicit flux vector splitting scheme which can be used in both explicit and implicit calculations is proposed and applied to a hypersonic thermochemical nonequilibrium flow problem. Since the consistency of these calculations must be remarkably good, the effort to make the present computational code has been greatly reduced. In this paper, consequently the numerical algorithm of the present scheme is weightedly explained.

Numerical simulation of supersonic mixing layers using a fourth-order accurate shock capturing scheme

The direct numerical simulations of the time-developing subsonic and supersonic mixing layers have been presented. We can conclude that the present scheme is considerably excellent than the existing second- and third-order TVD schemes to simulate high speed flows having slip surfaces and shock waves.