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Article

ERIK DICK

A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume…

Abstract

A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume method. In first order accurate form, a discrete set of equations is obtained which is both conservative and positive. Due to the positivity, the set of equations can be solved by collective relaxation methods in multigrid form. A full multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. Second order accuracy is obtained by the Chakravarthy‐Osher flux‐extrapolation technique, using the Roe‐Chakravarthy minmod limiter. In second order form, direct relaxation of the discrete equations is no longer possible due to the loss of positivity. A defect‐correction is used in order to solve the second order system.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 1 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

A.E. KANARACHOS and I.P. VOURNAS

An optimized multigrid method (NSFLEX‐MG) for the NSFLEX‐code (Navier‐Stokes solver using characteristic flux extrapolation) of MBB (Messerschmitt Bolkow Blohm GmbH) is…

Abstract

An optimized multigrid method (NSFLEX‐MG) for the NSFLEX‐code (Navier‐Stokes solver using characteristic flux extrapolation) of MBB (Messerschmitt Bolkow Blohm GmbH) is described. The method is based on a correction scheme and implicit relaxation procedures and is applied to two‐dimensional test cases. The principal feature of the flow solver is a Godunov‐type averaging procedure based on the eigenvalues analysis of the Euler equations by means of which the inviscid fluxes are evaluated at the finite volume faces. Viscous fluxes are centrally differenced at each cell face. The performance of NSFLEX‐MG is demonstrated for a large range of Mach numbers for compressible inviscid and viscous (laminar and turbulent) flows over a RAE‐2822 airfoil and over a NACA‐0012 airfoil.

Details

Engineering Computations, vol. 10 no. 2
Type: Research Article
ISSN: 0264-4401

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Article

S.A. Mohamed

The aim of the paper is to achieve textbook multigrid efficiency for some flow problems.

Abstract

Purpose

The aim of the paper is to achieve textbook multigrid efficiency for some flow problems.

Design/methodology/approach

The steady incompressible Euler equations are decoupled into elliptic and hyperbolic subsystems. Numerous classical FAS‐MG algorithms are implemented and tested for convergence. A full multigrid algorithm that costs less than 10 work units (WUs) is sufficient to reduce the algebraic error below the discretization error. A new algorithm “NUVMGP” is introduced. A two‐step iterative procedure is adopted. First, given the pressure gradient, the convection equations are solved on the computational grid for the velocity components by performing one Gauss‐Seidel iteration ordered in the flow direction. second, a linear multigrid (MG) cycle for Poisson's equation is performed to update pressure values.

Findings

It is found that algorithm “NUVMGP‐FMG” requires less than 6 WU to attain the target solution. The convergence rates are independent on both the mesh size and the approximation order.

Research limitations/implications

Lexicographic Gauss‐Seidel using downstream ordering is a good solver for the advection terms and provides excellent smoothing rates for relaxation. But it is complicated to maintain downstream ordering in case the flow directions change with location.

Originality/value

Although the scope of this work is limited to rectangular domains, finite difference schemes, and incompressible Euler equation, the same approaches can be extended for other flow problems. However, such relatively simple problems may provide deep understanding of the ideal convergence behavior of MG and accumulate experience to detect unacceptable performance and regain the optimal one.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 18 no. 6
Type: Research Article
ISSN: 0961-5539

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Article

Jian-Ming Fu, Hai-Min Tang and Hong-Quan Chen

The purpose of this paper is to develop a new approach for rapid computation of subsonic and low-transonic rotary derivatives with the available steady solutions obtained…

Abstract

Purpose

The purpose of this paper is to develop a new approach for rapid computation of subsonic and low-transonic rotary derivatives with the available steady solutions obtained by Euler computational fluid dynamics (CFD) codes.

Design/methodology/approach

The approach is achieved by the perturbation on the steady-state pressure of Euler CFD codes. The resulting perturbation relation is established at a reference Mach number between rotary derivatives and normal velocity on surface due to angular velocity. The solution of the reference Mach number is generated technically by Prandtl–Glauert compressibility correction based on any Mach number of interest under the assumption of simple strip theory. Rotary derivatives of any Mach number of interest are then inversely predicted by the Prandtl–Glauert rule based on the reference Mach number aforementioned.

Findings

The resulting method has been verified for three typical different cases of the Basic Finner Reference Projectile, the Standard Dynamics Model Aircraft and the Orion Crew Module. In comparison with the original perturbation method, the performance at subsonic and low-transonic Mach numbers has significantly improved with satisfactory accuracy for most design efforts.

Originality/value

The approach presented is verified to be an efficient way for computation of subsonic and low-transonic rotary derivatives, which are performed almost at the same time as an accounting solution of steady Euler equations.

Details

Engineering Computations, vol. 36 no. 9
Type: Research Article
ISSN: 0264-4401

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Article

S. SOLTANI, K. MORGAN and J. PERAIRE

An upwind unstructured grid cell‐centred scheme for the solution of the compressible Euler and Navier‐Stokes equations in two dimensions is presented. The algorithm…

Abstract

An upwind unstructured grid cell‐centred scheme for the solution of the compressible Euler and Navier‐Stokes equations in two dimensions is presented. The algorithm employs a finite volume formulation. Calculation of the inviscid fluxes is based on the approximate Riemann solver of Roe. Viscous fluxes are obtained from solution gradients computed by a variational recovery procedure. Higher order accuracy is achieved through performing a monotonic linear reconstruction of the solution over each cell. The steady state is obtained by a point implicit time integration of the unsteady equations using local time stepping. For supersonic inviscid flow an alternative space marching algorithm is proposed. This latter approach is applicable to supersonic flow fields containing regions of local subsonic flow. Numerical results are presented to show the performance of the proposed scheme for inviscid and viscous flows.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 3 no. 4
Type: Research Article
ISSN: 0961-5539

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Article

D. LEFEBVRE, J. PERAIRE and K. MORGAN

We investigate the application of a least squares finite element method for the solution of fluid flow problems. The least squares finite element method is based on the…

Abstract

We investigate the application of a least squares finite element method for the solution of fluid flow problems. The least squares finite element method is based on the minimization of the L2 norm of the equation residuals. Upon discretization, the formulation results in a symmetric, positive definite matrix system which enables efficient iterative solvers to be used. The other motivations behind the development of least squares finite element methods are the applicability of higher order elements and the possibility of using the norm associated to the least squares functional for error estimation. For steady incompressible flows, we develop a method employing linear and quadratic triangular elements and compare their respective accuracy. For steady compressible flows, an implicit conservative least squares scheme which can capture shocks without the addition of artificial viscosity is proposed. A refinement strategy based upon the use of the least squares residuals is developed and several numerical examples are used to illustrate the capabilities of the method when implemented on unstructured triangular meshes.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

He-yong Xu, Shi-long Xing and Zheng-yin Ye

The purpose of this paper is to investigate and improve a new method of unstructured rotational dynamic overset grids, which can be used to simulate the unsteady flows…

Abstract

Purpose

The purpose of this paper is to investigate and improve a new method of unstructured rotational dynamic overset grids, which can be used to simulate the unsteady flows around rotational parts of aircraft.

Design/methodology/approach

The computational domain is decomposed into two sub-domains, namely, the rotational sub-domain which contains the rotational boundaries, and the stationary sub-domain which contains the remainder flow field including the stationary boundaries. The artificial boundaries and restriction boundaries are used as the restriction condition to generate the entire computational grid, and then the overset grids are established according to the radius parameters of artificial boundaries set previously. The deformation of rotational boundary is treated by using the linear spring analogy method which is suitable for the dynamic unstructured grid. The unsteady Navier-Stokes/Euler equations are solved separately in the rotational sub-domain and stationary sub-domain, and data coupling is accomplished through the overlapping area. The least squares method is used to interpolate the flow variables for the artificial boundary points with a higher calculating precision. Implicit lower-upper symmetric-Gauss-Seidel (LU-SGS) time stepping scheme is implemented to accelerate the inner iteration during the unsteady simulation.

Findings

The airfoil steady flow, airfoil pitching unsteady flow, three-dimensional (3-D) rotor flow field, rotor-fuselage interaction unsteady flow field and the flutter exciting system unsteady flow field are numerically simulated, and the results have good agreements with the experimental data. It is shown that the present method is valid and efficient for the prediction of complicated unsteady problems which contain rotational dynamic boundaries.

Research limitations/implications

The results are entirely based on computational fluid dynamics (CFD), and the 3D simulations are based on the Euler equations in which the viscous effect is ignored. The current work shows further applicable potential to simulate unsteady flow around rotational parts of aircraft.

Practical implications

The current study can be used to simulate the two-dimensional airfoil pitching, 3-D rotor flow field, rotor-fuselage interaction and the flutter exciting system unsteady flow. The work will help the aircraft designer to get the unsteady flow character around rotational parts of aircraft.

Originality/value

A new type of rotational dynamic overset grids is presented and validated, and the current work has a significant contribution to the development of unstructured rotational dynamic overset grids.

Details

Aircraft Engineering and Aerospace Technology: An International Journal, vol. 87 no. 4
Type: Research Article
ISSN: 0002-2667

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Article

S. Sivaloganathan and J.M. Stockie

The purpose of this paper is twofold. Firstly, to present a detailedaccount of the generalized Lagrangian formulation of Hui and Zhao, in whichthe stream function ζ and…

Abstract

The purpose of this paper is twofold. Firstly, to present a detailed account of the generalized Lagrangian formulation of Hui and Zhao, in which the stream function ζ and Lagrangian distance λ, are used as independent variables, and secondly to assess and compare the performance of various flux limiters in this formulation with their corresponding performance in the Eulerian formulation. The generalized Lagrangian formulation is obtained by a transformation from the cartesian co‐ordinates (x, y) to the Lagrangian co‐ordinates (λ, ζ). In this manner, the number of independent variables for steady, 3‐D flow is reduced from four to three, placing this formulation on the same footing as the Eulerian formulation even for steady flows (as opposed to the conventional Lagrangian formulation which apparently still requires four independent variables even for steady flows). The generalized Lagrangian formulation with the Godunov scheme (using flux limiters) appears to have distinct advantages over the corresponding Eulerian formulation, particularly with respect to accuracy. Furthermore, the method requires no grid generation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

Arthur Rizzi and Charles J. Purcell

The large‐scale numerical simulation of fluid flow is described as a discipline within the field of software engineering. As an example of such work, a vortex flowfield is…

Abstract

The large‐scale numerical simulation of fluid flow is described as a discipline within the field of software engineering. As an example of such work, a vortex flowfield is analysed for its essential physical flow features, an appropriate mathematical description is presented (the Euler equations with an artificial viscosity model), a numerical algorithm to solve the mathematical equations is described, and the programming methodology which allows us to attain a very high degree of vectorization on the CYBER 205 is discussed. Four simulated flowfields with vorticity shed from wing edges are computed with up to as many as one million grid points and verify the realism of the simulation model. The computed solutions show all the qualitative features that are expected in these flows. The twisted cranked‐and‐cropped delta case is one where the leading‐edge vortex is highly stretched and unstable, displaying ultimately inviscid large‐scale turbulent‐like phenomena.

Details

Engineering Computations, vol. 2 no. 1
Type: Research Article
ISSN: 0264-4401

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Article

Kerk L. Phillips

The purpose of this paper is to infer the welfare of heterogeneous agents using a representative agent model.

Abstract

Purpose

The purpose of this paper is to infer the welfare of heterogeneous agents using a representative agent model.

Design/methodology/approach

It does so by partitioning the household into subunits and allocating consumption to each subunit proportionally to the income the subunit generates through wages and capital returns.

Findings

The author shows that for a simple dynamic general equilibrium model with immigration, the steady state utilities of these subunits correspond very closely to the utilities for an equivalent heterogeneous agent model. This is particularly true when labor–leisure decisions are made using slightly modified Euler equations.

Originality/value

More complicated models can be solved and simulated using fewer computational resources using this technique.

Details

Journal of Economic Studies, vol. 46 no. 7
Type: Research Article
ISSN: 0144-3585

Keywords

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