Convergence Acceleration Techniques Research Articles

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in-house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block-structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.

Development of an Improved Gas-Kinetic BGK Scheme for Inviscid and Viscous Flows

This paper deals with the development of an improved gas-kinetic BGK scheme for inviscid and viscous flow fields. As the first step toward efficient calculation, particle distribution functions in the general solution of the BGK model are simplified to the extent that the essential features of the standard gas-kinetic BGK scheme are not lost. Then, improved schemes are suggested, which overcome difficulties that may arise in the applications of BGK-type schemes to compressible viscous flow calculations. A Prandtl number correction method is also developed to allow the present schemes to work for arbitrary Pr number. For steady state problems, convergence acceleration techniques suitable for the present schemes are developed in the framework of an implicit time integration. Various numerical experiments ranging from one-dimensional shock tubes to viscous turbulent flows are performed to demonstrate accuracy, robustness, and other essential features of the present method.

An unsteady incompressible Navier-Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high-order-accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.

Throughflow model for design and analysis integrated in a three-dimensional Navier-Stokes solver

A multiblock multigrid Navier-Stokes solver has been extended to include a throughflow model for the design and analysis of turbomachines. The presence of the blades in the inviscid axisymmetric flow is modelled in the classical way through a distributed blade force to produce the desired turning, a blockage factor that accounts for the reduced area due to blade thickness, and a distributed frictional force representing the entropy increase due to viscous stresses and heat conduction. The exact blade geometry is not required. All features of the three-dimensional code concerning the physical fluid model, boundary conditions, spatial and time discretization, convergence acceleration techniques and data visualization are available to the throughflow module. This includes the capability to treat the entire range of relevant Mach numbers, from strictly incompressible (through a preconditioning technique) to supersonic, as well as any number of blade rows in any configuration, including, for example, bypass engines. Selected elements comprising the throughflow model are discussed, with special emphasis on the blade force and its discretization. The properties of analysis and design mode with respect to shocks and the associated losses are investigated. The methodology is demonstrated on a transonic compressor rotor and a four-stage low-speed turbine.

Computation of shock/boundary-layer interactions in bump channels with transport-type turbulence models

In this paper, an explicit time-marching finite-volume scheme has been used together with a number of convergence acceleration techniques such as the multigrid strategy. Two types of turbulence models, a Johnson—King (J—K) model and a two-layer k-ε/k-l model, have been incorporated and modified to model internal compressible flows with multiple walls. Some modifications have been made of the inner layer viscosity formulations of the J—K model in order to improve its predictive capability for flow separation. Partially implicit treatment of the transport-type equations of turbulence in the models is adopted, because the source terms in these equations can cause numerical stiffness when there are flow separation, sharp gradients and high cell-aspect ratio near the solid wall. A two-dimensional arc-bump flow investigated experimentally by Liu and Squire [X. Liu and L.C. Squire, Interaction on curved surface at transonic speed, in: Turbulent Shear/Shock Wave Interactions, IUTAM Symposium Palaiseau 1985 (Springer, Berlin Heidelberg 1985) 93–104.] was calculated using the J—K model with satisfactory agreement with the corresponding measurement. Although efficient and accurate, it is found that the J—K model lacks the theoretical generality to be extended to model three-dimensional (3D) complex internal flows with multiple walls. Therefore, a two-layer k-ε model is employed for 3D flow computation. Various measures are adopted to ensure stable and convergent numerical solution. A three-dimensional transonic channel flow with multiple shock/boundary layer interactions was studied with the aforementioned two-layer model and numerical methods. The results are compared with experimental measurements [J. Cahen, V. Couaillier, J. Delery and T. Pot, Validation of Navier—Stokes code using a k-ε turbulence model applied to a three-dimensional transonic tunnel, AIAA paper AIAA-93-0293, AIAA 1993] and numerical results obtained by using a Low-Reynolds-Number (LRN) k-ε model [VUB/FFA, Turbulence Models in EURANUS and the 3D Delery bump, Technical Report SNWP3.3/01, VUB, Pleinlaan 2, 1050 Brussels, Belgium and FFA, P.O. Box 11021, S-161 11 Bromma, Sweden 1993]. Compared with other (LRN) two-equation models, the two-layer model implemented is promising in modeling very complex 3D internal flows in terms of efficiency, robustness and accuracy. The two-layer model permits uniform distribution of flow properties to be specified as initial condition which makes the simulation easier to be carried out.

Calculation of the Electron Self-Energy for Low Nuclear Charge

We present a nonperturbative numerical evaluation of the one-photon electron self energy for hydrogenlike ions with low nuclear charge numbers Z=1 to 5. Our calculation for the 1S state has a numerical uncertainty of 0.8 Hz for hydrogen and 13 Hz for singly-ionized helium. Resummation and convergence acceleration techniques that reduce the computer time by about three orders of magnitude were employed in the calculation. The numerical results are compared to results based on known terms in the expansion of the self energy in powers of (Z alpha).

Convergence Acceleration for Hyperbolic Systems Using Semicirculant Approximations

The iterative solution of systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. The PDEs are discretized using a finite volume or finite difference approximation on a structured grid. A convergence acceleration technique where a semicirculant approximation of the spatial difference operator is employed as preconditioner is considered. The spectrum of the preconditioned coefficient matrix is analyzed for a model problem. It is shown that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of grid points and the artificial viscosity parameter. By linearizing the Euler equations around an approximate solution, a system of linear PDEs with variable coefficients is formed. When utilizing the semicirculant (SC) preconditioner for this problem, which has properties very similar to the full nonlinear equations, numerical experiments show that the favorable convergence properties hold also here. We compare the results for the SC method to those of a multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the MG scheme is sensitive to the amount of artificial dissipation added, while the SC method is not.

Convergence Acceleration for the Linearized Navier--Stokes Equations using Semicirculant Approximations

The iterative solution of systems of equations arising from partial differential equations (PDEs) governing boundary layer flow for large Reynolds numbers is studied. The PDEs are discretized using finite volume or finite difference approximations on tensor product grids. We consider a convergence acceleration technique, where a semicirculant approximation of the spatial difference operator is employed as preconditioner. A relevant model problem is derived, and the spectrum of the preconditioned coefficient matrix is analyzed. It is proved that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of gridpoints and the Reynolds number Re. The same type of result is also derived for finite size grids, where the solution fulfills a given accuracy requirement. By linearizing the Navier--Stokes equations around an approximate solution, we form a system of linear PDEs with variable coefficients. When utilizing the semicirculant preconditioner for this problem, which has properties very similar to the full nonlinear equations, the results show that the favorable convergence properties hold also here. We compare the semicirculant preconditioner to a multigrid scheme. The number of iterations and the arithmetic complexities are considered, and it clear that semicirculant method is much more efficient for problems where the Reynolds number is large. The number of iterations for the multigrid method grows like $\sqrt{Re}$, while the convergence rate for the scheme using semicirculant preconditioning is independent of Re. Also, the multigrid scheme is very sensitive to the level of artificial dissipation, while the method using semicirculant preconditioning is not.

New Three-Dimensional Inverse Method for High-Speed Vehicle Design

An inverse design method for fully three-dimensional supersonic and hypersonic bodies was developed for computercodesbasedontheEulerequations.Themethodisdesignedtobeeasilyincorporatedintoexistinganalysis codes.Theaim ofthemethodistoprovidetheaerodynamicdesignerwitha powerfultoolfordesign ofaerodynamic shapes of arbitrary cross section. Convergence acceleration techniques were developed and applied successfully to three-dimensional test cases. A three-dimensional body geometry modie cation method was developed with a new concept for the three-dimensional body slope angle. Examples are presented applying the method to a nonaxisymmetricfuselage-typepressuredistributionandacamberedwing-typeapplication.Themethodperforms equally well for both nonlifting and lifting cases. Nomenclature A ML = coefe cient of surface pressure-body shape rule; Eqs. (10) and (12) a = major axis length b = minor axis length Cp = pressure coefe cient C T p = target pressure coefe cient at the body surface c = coefe cient dee ned by Eq. (13) i = marching plane index jdim = number of grid points along the circumferential direction

Simplified spherical harmonic method for coupled electron-photon transport calculations

In this paper we apply the simplified spherical harmonic ${(SP}_{N})$ approximation to coupled electron-photon transport problems in two-dimensional cylindrical geometry in the energy range from roughly 10 keV to 10 MeV. The ${\mathrm{SP}}_{N}$ equations represent an asymptotic approximation that does not necessarily converge to the exact transport solution as $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ but can sometimes produce solutions that are much more accurate than diffusion theory at a fraction of the cost of a full transport treatment. To our knowledge, the ${\mathrm{SP}}_{N}$ approximation has previously been applied only to neutron transport problems. We investigate the applicability of the ${\mathrm{SP}}_{N}$ method to satellite electronics shielding calculations. In addition to applying the approximation, we generalize certain iterative convergence acceleration techniques originally developed for the one-dimensional ${S}_{N}$ (discrete ordinates) equations, and apply them to the two-dimensional ${\mathrm{SP}}_{N}$ equations. We present numerical comparisons with Monte Carlo calculations for the purpose of examining both the accuracy of the ${\mathrm{SP}}_{N}$ approximation and the computational efficiency of our solution techniques.

A 2D Navier-Stokes method for unsteady compressible flow calculations on moving meshes

AbstractA moving mesh method for the computation of compressible viscous flow past deforming and moving aerofoils is described. It is based on a well established time-marching finite-volume scheme, which has been widely used for steady compressible flows. An implicit real-time discretisation is used and the equations are integrated via a dual-time scheme. This involves the introduction of derivatives of a fictitious pseudo time. The solution at each real-time step involves seeking solutions which are steady-state solutions in pseudo time. This approach decouples the stability and accuracy limitations of the scheme and permits large real-time steps to be chosen. Also well-proven convergence acceleration techniques developed for steady flows such as local-time stepping, residual averaging and multigrid may be used in the pseudo-time stepping scheme without compromising real-time accuracy. A sequence of body-conforming grids and corresponding grid speeds is required, where the inner and outer boundaries of the grid move independently. The required grids and speeds are found using a transfinite interpolation technique. Applications of the method to external compressible flows are shown, including results from a parallel version of the computer program.

Multigrid Method for the Euler and Navier-Stokes Equations

Results are presented of a study to implement convergence acceleration techniques based on the multigrid concept in a three-dimensional computer code developed for solving Euler and Navier ‐Stokes equations. The multigrid method implemented is the full approximation storage ‐full multigrid algorithm, which is applicable to nonlinear equations. Multigrid convergence estimates based on a single-grid or bigrid stability analysis are performed. The latter was found to yield a better prediction of practical multigrid convergence rates. In typical applications, for both laminar and turbulent e ows, savings of up to 50 % in CPU time were obtained with the multigrid implementation.

An Integral Equation Solution for the Steady-State Current at a Periodic Array of Surface Microelectrodes

An integral equation approach is developed for the problem of determining the steady-state current to a periodic array of microelectrodes imbedded in an otherwise insulating plane. The formulation accounts for both surface electrode and bulk fluid reactions and evaluates Green's functions for periodic systems using convergence acceleration techniques. Numerical results are presented for disc-shaped microelectrodes at the center of rectangular periodic cells for a large range of dimensionless surface and bulk reaction rates and periodic cell sizes.

Blunt-fin induced interaction of wall shear layers in supersonic flow

The topic of this contribution is a numerical study of three-dimensional supersonic flow over a blunt fin mounted on a flat plate. The flow pattern is characterized by complex interaction of shock waves and the boundary layer along the plate surface. In this paper the steady-state results for two free stream Mach numbers M = 3 and 7 will be discussed and compared with experiments. For the numerical investigation a time-marching integration method has been applied based on a three-stage Runge-Kutta scheme. Several convergence acceleration techniques have been adopted. Since crisp shock fronts must be resolved for a proper computation of the interaction phenomena the mesh has been adapted in those regions where strong pressure gradients are expected. In addition, a special treatment of the artificial viscosity terms which enhances stability in regions with shocks has been introduced to achieve some upwind properties of the scheme which utilizes central differences for the spatial derivatives. For the simulation of turbulent flow algebraic models have been applied, namely, a Baldwin-Lomax model and a new turbulence model which is based on RNG-theory. The comparison with the experiments shows that for the case studied here the transitional character of the boundary layer has to be taken into account.

MULTIGRID CONVERGENCE ACCELERATION FOR TURBULENT SUPERSONIC FLOWS

A multigrid convergence acceleration technique has been developed for solving both the Navier-Stokes and turbulence transport equations. For turbulence closure a low-Reynolds-number q-ω turbulence model is employed. To enable convergence, the stiff non-linear turbulent source terms have to be treated in a special way. Further modifications to standard multigrid methods are necessary for the resolution of shock waves in supersonic flows. An implicit LU algorithm is used for numerical time integration. Several ramped duct test cases are presented to demonstrate the improvements in performance of the numerical scheme. Cases with strong shock waves and separation are included. It is shown to be very effective to treat fluid and turbulence equations with the multigrid method.

MULTIGRID CONVERGENCE ACCELERATION FOR TURBULENT SUPERSONIC FLOWS

A multigrid convergence acceleration technique has been developed for solving both the Navier–Stokes and turbulence transport equations. For turbulence closure a low-Reynolds-number q–ω turbulence model is employed. To enable convergence, the stiff non-linear turbulent source terms have to be treated in a special way. Further modifications to standard multigrid methods are necessary for the resolution of shock waves in supersonic flows. An implicit LU algorithm is used for numerical time integration. Several ramped duct test cases are presented to demonstrate the improvements in performance of the numerical scheme. Cases with strong shock waves and separation are included. It is shown to be very effective to treat fluid and turbulence equations with the multigrid method. A comparison with experimental data demonstrates the accuracy of the q–ω turbulence closure for the simulation of supersonic flows. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1019–1035, 1997.

Numerical Simulation of Receptivity Phenomena in Transitional Boundary-Layer Flows

A computational fluid-acoustic methodology is described and used to simulate induced and acoustic receptivity in unsteady boundary-layer flows. The methodology solves a finite compressibility form of the Navier-Stokes equations using a flux differenced, finite volume technique. Special attention is paid to nonreflective boundary conditions appropriate for unsteady, multidimensional problems including those involving viscous shear and propagating waves. The numerical experiments include tbe simulation of acoustic receptivity due to surface inhomogeneity in which the acoustic phenomena are modeled using physically appropriate wavelengths. Required steady solution accuracy, convergence acceleration techniques, boundary condition/flowfield interactions and the challenges of scale resolution vs computational cost are addressed in a series of simulations.

Euler summation for fourier series and laplace transform inversion

Euler summation is a convergence-acceleration technique which has proved very effective in Fourier-series methods for Laplace transform inversion. We present an analysis of the effect of Euler summation that explains its excellent performance. The central result is a bound on the truncation error when Euler summation is used. Our analysis supports some of the parameter-selection strategies proposed in Abate and Whitt's recent comprehensive treatment of Fourier series methods [2]. Specifically, we show that if our goal is to minimize the required number of terms of the Fourier series subject to the truncation error bound being no more than a specified target accuracy, then the “degree of averaging parameter” m should depend only on the desired accuracy, and not on properties of the function in question (so long as that function is sufficiently smooth). We also present a framework, here called “product smoothing, ” for constructing related summation methods with desirable properties

Computational methods in chemical engineering

From the Publisher: Presenting a clear, concise exposition of practical computational methods in chemical engineering, this book provides a bridge between introductory applications and more advanced levels of mathematics. Authors Owen Hanna and Orville Sandall include broad use of convergence acceleration techniques such as Pade approximation for series; Shanks transformation for series; linear and nonlinear systems of algebraic equations; systematic use of global Richardson extrapolation for integrals and ODE systems to monitor the overall error; and discussion of methods for the solution of stiff ODE. In addition, some 25 user-friendly microcomputer programs which are demonstrated and used for many of the problems are available in IBM format on the accompanying diskette in FORTRAN 77 and True BASIC.

Acceleration of Convergence and Spectrum Transformation of Implicit Finite Difference Operators Associated with Navier-Stokes Equations

Eigensystem analysis techniques are applied to finite difference formulations of Euler and Navier-Stokes equations in two dimensions. Spectrums of the resulting implicit difference operators are computed. The convergence and stability properties of the iterative methods are studied by making into account, the effect of grid geometry, time-step, numerical dissipation,, viscosity, boundary conditions, and the physics of the underlying flow. The largest eigenvalues are computed by using the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the dominant eigenvalues with the rate of convergence of the iterative method. Based on the pattern of eigenvalues distributions for various flow configurations, the feasability of applying existing convergence-acceleration techniques like eigenvalue annihilation and relaxation are discussed. Finally a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. The procedure of shifting the spectrum is applied to ARC2D, a flow code developed and being used at NASA Ames Research Center. When compared to eigenvalue annihilation, the shifting method clearly establishes its superiority. For the ARC2D code, an efficiency of 20 to 33% has been achieved by this method.