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Numerical Solution of the Steady Flow in Turbomachine Blades and Ducts of Arbitrary Shape

A solution procedure for solving partial diffential equations of elliptic type in an arbitrary shaped region is presented. The interesting region is solved by a body oriented grid and transformed in a rectangle with square mesh by numerical evaluated transformation functions. With this method the numerical solution of a partial differential system may be done in this rectangular field with no interpolation required regardless of the shape of the physical boundaries and of the spacing in the physical field. This new solution technique has been used to analyse the compressible inviscid flow through turbomachine blades and the compressible viscid flow through an interstage return bend of a radial compressor.

M. von Allmen, J. Wachter, B. Schulz

Computation of Strong Interactions in Transonic Flows

Compressible laminar separated boundary layer is computed. The boundary layer equations are solved by a second order finite difference technique. The solution is obtained by using an under-relaxation procedure on profiles and pressure coefficient. The inverse problem with prescribed Cf is first considered. Then a shock wave boundary layer interaction is computed by coupling viscous-inviscid flows. The results for the interaction on a 10° ramp at M = 3 are presented. Extensions to transonic turbulent flows are considered.

D. Aymer de la Chevalerie, R. Leblanc

Natural Convection in Cavities for High Rayleigh Numbers

An analysis has been made of the motion in a long shaped vertical (or nearly vertical) cavity. Particular emphasis has been devoted to the stability, at large Ra, of an hermitian finite difference method to solve the 2D Navier-Stokes and energy equations. A secondary flow causing multicellular motion has been studied for Ra ranging from 2 × 105 to 6.8 × 105, for Pr = 0.7 and ℓ = 2.5, and to 1.2 × 106, for Pr = 480 and ℓ = 15.

P. Bontoux, B. Gilly, B. Roux

Computational Schemes in General Curvilinear Coordinates for Navier-Stokes Flows

The geometrical versatility obtained through boundary fitted mappings, requires an appropriate extension of the Cartesian F.D. schemes in order to obtain accurate solutions. The present study shows that the contravariant velocity components located at the cell midsides, generalize in the proper way the MAC scheme to curvilinear meshes. The results indicate also that the assumption of the Cartesian components as variables brings, for distorted grids, to wiggles characterizing a loss of accuracy, peculiar of models based on an improper location of the variables.

A. Di Carlo, R. Piva, G. Guj

Studies of Turbulent Confined Jet Mixing

The paper presents recent work in the analysis of confined jet mixing. The theoretical part extends previous work of the authors [4] in predicting flow behaviour by now using a system of general 2-D orthogonal axisymmetric co-ordinates; this enables flows in the non-uniform mixing duct to be calculated. The prediction procedure, via a primitive pressure-velocity finite difference code is more efficient and accurate than its stream function-vorticity predecessor and is simpler to modify in the simulation of boundary conditions, for example. The final developed mathematical model provides an extremely useful tool for jet pump and ejector design. The designer needs only to specify geometry and primary inlet conditions together with operating pressures to obtain performance information such as thrust augmentation, pressure rise, entrainment and efficiency.

D. R. Croft, P. D. Williams, S. N. Tay

A Mixte Compact Hermitian Method for the Numerical Study of Unsteady Viscous Flow Around an Oscillating Airfoil

Unsteady viscous flow around an oscillating elliptic cylinder is studied in this paper by a so-called “combined” or “mixte” method, which uses a second order finite difference scheme to solve the vorticity transport equation and a fourth order compact hermitian method to calculate the stream function equation. Specific aerodynamic properties of the phenomenon 3 4 are analyzed in detail for Reynolds numbers equal to 10 and 10 and for different values of the period and amplitude of the oscillation.

Olivier Daube, Phuoc Loc Ta

Finite Element Methods for Transonic Flow Calculations

An artifical compressibility formulation of the transonic full potential equation is discretized with bilinear Finite Elements and the resulting non linear system of equations is solved with a relaxation method and with approximate factorization methods. A finite element approach to the AF1 and AF2 schemes is presented. Results containing shocks are presented for different types of geometry, including turbine and compressor cascades.

Herman Deconinck, Charles Hirsch

Evaluation of a Minimum Principle for Transonic Flow Computations by Finite Elements

A simple Finite Element scheme is proposed for the solution of the full potential equation of gasdynamics which is based on a variational principle stating that the static pressure forms a minimum relief with respect to prescribed boundary conditions. The introduction of an artificial density allows for the computation of transonic flow past aircraft components of engineering interest. Selected results will show benefits and present limitations of the method.

Albrecht Eberle

Compressible Flow in Arbitrarily Connected Fluid Domains

A computer code FLUST has been developed that is able to calculate fluid flow in arbitrarily connected areas. Models of different dimensionality and numeric solution methods can be integrated into the code framework. Presently one- and two-dimensional compressible flow can be treated, using finite difference methods. Pressure, density, internal energy and velocity are integrated semi-implicitely. Homogeneous equilibrium two-phase flow can be treated. The code FLUST has been used for precalculations of the HDR blowdown experiments. Modelling of the expe-mental setup and results of the precalculations are presented.

Günter Enderle

A Numerical Investigation of a Two-Dimensional Shock Structure

The paper presents an application of the discrete ordinates method to the investigation of a two-dimensional shock structure close to a wall. The region of a shock wave is investigated from kinetic theory view point. The flow of a monatomic gas in a stationary coordinates system moving with the wave is described by the Boltzmann model kinetic equations (BGK and Ellipsoidal type). The distributions of number density, velocity and temperature in the flow field are obtained as the result of the calculations which are carried out. The present paper is work in progress.

Piotr Gajewski, Bernd Schmidt

Numerical Solution of Linear and Non-Linear Parabolic Differential Equations by a Time-Discretisation of Third Order Accuracy

Solving parabolic differential equations by the well-known explicit one-step method the stability condition imposes a rigorous restriction on the time step. But this condition is only a sufficient one if one considers more than one time step, say n. A slight modification of this method dramatically enlarges the time steps. In this paper a n-step method is presented which is n times faster than the known explicit one. Subsequently an extension yields a method of third order accuracy which seems to be an useful explicit alternative to the implicit Crank-Nicolson method of second order accuracy.

W. Gentzsch

Mixed Eulerian — Lagrangian Formulation and Finite Element Solution of 2D Moving Boundary Problems in Compressible Fluid Dynamics

Presented here is a numerical method to analyze the unsteady states of perfect compressible fluid motions in 3D axisymmetric geometries for those complicated cases where part of the configuration is bounded by continuously changing surfaces, such as free surfaces and material de-formable boundaries. The following topics are especially emphasized:1)the introduction of a reference motion and the subsequent definition of mixed Eulerian-Lagrangian coordinates,2)the development of a powerful numerical technique based on variable domain finite elements for the spatial discretization and finite differences for marching in time.Asshown by sample calculations, appropriate selection of the mesh reference motion simplifies, in particular, the treatment of deformable sliding surfaces and the description of impact/rebounding phenomena.

G. Van Goethem

Transonic Flow over Airfoils with Tangential Injection

The influence of tangential slot injection in an attached, turbulent boundary layer on the pressure distribution of an airfoil is investigated for transonic freestream Mach numbers. The flow field is calculated by matching finite-difference solutions of the transonic small disturbance equation and the boundary layer equations. The results, obtained for different slot positions and injection rates, indicate that the surface pressure distribution can significantly be changed by tangential injection.

D. Hänel, H. Henke, A. Merten

On the Finite Element Modelling of Viscous-Convective Flow

The paper discusses relationship, performance and stability properties of the Ritz-Galerkin and least squares finite element methods, with respect to the solution of viscous-convective flow problems.

A. Kanarachos

On Steady Shock Computations Using Second-Order Finite-Difference Schemes

A perturbation analysis has been made for steady state discontinuities as computed by second-order finite-difference schemes. It is shown that steady state eigenvectors exist for such perturbations for initial-boundary value problems involving a simple non-linear scalar equation and the system of conservation laws for inviscid, compressible flow. Results are given for the Lax-Wendroff scheme, the MacCormack scheme, and the hybrid scheme of Warming and Beam.

Lasse K. Karlsen

Vector Processing on the Cyber 200 and Vector Numerical Linear Algebra

The CDC CYBER 200 series computer provides computational capability measured in the hundreds of mflops (millions of floating point operations per second). This capability is due to its vector processing hardware. The goal of this paper is twofold:1.To illustrate this hardware capability by a selected set of syntactic kernels, and2.To provide an introduction to the applicability of vector processing to some basic algorithms of numerical linear algebra.

Michael J. Kascic

Treatment of Incompressibility and Boundary Conditions in 3-D Numerical Spectral Simulations of Plane Channel Flows

A spectral method for numerical computation of 3-D time-dependent incompressible flows between two plane parallel plates is presented. Fourier expansions in the coordinates parallel to the walls and expansions in Chebyshev polynomials in the normal coordinate are used. The time coordinate is discretized with second order finite differences, treating the viscous terms implicitly. An efficient direct solution procedure for the implicit equations is developed which reduces the 3-D problem to a set of essentially tridiagonal linear equations in one space coordinate. Boundary and continuity conditions are satisfied exactly, apart from round-off errors.

L. Kleiser, U. Schumann

Design of a Calculation Method for 3D Turbulent Boundary Layers

An engineering method is presented to compute the 3D laminar and turbulent boundary layer flow over non-developable surfaces. Turbulence phenomena are described by a simple eddy viscosity model. Much attention was paid to the organization of the method to enhance its users-oriented properties. The organization and the numerics are discussed in the framework of relational diagrams which elucidate the inter-relations between information-sets and processes. Results are shown for the laminar flow over a flat plate with attached cylinder and for the turbulent flow over the rootsection of an aircraft wing.

J. P. F. Lindhout, B. van den Berg

A Class of Diagonally Dominant Implicit Schemes with Arbitrary Numerical Dissipation

A class of implicit finite difference schemes are proposed for time dependent solution of an unsteady partial differential equation. These schemes are temporally inconsistent to the PDE, only in the steady state, the difference equations become consistent to the steady state differential equation. The coefficient matrix of difference equations is tridiagonal and always diagonally dominant. An important feature of these schemes is that arbitrary amount of numerical dissipation can be incorporated in difference equations by virtue of difference formulas. Calculations with nonlinear Burgers’ equation show fast convergence.

Nimai Kumar Mitra, Martin Fiebig

The “Post-Correction” Technique for the Fitting of Shocks and Other Boundaries

A new technique for shock fitting is presented, which combines the advantages of a proper physical formulation with a high degree of simplicity.

Gino Moretti, Tom de Neef

Finite Element Approximation of a Variational Principle for Perfect Fluid Flows with Free Boundaries

The subsonic flow of several perfect fluids with free boundaries can be described by a variational principle in streamfunction formulation for the two-dimensional case. The variational unknowns are the stream-function, the density and the shapes of the free boundaries. The finite element approximation of this variational principle combined with an automatic mesh generation of the variable domains leads to an optimization problem which is solved by powerful minimization techniques. Three aerodynamic applications are presented in the irrotational axisymmetric or plane case.

Philippe Morice

A Fourth-Order Compact Implicit Scheme for Solving the Non-Linear Shallow-Water Equations in Conservation-Law Form

A Fourth-order compact implicit finite-difference scheme is applied for solving numerically the nonlinear shallow-water equations in conservation-law form. The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored (ADI) form. Third-order uncentered boundary conditions which preserve the overall fourth-order convergence are experimented with and compared. Von Neuman linearized stability analysis as well as Kreiss-type normal-mode analysis are performed. The integral invariants of the shallow-water equations are well conserved during the numerical integration. Accuracy tests confirm the fourth-order accuracy of the scheme.

I. M. Navon

Cømparisøn øf Søme Numerical Methøds før Sølving Hyperbølic Differential Equatiøns with Discøntinuøus Initial Values

Numerical solutions of the hyperbolic equations of unsteady compressible fluid flow obtained by the method of Lax-Wendroff show non-physical overshoot in case of discontinuous initial values. We examine error and overshoot of several proposed methods to improve this: strongly amplifying physical viscosity, insertion of damping terms due to Lax-Wendroff [1], modifications due to Abarbanel-Zwas [2] and Lerat-Peyret [3], fourth order smoothing, and flux correction due to Book, Boris and Hain [4]. A variation of this last method, which we call “naive flux correction”, does not alter either accuracy or stability of the Lax-Wendroff method if clipping included in flux correction (see equ. 28) is not active.

Herbert Niessner, Tomas Bulaty

Plane Unsteady Flow of Inviscid and Incompressible Fluid Around a System of Profiles

The method of conformai representation has been applied to investigation of plane, unsteady, inviscid and incompressible flow around an arbitrary system of profiles, moving in a known manner, It has been shown, that the formerly developed algorithm [1,2] for determination of the mapping function can be utilised also in the present unsteady case. It has been demonstrated, too, that the system of linear equations enabling one to determine the time-depending complex potential does not differ formally from the one established in the steady case [3]. General formulae for pressure, force and moment, acting on an arbitrary profile of the system have been derived.

W. J. Prosnak, M. E. Klonowska

Three-Dimensional Numerical Evaluation of Heat Loss through Natural Convection in a Solar Boiler

An implicit finite difference method is presented for the integration of compressible Navier Stokes equations. The scheme involves a generalized linearization process and the Douglas-Gunn ADI technique. The variables are defined on a variable mesh staggered grid and the spatial differencing satisfies some integral conservation properties for mass, momentum and energy. Stability is demonstrated for time steps much larger than the CFL limit. The method is applied to the computation of free convection flows in two and three dimensions with high values of the Grashoff number.

P. Le Quere, T. Alziary de Roquefort

Boundary Conditions in Difference Schemes for Hyperbolic Systems

The general method for approximation of the boundary conditions in mixed problems for the hyperbolic systems is proposed. The method is fitted for the approximation of any order and for the discontinuous solutions. The examples of the difference schemes of Ith, 2d and 3d order are given. The stability of schemes are investigated by means of theory developed by Kreiss [1,2]. The examples of computation of the reflecting shock wave are given.

V. V. Rusanov, E. I. Nazhestkina

Computational Methods for the Design of Adaptive Airfoils and Wings

A new design principle for transonic aerodynamics has led to several operational computer codes for shock-free flow. The method is described with emphasis on theoretical variable geometry definition of transonic airfoils and wings for efficient supercritical flight.

Helmut Sobieczky

A Generalized Grid-Free Finite Difference-Method

A generalized explicit finite difference method for arbitrarily distributed nodes is presented. It is developed for the calculation of two-dimensional unsteady inviscid flows. The advantage of the method is its independence from a regular grid, so that geometrically difficult domains with complicated boundaries can be discretized easily. Numerical results from a test problem (Ringleb Flow) are shown.

L. Theilemann

A Calculation Method for Incompressible Boundary Layers with Strong Viscous-Inviscid Interaction

This paper presents a method for the calculation of boundary layers with strong viscous inviscid interaction. The method differs from the classical methods for solving boundary layer equations through the use of an interactive boundary condition which replaces the usually prescribed pressure. This boundary condition describes how the outer potential flow reacts on the presence of the boundary layer. The method is demonstrated on two problems: i) a laminar boundary layer with a separation bubble, and ii) the laminar flow near the trailing edge of a flat plate.

A. E. P. Veldman

Numerical Solution of Advection-Diffusion Problems by Collocation Methods

Pseudo-spectral and orthogonal collocation methods are compared in general terms, and their use is demonstrated solving a test-problem which describes the time dependent development of a plume behind an elevated crosswind line source. Problems and recent developments of the numerical methods are discussed. Preliminary results from a pseudo-spectral numerical simulation of a point source plume in a large-eddy simulation are given.

Hans Wengle

Numerical Analysis of the Stability and Non-Uniqueness of Spherical Couette Flow

Numerical study of the non-linear axisymmetric viscous flow in a spherical layer is presented. The boundary spheres may rotate with constant but different angular velocities about the same axis. The flow is governed by a non-linear boundary value problem of the Navier-Stokes equations. The solution of this problem is found by the method of stabilization. The unknown functions are represented as series of Legendre associated functions with the coefficients depending on z and t. Two different procedures are used for non-linear terms computation. The numerical results show the existence of several different types of steady state flows in the same supercritical regions of the similarity parameters. The stability curves of the basic flow in the (Re 1 , δ) and (Re 1 , Re 2 )-planes are obtained.

I. M. Yavorskaya, N. M. Astaf’eva
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