Tuncer Cebeci

Accurate numerical methods for boundary layer flows I. Two dimensional laminar flows

A very simple and accurate numerical scheme which is applicable t o quite general boundary layer flow problems has been devised. It has been tested extensively on laminar flows, turbulent flows (using the eddy viscosity and eddy conductivity formulations), wake flows and many other such problems. In the brief space alloted to us here we shall illustrate the method by showing its application in some detail to nonsimilar plane laminar incompressible boundary layers and in particular to the well known case of Howarths flow [3] .

Calculation of Separation Points in Incompressible Turbulent Flows

The purpose ohis paper ivaluate tccuracy with which tocation ourbulent separation c be predicted owo-dimensional ac bodies. Tvaluation wade btudying aonsiderable number of flows that had separation. Calculate d separation points were compared with the experimentally measured location. Fou r methods of predicting separation in turbulent flow were evaluated. The y were Goldschmied s method, Stratfords method, Heads method, and the Cebeci-Smit h method. I t wa s concluded from the study that the last three listed methods predict separation points with the reliability and accuracy needed for aerodynamic design purposes.

Three-dimensional laminar boundary layers and the ok of accessibility

Abstract : An investigation is carried out into the structure of the laminar boundary layer originating from the forward stagnation point of a prolate spheroid at incidence in a uniform stream, assuming that the external velocity distribution is given by attached potential theory. The principal new results of the study are: (1) A new transformation of the body coordinates is devised which facilitates the computation of the solution near the nose, (2) Two variations of the standard box method of solving the equations are devised to enable solutions to be computed in regions of cross-flow reversal, (3) Whereas in two dimensional flows the effect of the boundary layer approaching separation on the external flow may be represented by a blowing velocity, in the present study we find that this is only true near the windward line of symmetry, (4) The boundary layer over the whole of the spheroid cannot be computed in an integration from the forward stagnation point. (5) For alpha or = 15 deg the accessible region on the leeward side of the ok is largely determined by the external streamline through the ok.

Separation of three-dimensional laminar boundary layers on a prolate spheroid

The laminar flow around a prolate spheroid at 6° angle of attack has been determined by the numerical solution of steady, three-dimensional boundary-layer equations with the external-pressure distribution obtained from an analytic solution of the inviscid-flow equations. The flow is shown to comprise a region of positive crossflow, followed by a substantial region of negative crossflow, a separation line and two terminal lines beyond which solutions of the boundary-layer equations could not be obtained. The separation line defines one boundary of a region of open separation and accords with the argument of Lighthill in that separation of three-dimensional boundary-layer flows is defined by a skin-friction line. A procedure is described that permits the identification of this skin-friction line and requires that it passes through the first location at which the longitudinal component of the wall shear is zero and the circumferential component negative. The numerical tests show that the finite-difference scheme based on the characteristic box allows calculations against the circumferential flow and with an accuracy equal to that of the regular box provided that a stability criterion is used to choose the grid intervals. This stability criterion is shown to be essential for accurate solutions in the vicinity of the separation and terminal lines and implies the need for extremely fine grids. It is evident that similar numerical constraints will apply to calculations performed with an interactive boundary-layer procedure or with higher-order forms of the Navier-Stokes equations.

Further comparisons of interactive boundary-layer and thin-layer Navier-Stokes procedures

It is generally accepted that the Navier—Stokes equations correctly represent fluid-flow phenomena. Since the unsteady three-dimensional equations can generally be solved for flows where small-scale fluctuations are unimportant, emphasis has been placed on particular reduced forms such as those appropriate to regions of inviscid flow and boundary layers. In recent years, and with the application of numerical solution procedures in mind, attention has also been paid to the Reynolds-averaged Navier—Stokes equations and various further-reduced forms, including their so-called parabolized forms and the thin-layer Navier—Stokes (TLNS) equations.

Laminar Free‐Convection Heat Transfer from a Needle

The partial differential equations for laminar free convection over a needle are reduced to ordinary differential equations by a similarity analysis, and the values of local skin friction, heat transfer for various needles are obtained.

Studies of Numerical Methods for the Plane Navier-Stokes Equations.

Abstract : In the present report and in our most recent work we have concentrated on three basic formulations: primitive variables vorticity-stream function, and stream function-biharmonic formulation. With each such formulation there are a variety of difference equations that could be used and then there are a large number of iteration schemes that could be employed to solve these difference equations. The ultimate goal, of course, is to find the best combination of all these techniques.

An inverse boundary-layer method for compressible laminar and turbulent boundary layers

This paper presents an efficient two-point finite-difference method for solving the compressible laminar and turbulent boundary-layer equations for a given external velocity distribution (standard problem) as well as an efficient method for solving the same equations for a prescribed positive wall shear or displacement thickness (inverse problem). In the equations the Reynolds stress terms are modeled by using the eddy-diffusivity formulas developed by Cebeci and Smith. The accuracy of the method is investigated for both incompressible and compressible turbulent flows. A cf C f f g h H K L M P Pr Pr< Re u,v UT

Prediction of Post-Stall Flows on Airfoils

The calculation of the performance of airfoils requires the solution of the conservation equations, which, in general, can be accomplished in different ways. An important approach, where large scale computational facilities are available, is to solve the Reynolds-averaged Navier-Stokes equations or a reduced form such as the so-called parabolized and thin-layer Navier-Stokes equations. Two alternative and successful computational methods have been proposed by Maskew and Dvorak [126] and Gilmer and Bristow [79] in which an empirical inviscid flow model is used to represent the effects of flow separation with solutions of the boundary-layer equations in standard form up to the point of separation and a free surface assumption to model the separated flow region. The shape and the length of the separation are computed by satisfying a constant pressure boundary condition on the surface, and very good results have been obtained for airfoils over a wide range of angles of attack including stall and post-stall.

A general method for calculating three-dimensional laminar and turbulent boundary layers on ship hulls

Abstract : This report describes the progress made during the past year towards the development of a general method for computing three-dimensional incompressible laminar and turbulent boundary layers on ship hulls. The method employs an implicit two-point finite-difference method and an algebraic eddy-viscosity formulation. During the past year the efforts concentrated on the choice of an appropriate coordinate system; the calculation of the geometric parameters of this coordinate system; the numerical solution of the governing equation of the orthogonal curvilinear system; and obtaining preliminary results for simple ship forms. Further studies are in progress and will be reported in a forthcoming report.