# Showing papers in "Computers & Fluids in 1989"

••

[...]

TL;DR: In this article, the Navier-Stokes equations in an implicit flux-split difference formulation are solved numerically using a Gauss-Seidel line-relaxation procedure.

Abstract: The Navier-Stokes equations in an implicit flux-split difference formulation are solved numerically using a Gauss-Seidel line-relaxation procedure. Particular attention is given to the selection of flux-vector splitting method and flux splitting in boundary layers. Results for sample problems involving (1) turbulent supersonic flow over a cone and (2) the viscous hypersonic flow of a chemically reacting gas in thermal nonequilibrium past a blunted cone are presented in extensive graphs and briefly characterized. The present flux-split procedures are shown to provide accurate shear-layer calculations.

458 citations

••

[...]

TL;DR: In this paper, the Navier-Stokes equations for incompressible fluids are used to model the steady and unsteady flows past a circular cylinder, which is assumed to be two-dimensional.

Abstract: Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.

119 citations

••

[...]

TL;DR: In this paper, the flow characteristics of a Newtonian fluid in a two-dimensional, planar, right angled Tee branch are studied over a range of inlet Reynolds number of 10-800 by solving the Navier-Stokes equations using a finite element discretization.

Abstract: The flow characteristics of a Newtonian fluid in a two-dimensional, planar, right angled Tee branch are studied over a range of inlet Reynolds number of 10–800 by solving the Navier-Stokes equations using a finite element discretization. The effects of the branch length and the grid size on the interior flow field are examined to assess the accuracy of the solutions. In one case the computed velocity field is compared with the Laser Doppler anemometry measurements available in the literature and excellent agreement has been obtained. The computed velocity field is believed to be accurate within about 5%. Results are presented for two types of experimentally realizable boundary conditions—viz. equal exit pressure at the outlet of each branch and specified flow split between the branches. For the case of equal exit pressures the fractional flow in the main duct increases with increasing Reynolds number and the flow characteristics in the side branch become akin to that in a cavity. For the case of specified flow split, the number, size and strength of the recirculation zones increase as more fluid is forced to go into the side branch. The length of the side branch appears to have very little influence on the interior flow field, particularly at higher Reynolds number. This observation is rationalized as being due to the parabolized approximation becoming more valid at higher Reynolds numbers. The critical Reynolds number at which the first recirculation zone appears in the side branch increases with increasing fractional flow in the side branch and with decreasing side branch width.

54 citations

••

[...]

TL;DR: In this paper, a particle discretisation of the flow is used to conserve the total vorticity of the Navier-Stokes equations, and numerical results concerning the shear layer problem are presented and the influence of the Reynolds number on the numerical solution is explored.

Abstract: In this paper, we present a new method for solving the Navier-Stokes equations. This method is based on a particle discretisation of the flow. This viscous terms are accounted for by using an elementary solution of the diffusion equation with a Dirac'function for the initial data. The algorithm is especially designed to conserve the total vorticity. Numerical results concerning the shear layer problem are presented and the influence of the Reynolds number on the numerical solution is explored.

52 citations

••

[...]

TL;DR: In this paper, a new asymptotic theory of turbulent boundary layers is proposed to determine the singular behavior of solutions of the boundary layer equations at the point of zero skin friction in a prescribed pressure distribution.

Abstract: The problem addressed in this paper is the turbulent counterpart of the classical Goldstein analysis of laminar separation points. In the present paper we present a new asymptotic theory of turbulent boundary layers and we use it to determine the singular behavior of solutions of the boundary layer equations at the point of zero skin friction in a prescribed pressure distribution. In the first part of the paper we describe a new asymptotic theory of turbulent boundary layers which is based on formal expansions in terms of two parameters: Reynolds number Re → ∞ and α → 0 , where α is a turbulence model constant appearing in the zero equation turbulence model employed in the present study. In order to establish the accuracy of the theory, solutions obtained with the new theory for equilibrium turbulent flow are compared with “exact” solutions of Mellor and Gibson. In the second part of the paper the local behavior of solutions of the turbulent boundary layer equations with a zero equation turbulent model is established using the new asymptotic theory. It is demonstrated that the solution of the turbulent boundary layer equations are singular at separation points if the pressure distribution is prescribed just as in the laminar case. However, the nature of the singularity is different in laminar and turbulent flow. We show that the approach to zero skin friction is linear in turbulent flow as compared to the square root behavior of the laminar “Goldstein” singularity.

45 citations

••

[...]

TL;DR: In this article, a symmetric splitting for both the Euler and Navier-Stokes equations is presented which removes the stiffness of these equations when the Mach number is small.

Abstract: Examined are some splitting techniques for low Mach number Euler flows. Shortcomings of some of the proposed methods are pointed out and an explanation for their inadequacy suggested. A symmetric splitting for both the Euler and Navier-Stokes equations is then presented which removes the stiffness of these equations when the Mach number is small. The splitting is shown to be stable.

36 citations

••

[...]

TL;DR: The behavior of gas dynamic flows which are perturbations of a uniform stream in terms of information transfer across artificial (computational) boundaries remote from the source of disturbance is discussed in this article.

Abstract: The behavior of gas dynamic flows which are perturbations of a uniform stream in terms of information transfer across artificial (computational) boundaries remote from the source of disturbance are discussed. A set of boundary conditions is derived involving vorticity, entropy, and pressure-velocity relationships derived from bicharacteristic equations.

35 citations

••

[...]

TL;DR: In this article, the incompressible axisymmetric steady Navier-Stokes equations are written using the stream function-vorticity formulation and the resulting equations are discretized using a second-order central difference scheme.

Abstract: The incompressible axisymmetric steady Navier-Stokes equations are written using the streamfunction-vorticity formulation. The resulting equations are discretized using a second-order central-difference scheme. The discretized equations are linearized and then solved using an exact LU decomposition, Gaussian elimination, and Newton iteration. Solutions are presented for Reynolds numbers (based on vortex core radius) 100-1800 and swirl parameter 0.9-1.1. The effects of inflow boundary conditions, the location of farfield and outflow boundaries, and mesh refinement are examined. Finally, the stability of the steady solutions is investigated by solving the time-dependent equations.

34 citations

••

[...]

TL;DR: In this article, the PISO and SIMPLE algorithms were compared by comparison of the computational effort required for reaching the same convergence criterion in two test problems, and the results for swirling flow problem were verified on the basis of available experimental data.

Abstract: Properties of the PISO and SIMPLE algorithms of solution of momentum equations are examined by comparison of the computational effort required for reaching the same convergence criterion in two test problems. In the present paper swirling and non-swirling flows are considered in axisymmetric geometry. In the computations both, interative and time marching versions of these methods are considered. Optimal values of numerical parameters, on which effectivity of both algorithms depends for various differential grids, were found and the results for swirling flow problem were verified on the basis of available experimental data.

33 citations

••

[...]

Virginia Tech

^{1}TL;DR: In this article, upwind algorithms for the numerical solution of the multidimensional Euler equations for real gases are developed for high-Mach-number high-temperature flows.

Abstract: Upwind algorithms are developed for the numerical solution of the multidimensional Euler equations for real gases. Flux-splitting methods are derived which account for a general equation of state. Approximations to the state equation based on physical arguments result in simplified algorithms which may be implemented into existing perfect-gas codes. Applications of the method to several high-Mach-number high-temperature flows are presented for two and three space dimensions.

22 citations

••

[...]

TL;DR: In this article, a finite-difference method based upon an upwind hybrid formulation, a blend of the "lambda" and the "fluxdifference splitting" formulations, is used to study transonic flows governed by the Euler equations, about half and full circular cylinders.

Abstract: Transonic flows governed by the Euler equations, about half and full circular cylinders, are investigated numerically. The tool to carry out this study is a finite-difference method based upon an upwind hybrid formulation, a blend of the “lambda” and the “flux-difference splitting” formulations. In the case of the half cylinder, a separation occurs behind the shock, with the generation of a circulating bubble. It is worthwhile mentioning that for M ∞ = 0.5 we have not been able to reach a steady configuration, instead a barely noticeable and perfectly periodic unsteadiness develops. The phenomenon is much more evident at M ∞ = 0.60 . For the full cylinder case, we have found that the symmetrical configuration with respect to the longitudinal geometrical symmetry plane is not stable. Rather, asymmetric, unsteady, periodic flows are predicted with the shedding of eddies behind the cylinder, which trap the vorticity generated by the shocks. Numerical experiments with different computational parameters and grid sizes seem to remove any doubt about the reliability of the present numerical results to represent solutions of the Euler equations in these particular problems.

••

[...]

TL;DR: In this article, a finite-volume procedure is introduced for calculating the 3D turbulent flow in complex junctions consisting of a main duct and a side branch which may intersect at any angle.

Abstract: A finite-volume procedure is introduced for calculating the 3-D turbulent flow in complex junctions consisting of a main duct and a side branch which may intersect at any angle. Any combination of flow directions in the junction ducts can be accommodated, and regions of separation are permitted. The procedure is folmulated with economy of storage being an important consideration. A characteristic feature of the scheme is a quasi-uncoupled, zonal treatment of the main duct and the side branch, whereby each is covered by a separate non-orthogonal mesh and computed separately. Coupling is then achieved iteratively within the overall solution process. Regions of separation are treated by means of “elliptic” patches embedded within a “partially parabolic” flow domain. The capabilities of the schemes are demonstrated through comparisons between computations and experimental LDA data for a range of laboratory junction flows. The procedure is then applied to practical junctions used in real IC-engine manifolds.

••

[...]

TL;DR: In this article, a two-dimensional rectangular box is partially filled with a fluid containing a solute which evaporates at the upper surface, and the system is considered under zero-gravity.

Abstract: A two-dimensional rectangular box is partially filled with a fluid containing a solute which evaporates at the upper surface. The system is considered under zero-gravity. For sufficiently large Marangoni number the quiesent state becomes unstable due to surface tension effects. By aid of a Galerkin method using splines the eigenvalues and eigenvectors of the linearized system are determined. Each eigenvalue corresponds to a critical Marangoni number for a certain mode (eigenvector). The eigenvalues have been investigated as functions of the aspect ratio of the box. Two different symmetries are possible for the modes and it is shown that only eigenvalues pertaining to modes of different symmetry can coincide.

••

[...]

TL;DR: In this article, the Navier-Stokes equations are solved by means of a robust multigrid block-line Gauss-Seidel method, and the corresponding boundary layer equations at every longitudinal station are solved using an incremental block-implicit scheme combined with a deferred correction strategy to achieve fast convergence on the nonlinear terms.

Abstract: The present study is concerned with the numerical prediction of planar and axially-symmetric sudden expansion flows, using the Navier-Stokes as well as the boundary layer equations. The vorticity-steam function Navier-Stokes equations are solved by means of a robust multigrid block-line-Gauss-Seidel method. The corresponding boundary layer equations are solved at every longitudinal station by means of an incremental block-implicit scheme, using the Newton method combined with a deferred correction strategy to achieve fast convergence on the nonlinear terms. Accurate solutions to the Navier-Stokes equations and to the boundary layer equations with and without the FLARE approximation are provided for both planar and axially-symmetric expansions.

••

[...]

TL;DR: In this article, the transonic flow around a low-aspect-ratio infinite wing in a wind tunnel is investigated by means of numerical simulations, with a focus on shock-induced separated flows.

Abstract: The transonic flow around a low-aspect-ratio infinite wing in a wind tunnel is investigated by means of numerical simulations, with a focus on shock-induced separated flows. A coarse global grid with far-field boundaries matching those of the test section is subdivided into zones: the flow in the clustered zones near the wing is analyzed by solving the Reynolds-averaged Navier-Stokes equations, while that farther from the wing is modeled with the Euler equations. The results are presented graphically, and it is shown that a mushroomlike separated flow with two counterrotating vortices can be simulated when the correct shock strength is imposed (by careful selection of the artificial dissipation, the boundary conditions, the grid refinement, the algebraic turbulence model, and the geometry representation).

••

[...]

TL;DR: In this paper, the analysis of finite difference methods for describing the unsteady motion of a compressible inviscid fluid with emphasis for the modelling of the wave propagation phenomena is discussed.

Abstract: The paper concerns the analysis of finite difference methods for describing the unsteady motion of a compressible inviscid fluid with emphasis for the modelling of the wave propagation phenomena. The main questions addressed are how the multidimensional nature of the flow is retained by the numerical process and how the treatment of boundaries is affected by the choice of the numerical formulations.

••

[...]

TL;DR: In this article, the velocity components of the compressible flow equations are written in terms of a multiplicative composite consisting of a viscous or rotational velocity and an inviscid, irrotational, potential-like function.

Abstract: Solutions for transonic viscous and inviscid flows using a composite velocity procedure are presented. The velocity components of the compressible flow equations are written in terms of a multiplicative composite consisting of a viscous or rotational velocity and an inviscid, irrotational, potential-like function. This provides for an efficient solution procedure that is locally representative of both asymptotic inviscid and boundary layer theories. A modified conservative form of the axial momentum equation that is required to obtain rotational solutions in the inviscid region is presented and a combined conservation/nonconservation form is applied for evaluation of the reduced Navier-Stokes (RNS), Euler and potential equations. A variety of results is presented and the effects of the approximations on entropy production, shock capturing, and viscous interaction are discussed.

••

[...]

TL;DR: In this article, a collection of experiments in which the question of grid adaptation for flame propagation problems is investigated, and local refinement or mesh deformation procedures are applied to a sample of spatial approximations, including finite elements, finite differences and spectral methods.

Abstract: This paper presents a collection of experiments in which we investigate the question of grid adaption for flame propagation problems. Local refinement or mesh deformation procedures are applied to a sample of spatial approximations, including finite elements, finite differences and spectral methods. Comparisons are performed using the simplified thermo-diffusive model and a more realistic model involving compressible aerodynamics and combustion.

••

[...]

TL;DR: In this article, a complex three-dimensional shockwave/turbulent boundary layer interaction at Mach 4 has been investigated experimentally and computationally, using two turbulence models with substantial refinement.

Abstract: A complex three-dimensional shock-wave/turbulent boundary layer interaction at Mach 4 has been investigated experimentally and computationally, using two turbulence models with substantial refinement. With the use of a fine grid model, secondary flow separation was successfully computed, and grid changes did not improve agreement with experiment for the extent of upstream influence. The use of a non-isotropic turbulence model gave a slight improvement in upstream influence, but the size of the interaction was still significantly less than in the experiment.

••

[...]

TL;DR: A fast Euler solver for the computation of internal flows, based on the integration of the compatibility conditions along bicharacteristic lines, thus reducing multi-dimensional flow problems to a sequence of simple quasi one-dimensional problems.

Abstract: The present paper provides a fast Euler solver for the computation of internal flows. This methodology is a natural derivation of the classical lambda formulation and is based on the integration of the compatibility conditions along bicharacteristic lines, thus reducing multi-dimensional flow problems to a sequence of simple quasi one-dimensional problems. Starting from the time dependent Euler equations in vector form, the governing equations are derived for a general orthogonal coordinate system, for both two- and three-dimensional flows. The Compressible Over INcompressible (COIN) variant is also used, in order to improve the accuracy of results, and a shock fitting technique is applied to compute transonic flow cases. The resulting code is much simpler than other implicit codes developed for the lambda formulation, and provides a faster convergence to steady state conditions. The merits of the present approach are demonstrated by means of a few applications.

••

[...]

TL;DR: In this paper, a finite difference method based on boundary-fitted coordinates is combined with a Lagrangian description of the free surface to solve the nonlinear fluid-motion problem in a tank.

Abstract: A Finite-difference method based on boundary-fitted coordinates is combined with a Lagrangian description of the free surface to solve the nonlinear fluid-motion problem in a tank. Numerical results are obtained for the impulsive response of a tank and a forced heaving (vertical) motion of a body in a tank, for both linear problem. Lone-time solutions are found to have excellent stability characteristics and fine convergence properties.

••

[...]

TL;DR: In this article, the first stage in the development of a code for modelling the detonation stage of vapour explosions is described, and the equations governing this situation and the numerical scheme developed to solve them are presented.

Abstract: In this paper we report on a computer code developed to model the flow of a mixture of two gases. This is the first stage in the development of a code for modelling the detonation stage of vapour explosions. We describe the equations governing this situation and the numerical scheme developed to solve them. Calculations are presented for a steady-state shock, transient simulations of shock tubes (in one case containing different gases in each section) and detonations. We conclude that the numerical scheme presented in this paper is suitable for the simulation of compressible flows encountered in the detonation stage of vapour explosions.

••

[...]

TL;DR: In this paper, a method for the numerical solution of inviscid flows with discontinuities in quasi-one-dimensional unsteady problems is presented, which is based on the method of characteristics for the calculation of discontinuity points, boundary points and, when required, of grid points close to them.

Abstract: A method is presented for the numerical solution of inviscid flows with discontinuities in quasi-one-dimensional unsteady problems. The numerical technique, belonging to the family of the fitting techniques, is based on the method of characteristics for the calculation of discontinuity points, boundary points and, when required, of grid points close to them. The integration of the remaining grid points, enclosed between the discontinuities, is performed by a finite difference scheme following the “λ formulation”. The time step adopted for the numerical integration of the two different sets of points can be substantially decoupled. A simple and effective criterium for the unsteady shock detection is also proposed. Both the general philosophy and the details of the formulation of the method are illustrated and the efficiency of the numerical procedure is analyzed by means of suitable test cases involving shocks, contact and gradient discontinuities, and their interactions. Moreover, as applicative examples, the results of the simulation of complicated flow transients in convergent/divergent nozzles and in closed-end tubes are presented.

••

[...]

TL;DR: In this paper, an asymptotic analysis of the 2D counter rotating vorctices is presented, assuming small ratio between vortex core radius and the initial vortex spacing, which indicates that for the short time span the numerical solutin is of a fluctuating nature resulting from the distribution of the initial condition.

Abstract: An asymptotic analysis of the 2-D counter rotating vorctices is presented in this work, assuming small ratio between vortex core radius and the initial vortex spacing. This analysis indicates that for the short time span the numerical solutin is of a fluctuating nature resulting from the distribution of the initial condition. This short time period is proportional to the ratio between the initial vortex core area and the kinematic viscosity. In the limit case the asymptotic analysis yields a vorticity distribution that does not exhibit these fluctuations and can be used as a modified approximation of the numerical initial condition. In order to verify this conclusion, the sensitivity of the numerical problem is checked against the traditional initial condition (Oseen solution), in which a pseudo-spectral approach is chosen as a numerical algorithm. The pseudo-spectral method is shown to be an efficient means of handling the numerical boundary conditions and for validating the conclusions that have been obtained by the asymptotic analysis.

••

[...]

TL;DR: A new approach based on a generalized Euler's transformation is established and its efficiency is verified by computing the exact transonic flow over a smooth bump, which could be profitably used as a test for operational computer codes.

Abstract: The evaluation of the hypergeometric functions has always been a main problem of the computation of isentropic compressible flows by hodograph method, particularly because of the ill-conditioning of the involved series. In this work a new approach based on a generalized Euler's transformation is established and its efficiency is verified by computing the exact transonic flow over a smooth bump. This flow is also a new one and could be profitably used as a test for operational computer codes.

••

[...]

TL;DR: The two-equation turbulence models of Harlow and Nakayama and Spalding were used to predict the spreading rates of the radial free jet and the radial wall jet as mentioned in this paper.

Abstract: The two-equation turbulence models of Harlow and Nakayama ( κ - e ) and Spalding ( κ - W ) are used to predict the spreading rates of the radial free jet and the radial wall jet. Spalding's model appears to fare best for these flows, but both models tend to underestimate the measured spreading rates, especially for the wall-jet configuration. It is shown that improved predictions can be obtained by making the length-scale-determining equation much more sensitive to the generation rates of turbulence energy associated with the turbulent normal stresses. Calculations obtained with the modified scale equation are also presented for the plane and round free jet. The plane-jet growth rate is still predicted reasonably well, and the round-jet predictions show some improvement over those obtained with the unmodified turbulence models.

••

[...]

TL;DR: In this paper, the effects of shock vorticity on the supersonic, inviscid flow about circular, cross-sectional cones is considered, while the Euler equations are solved, so that no viscous effects are included.

Abstract: This work represents an application of Professor Moretti's computational procedures to an investigation of fluid physics. Specifically, the effects of shock vorticity on the supersonic, inviscid flow about circular, cross-sectional cones is considered. While the Euler equations are solved, so that no viscous effects are included, the shock vorticity can cause flow separation and vortex formation. In the computational results to be shown here the interaction of these vortices and shocks produces a number of interesting phenomena which will be discussed detail.

••

[...]

TL;DR: In this paper, the effect of various types of forcing on the vortical evolution of mixing layers was simulated using pseudospectral methods, and it was shown that combining harmonic and subharmonic forcing can both augment and later destroy streamwise vorticity.

Abstract: The vortical evolution of mixing layers subject to various types of forcing is numerically simulated using pseudospectral methods. The effect of harmonic forcing and random noise in the initial conditions is examined with some results compared to experimental data. Spanwise forcing is found to enhance streamwise vorticity in a nonlinear process leading to a slow, secondary growth of the shear layer. The effect of forcing on a chemical reaction is favorably compared with experimental data at low Reynolds numbers. Combining harmonic and subharmonic forcing is shown to both augment and later destroy streamwise vorticity.

••

[...]

TL;DR: A fast numerical solver of the Euler equations is presented, based on the Lambda formulation, and results for internal flows are presented, obtained with both orthogonal and non-orthogonal grids.

Abstract: A fast numerical solver of the Euler equations is presented. The method is based on the Lambda formulation and is an improved version of the scheme proposed by Moretti [1, 2], further revised by Moretti—Onofri [3] and Dadone—Moretti [4]. Results for internal flows are presented, obtained with both orthogonal and non-orthogonal grids. The solutions show second order accuracy and high rate of convergence.

••

[...]

TL;DR: In this paper, the authors developed an eigenvalue analysis for shock stability using the algebraic symbolic manipulator MACSYMA, which is then applied on the evolving shock contour to determine the stability limit on the PNS stepsize.

Abstract: Parabolized Navier-Stokes (PNS) solution techniques for high speed compressible flow involve marching in a “time like” spatial coordinate but have been observed in practice to be sensitive to flow conditions and require stabilization by numerical dissipation techniques. In this study we develop an eigenvalue analysis for shock stability. Using the algebraic symbolic manipulator MACSYMA, an explicit closed-form solution for the shock system eigenvalues is developed. This is then applied on the evolving shock contour to determine the stability limit on the PNS stepsize. Numerical experiments confirm the validity of the approach and demonstrate that the approach can be implemented explicitly to yield an adaptive stepsize algorithm and thereby a more robust PNS scheme.