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## Über dieses Buch

The Third Symposium on Numerical and Physical Aspects of Aerodynamic Flows, like its immediate predecessor, was organized with emphasis on the calculation of flows relevant to aircraft, ships, and missiles. Fifty-five papers and 20 brief communications were presented at the Symposium, which was held at the California State University at Long Beach from 21 to 24 January 1985. A panel discussion was chaired by A. M. O. Smith and includeq state­ ments by T. T. Huang, C. E. lobe, l. Nielsen, and C. K. Forester on priorities for future research. The first lecture in memory of Professor Keith Stewartson was delivered by J. T. Stuart and is reproduced in this volume together with a selection of the papers presented at the Symposium. In Volume II of this series, papers were selected so as to provide a clear indication of the range of procedures available to represent two-dimensional flows, their physical foundation, and their predictive ability. In this volume, the emphasis is on three-dimensional flows with a section of five papers concerned with unsteady flows and a section of seven papers on three­ dimensional flows: The papers deal mainly with calculation methods and encompass subsonic and transonic, attached and separated flows. The selec­ tion has been made so as to fulfill the same purpose for three-dimensional flows as did Volume II for two-dimensional flows.

## Inhaltsverzeichnis

### 1. Calculation Methods for Aerodynamic Flows—a Review

Abstract
The present volume has been arranged so as to emphasize present abilities to calculate flows involving three independent variables and, at the same time to provide a background of the two-dimensional approaches upon which they are based. In the earlier review by Cebeci, Stewartson, and Whitelaw [1], two-dimensional flows were considered in some detail, and the corresponding volume included 22 papers which described related calculation methods and their achievements. Thus, the following section makes clear the background of advances which have been made since the Symposium of [1], and Sections 3 and 4 review the present status of unsteady three-dimensional methods. In each section new results are reported where they help to clarify the abilities of calculation methods.
Tuncer Cebeci, J. H. Whitelaw

### Chapter 2. Stewartson Memorial Lecture: Hydrodynamic Stability and Turbulent Transition

Abstract
Professor Keith Stewartson, who died on 7 May 1983, was a towering figure in applied mathematics and theoretical fluid mechanics. Indeed, elsewhere (The Times, London, 18 May 1983) I compared him in stature to three giants of mathematics and its applications: Sir George Stokes, Lord Kelvin, and Lord Rayleigh. And at a ceremony held at the University of East Anglia in 1979, when he was recipient of an Honorary D.Sc. degree, the public orator compared Stewartson to L. Euler (see [115]). Such is the esteem in which Keith Stewartson and his researches are held by his contemporaries.
J. T. Stuart

### Chapter 2. Transition Calculations in Three-Dimensional Flows

Abstract
The most advanced methods currently available for predicting transition are based upon boundary-layer stability theory. The main idea of that theory is that initially infinitesimal disturbances, internalized by the boundary layer, are selectively amplified by the viscous flow as they propagate downstream. The amplification of a disturbance depends upon its frequency and propagation direction.
R. Michel, E. Coustols, D. Arnal

### Chapter 3. Transitional Spot Formation Rate in Two-Dimensional Boundary Layers

Abstract
In two-dimensional constant pressure flows, the distribution of transitional intermittency is [226]
$$\gamma = \left\{ {\begin{array}{*{20}{c}} {1 - \exp \left[ { - \frac{{{{\left( {x - {x_t}} \right)}^2}n\sigma }}{U}} \right]ifx > {x_t}} \\ {0,ifx > {x_t}} \end{array}} \right\}$$
(1.1)
R. Narasimha, J. Dey

### Chapter 4. The Computation of Viscid-Inviscid Interaction on Airfoils with Separated Flow

Abstract
In this paper we are concerned with the computation of viscous subsonic and transonic flow over two-dimensional airfoils at high Reynolds numbers where the boundary layers are thin and turbulent over most of the airfoil and wake. Our objective in this work is to develop a fast viscid-inviscid-interaction method that can compute viscous flows over airfoils with extensive regions of flow separation and that can be used to predict the stalling characteristics of airfoils with reasonable accuracy.
R. E. Melnik, J. W. Brook

### Chapter 5. Laminar Separation Studied as an Airfoil Problem

Abstract
Recent work on windmills, swimming and flying propulsion of animals, and high-altitude cruising of miniature aircraft has aroused great interest in airfoils operating in the Reynolds-number (Re) range well below one million. This paper presents an aerodynamic theory of the laminar separation to further our understanding of the aerodynamic-flow behavior at Re = 104–105, that is, at the lower end of this flight regime.
H. K. Cheng, C. J. Lee

### 7. A Quasi-simultaneous Finite Difference Approach for Strongly Interacting Flow

Abstract
Viscous effects can have a substantial impact on the aerodynamic performance of internal and external flow configurations. For a significant number of flows of practical interest, the Reynolds number is sufficiently large for the flow field to be represented by interacting boundary layer theory (IBLT), whereby the flow is divided into viscous and in viscid flow regions with the two regions coupled through the viscous displacement thickness. The formal justification for the use of IBLT in the prediction of high Reynolds number strong interaction laminar flows is provided by triple deck theory [316], which is obtained from an asymptotic expansion of the Navier—Stokes equations. In IBLT the viscous region is represented by the Prandtl boundary layer equations; the inviscid flow can be represented in a number of different ways which depend on the flow configuration and the Mach number. It is clear from previous work (see [340] for a review) that many flows with separation can be solved accurately (neglecting the problems of modeling the turbulence and transition processes) through the use of IBLT. A critical step in the numerical solution of the governing equations in IBLT is the coupling mechanism which connects the viscous and inviscid flow equations. Several coupling algorithms currently exist for strongly interacting flows; however, the efficiency of these procedures, which are discussed below, generally decreases, along with the possibility of unstable behavior, as the size of the separation region increases.
David E. Edwards, James E. Carter

### Chapter 7. Newton Solution of Coupled Euler and Boundary-Layer Equations

Abstract
At present most methods for the numerical calculation of steady viscous-inviscid interactions iterate between an inviscid solver, which calculates the outer flow using either potential or time marching Euler methods, and a viscous solver, which calculates the boundary-layer flow using integral or finite-difference methods. In direct-coupling schemes the inviscid solver calculates the edge pressure gradient dp/, which is passed to the viscous solver, which then calculates the surface displacement thickness δ*, which is passed back to the inviscid solver. With adverse pressure gradients present, this iterative procedure either is very slow or fails outright. A faster and more robust method is the semiinverse coupling scheme, in which δ* is specified for both the inviscid and the viscous solvers as developed by Carter [13] and Wigton and Holt [343]. The resultant mismatch in the pressure gradients dp/, from the two solutions is used to update the specified values of δ*. The quasi-simultaneous technique of Veldman [358] is a more refined method where the viscous equations and a small-perturbation representation of the inviscid flow are solved simultaneously at each iteration. The convergence rate in all these schemes is limited by the accuracy of the δ* updating formula or the small-perturbation assumption, the latter becoming invalid in shocked flows.
M. Drela, M. Giles, W. T. Thompkins

### Chapter 8. Aerofoils at Low Reynolds Numbers—Prediction and Experiment

Abstract
Experimental data have been obtained for three widely different aerofoils. In historical order they are the Göttingen 797, NACA 643–418, and Wortmann FX63-137. Four modified versions of the Wortmann section have also been tested in which the undercamber was progressively removed. Force and pressure measurements have been taken to determine aerofoil performance, whilst flow visualisation has been used to detect transition and separation. The Reynolds numbers ranged from 3 × 105 to 1 × 106.
P. M. Render, J. L. Stollery, B. R. Williams

### Chapter 9. Comparison of Interactive and Navier—Stokes Calculations of Separating Boundary-Layer Flows

Abstract
In their review of calculation methods for flow on airfoils, Cebeci, Stewartson, and Whitelaw [1] discussed the relative merits of procedures which solve Reynolds-averaged two-dimensional forms of the Navier—Stokes equations and those which solve potential-flow and boundary-layer equations. They concluded that the latter method is more economical and, with interaction between the inviscid and viscous flows, is likely to provide more accurate results up to some limiting angle of attack at which the suction-side separation is so large that the boundary-layer assumptions result in greater inaccuracy than do the numerical assumptions required to solve the Navier—Stokes equations. This conclusion must be regarded as tentative, since interactive calculations, for example those of [375, 340, 369], show significant differences from measurements even in the absence of separation, and some solutions of the Navier—Stokes equations, for example [376, 377], suggest close agreement with measurements even for transonic flow.
D. Adair, B. E. Thompson, J. H. Whitelaw, B. R. Williams

### Chapter 10. Significance of the Thin-Layer Navier—Stokes Approximation

Abstract
A versatile technique has evolved for formulating the governing equations for inviscid and viscous compressible flows. The Navier—Stokes equations are first written in Cartesian coordinates in divergence or conservation form. With an arbitrary time-dependent transformation of coordinates, Peyret and Viviand [388] have obtained a general conservation form of the Navier—Stokes equations in the computational coordinates. This approach has also been used to solve the Euler equations. In many cases the Navier—Stokes equations are simplified by retaining only the viscous terms with derivatives in the coordinate direction normal to the body surface. This is the thin-layer Navier—Stokes-equation approximation, with the initial development of these equations given by Steger [389]. The thin-layer approximation along with additional assumptions is also used in the parabolized Navier—Stokes solution procedure.
F. G. Blottner

### Chapter 11. A Comparison of Interactive Boundary-Layer and Thin-Layer Navier—Stokes Procedures

Abstract
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.
Unmeel Mehta, K. C. Chang, Tuncer Cebeci

### Chapter 12. Development of a Navier—Stokes Analysis to Investigate the Mechanism of Shock-Wave—Boundary-Layer Interactions

Abstract
The flowfield resulting from the interaction of a shock wave and a boundary layer on a projectile or missile remains a major problem which has yet to be completely analyzed. Since the interaction causes an abrupt pressure rise and boundary-layer thickening and may be accompanied by regions of local separation, the interaction flow can be a major contributor to the overall body drag and can cause substantial changes in the pressure distribution from that expected from potential-flow considerations. Furthermore, since the interactions are often three-dimensional due to either geometric or flow-incidence effects, the generated forces may not be symmetric and may result in significant side forces. These side forces, in turn, can lead to serious control and guidance problems. Finally, the shock-wave—boundary-layer interaction zone may be a region of severe heat transfer. The problems associated with interactions are particularly troublesome at transonic speeds, where both the shock location and its shape are sensitive to changes in body geometry. As a consequence, in recent years there has been considerable interest in the development of accurate and efficient prediction techniques for shock-wave—boundary-layer interaction problems, in order to better predict the effects on performance of this phenomenon.
D. V. Roscoe, H. J. Gibeling, H. McDonald, S. J. Shamroth

### Chapter 13. Unsteady Airfoil Boundary Layers—Experiment and Computation

Abstract
Improved understanding of unsteady aerodynamics offers the potential to design more effective helicopter rotors, gas turbine engines, and aircraft control surfaces. Several aspects of a representative unsteady flow about a fixed airfoil have been studied at M.I.T. The work has concentrated on obtaining experimental measurements of boundary layer and wake velocities in attached flow and surface pressure distributions in both attached and separated flows. Computational studies have also been made of portions of these flows. This paper is concerned with describing some of the boundary layer results and comparing them with a numerical computation.
Peter F. Lorber, Eugene E. Covert

### Chapter 14. A Viscous—Inviscid Interaction Method for Computing Unsteady Transonic Separation

Abstract
In steady separated flows, computing capability at high Reynolds number is now developed either with direct solvers, usually simulating the Navier-Stokes (NS) and Thin-layer Navier-Stokes (TNS) equations [402, 431], or with indirect solvers, defined as Viscous-Inviscid interaction Solvers (VIS) [273, 432]. As compared to the direct NS-TNS methods, the indirect VIS methods differ simply in splitting the numerical treatment into viscous plus inviscid-like plus interaction solvers. This splitting is intended primarily to generalize the well-conditioned numerical methods designed for boundary layers, which are efficient and accurate at high Reynolds numbers, and to recover the boundary-layer technique as a means of numerical conditioning for solving the higher-order equations.
J. C. Le Balleur, P. Girodroux-Lavigne

### Chapter 15. Computations of Separated Subsonic and Transonic Flow about Airfoils in Unsteady Motion

Abstract
In studies of the unsteady aerodynamic and aeroelastic properties of wings, control surfaces, and rotating blades of helicopters, propellers, and com-pressors, a challenging problem of practical interest is the prediction of unsteady airloads at separated flow conditions. Until recently a theoretical prediction of these airloads was generally considered to be possible only by using methods based on the Navier-Stokes equations. From such methods various successful applications have been made to attached and separated unsteady transonic flows [443, 463-466]. However, because of the large computational effort required, these methods are not yet feasible for engineering purposes.
R. Houwink

### Chapter 16. Massive Separation and Dynamic Stall on a Cusped Trailing-Edge Airfoil

Abstract
In recent years, major steps have been taken towards a complete theoretical description of two-dimensional, laminar, incompressible high-Reynolds-number viscous flows. The asymptotic analysis used to develop these descriptions has shown that high-Reynolds-number flows may involve many complicated phenomena. Among these are: highly disparate length scales, nonuniqueness and hysteresis, and the possibility of spontaneously developing local, or free, viscous-inviscid interactions. These phenomena are usually associated with some form of separation.
A. P. Rothmayer, R. T. Davis

### Chapter 17. Analysis of Two-Dimensional Incompressible Flow Past Airfoils Using Unsteady Navier-Stokes Equations

Abstract
The flow over streamlined lifting airfoils has been a subject of considerable interest to fluid dynamicists, and to date, significant progress has been made towards the design of airfoils, wings, etc., by drawing together resources from experimental, numerical, analytical, and empirical studies. The detailed flow structure of airfoils and wings near maximum lift in low-to-high Reynolds-number (Re) flows still remains unresolved. The increasing interest in these flows stems from the desire for better control in civilian aircraft, and for high maneuvering capability in high-performance military aircraft. The improved performance can be realized from the potential of increasing maximum lift and simultaneously reducing drag under this condition. For some combination of flow parameters, the flow field around an airfoil experiences significant separation, which degrades its performance and leads to stall. The nature of the stall may be characterized by various phenomena such as separation, unsteadiness, transition, and turbulence. The present study is directed towards accurately simulating this flow field and providing further insight into this class of flows. Other important fluid-dynamics applications involving unsteady flows include blade rows in turbomachinery, marine propellers, helicopter rotor blades, and bluff bodies such as buildings, towers, underwater cables, etc., in cross flows. For this class of bluffy-body flows, understanding the vortex-shedding characteristics is very significant. The simulation technique presented here can also provide guidelines for analyzing some of these flow fields.
K. N. Ghia, G. A. Osswald, U. Ghia

### Chapter 18. Computation of Velocity and Pressure Variation Across Axisymmetric Thick Turbulent Stern Flows

Abstract
Many propellers and appendages are located inside of ship stern boundary layers. Therefore, it is essential for naval designers to obtain a fundamental understanding and accurate predictions of this special class of external thick turbulent stern flows. A series of experiments has been conducted at David W. Taylor Naval Ship R & D Center to determine the unique turbulence structure and viscous-inviscid interaction of thick axisymmetric [526–529] and simple three-dimensional [530–532] stern flows. The Lighthill [533] displacementbody concept has been proven experimentally to be an accurate approach for computing viscous-inviscid stern flow interaction. The measured static pressure distributions on the body and across the entire boundary layers were predicted by the displacement-body method to an accuracy within one percent of dynamic pressure.
Thomas T. Huang, Ming-Shun Chang

### Chapter 19. Inverse-Mode Solution of the Three-Dimensional Boundary-Layer Equations about a Shiplike Hull

Abstract
The development of prediction techniques for a flow field containing separated regions is of fundamental importance, since separation influences the performance of engineering devices such as wings, compressors, and inlets; it also concerns heat-transfer applications, as the location of separation greatly influences the values of heat-transfer coefficients. Prediction techniques for separated flows can also lead to a better understanding of trailing-edge flow phenomena and their influence on the configuration of flow past wings. Finally, they can be a valuable tool for the investigation of ship stern flows, which must be accurately computed if hull-propeller interactions are considered.
J. Piquet, M. Visonneau

### Chapter 20. Prediction of Dynamic Separation Characteristics Using a Time-Stepping Viscid-Inviscid Approach

Abstract
Flow separation on the lifting surfaces of a vehicle at high angle of attack is always complicated by a certain degree of unsteadiness, but when the vehicle itself is undergoing unsteady motion or deformation, or if it enters a different flow field rapidly, then the complexity of the separated flow is even greater, and culminates in the phenomenon of dynamic stall. If the angle of attack oscillates around the static stall angle, the fluid-dynamic forces and moments usually exhibit large amounts of hysteresis, and a condition of negative aerodynamic damping often develops during part of the cycle. This can lead to the condition of flutter in single-degree-of-freedom oscillating rigid-body motion. (Normally, in attached flow, flutter only occurs when the body motion includes multiple degrees of freedom—e.g., combined bending and torsion of an aircraft wing.) During a rapid increase in angle of attack, the static stall angle can be greatly exceeded, resulting in excursions in the dynamic force and moment values that are far greater than their static counterparts. The consequences of dynamic stall are far-reaching and lead to such problems as wing drop, yaw (sometimes leading to spin entry), wing rocking, and buffeting as well as stall flutter.

### Chapter 21. Computation of Turbulent Separated Flows over Wings

Abstract
Two cases of separated flow at high Reynolds number are considered, the first for a swept wing in the transonic range and the second for a sharp-nosed highly swept delta wing at a large angle of attack and low speed. Common to the two cases is the treatment of the boundary layer by a 3D integral boundary-layer-wake code in the direct mode. The viscous transport is turbulent in these cases and confined to a thin layer wetting the configuration surface and its downstream extension. The problem then reduces to the coupling of an equivalent inviscid problem and the boundary-layer-wake problem. In the former, sources are added along the configuration surface and the trailing wake surface to reflect the viscous displacement effects. The latter establishes the source strength in terms of the inviscid edge flow.
J. C. Wai, J. C. Baillie, H. Yoshihara

### Chapter 22. An Interactive Scheme for Three-Dimensional Transonic Flows

Abstract
For reliable calculation of the flow over aerodynamic bodies, the viscous boundary-layer solution must be allowed to influence the inviscid-flow solution. This is the basis for the recent emphasis on the iterative coupling of the inviscid- and viscous-flow equations for aerodynamic problems. However, for a prescribed pressure distribution, the boundary-layer equations tend to become singular as separation is approached. This, in turn, has led to the development of procedures for solving the boundary-layer equations in an inverse form.
Tuncer Cebeci, L. T. Chen, K. C. Chang

### Chapter 23. x-Marching Methods to Solve the Navier-Stokes Equations in Two- and Three-Dimensional Flows

Abstract
In many aerodynamic problems, the general direction of the main flow is a preferred direction: convection phenomena are important along it and dif-fusion is effective normal to it. This leads to simplified equations and to particular numerical procedures which can be viewed as extensions of boundary-layer calculation techniques, but which offer an extended range of applications [205,419,6201].
J. Cousteix, X. de Saint-Victor, R. Houdeville

### Chapter 24. Computation of Three-Dimensional Flows with Shock-Wave—Turbulent-Boundary-Layer Interaction

Abstract
Three-dimensional flows with shock-wave—turbulent-boundary-layer interaction represent a class of complex fluid-dynamics problems of practical significance. In many cases the understanding and prediction of these flow fields are essential to the successful design of high-speed aircraft or entry vehicles. Detailed computations of these flow fields are now feasible with the advent of large and fast vectorized computers and efficient numerical algorithms. However, before computational fluid dynamics can be used in the aerodynamic design process, the computer codes and their associated turbulence models must be evaluated for these complex flow fields.
C. C. Horstman, M. I. Kussoy, W. K. Lockman

### Backmatter

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