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Thèse de doctorat/ PhD Thesis Citation APA:

Monnoyer, F. (1985). The effect of surface curvature on three-dimensional, laminar boundary layers (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/213617/1/d8116294-e46a-4e70-815a-a292684fb8e9.txt

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Exemplaire destiné à la Bibliothèque

Consultation autorisée Consultation rofuoéo Institut von Karman

de Dynamique des Fluides

Université Libre de Bruxelles Faculté des Sciences Appliquées

THE EFFECT OF SURFACE CURVATURE ON THREE-DIMENSIONAL. LAMINAR BOUNDARY LAYERS

F. FI onnoyer

I ngénieur civil mécanicien A. I.Lv.

Dissertation présentée en vue de l'obtention du grade de Docteur en Sciences Appliquées

NOVEMBRE 1985

Institut von Karman de Dynamique des Fluides

U.L.B.

BIBLIOTHEQUE DES SCIENCES ET TECHNIQUES

Université Libre de Bruxelles Faculté des Sciences Appliquées

THE EFFECT OF SURFACE CURVATURE ON THREE-DIMENSIONAL, LAMINAR BOUNDARY LAYERS

F. ^{M onnoyer}

I ngénieur civil mécanicien

A.I.Lv.

6io, 00^-3 M.

Dissertation présentée en vue de l'obtention du grade de Docteur en Sciences Appliquées

NOVEMBRE 1985

Institut von Karman

de Mécanique des Fluides

Université Libre de Bruxelles Faculté des Sciences Appliquées

Institut de Mécanique Appliquée

Il est classe traînée

Thèse Annexe:

possible d'améliorer les performances d'un planeur de standard au moyen de dispositifs de réduction de la induite.

F. Monnoyer

;

AVANT-PROPOS

A l'issue de ce travail, je tiens à remercier le Prof. J.J. Ginoux, directeur de l'Institut von Karman, qui a accepté de promouvoir cette thèse de doctorat et m'a permis de la mener à bien à l'IVK.

Je remercie également les Prof. J. Sandford et M. Carbonaro, qui m'ont accueilli dans le département d'aéronautique et m'ont aidé à accomplir ce travail dans les

meilleures conditions. J'adresse mes remerciements au Prof. J. Wendt, qui a consacré un temps précieux à la lecture et la correction d'un manuscrit difficile.

J'exprime ma profonde gratitude au Prof. Dr.

E. H. Hirschel, de Messerchmitt-Bblkow-Blohm, pour m'avoir orienté et conseillé de la manière la plus fructueuse depuis le début de mes recherches. Il m'a proposé le sujet de cette thèse, a suivi avec intérêt son développement et m'a prodigué de nombreux conseils avec toute sa compétence et toute son amabilité. Sans la compréhension dont il a fait preuve à mon égard, ce travail n'aurait pu être mené à bien.

J'adresse mes remerciements à Mr. J. van der Blieck, directeur du Nationaal Lucht- en Ruimtevaartlaboratorium, qui m'a donné l'autorisation d'utiliser le programme de calcul de couches limites tridimensionnelles développé par les chercheurs du NLR. Parmi ceux-ci, je remercie particu- lirement Mr. J. Lindhoüt, qui m'a chaleureusement encouragé

et m'a aidé de toute sa compétence.

Les nombreux échanges d'idées avec les doctorants de l'IVK ont également contribué à l'accomplissement de ce travail, ainsi que l'ambiance de travail qui fut toujours agréable.

Enfin, je suis reconnaissant à l'Institut pour l'Encouragement de la Recherche Scientifique dans l'Industrie et l'Agriculture pour le support financier qu'il m'a accordé durant la réalisation de cette thèse.

528393

ABSTRACT

The study of second-order three-dimensional compressible laminar boundary-layer theory is performed which includes the effects of surface curvature. In the theoretical analysis, the governing équations are determined with their domain of application, the boundary conditions are discussed, and a définition is given for the displacement thickness, valid to second order. This theory is applied to the quasi two-dimensiona1 flow around infinité swept elliptical cylinders. Calculations of boundary layers on prolate spheroids at incidence are also presented and compared with experimental and first-order results.

SOMMAIRE

L’étude des couches limites laminaires compressibles tridimensionnelles est réalisée de manière à tenir compte des effets de la courbure de paroi. Dans 1’ analyse théorique, les équations sont établies ainsi que leur domaine d’application. On définit également les conditions limites et les équations régissant l’épaisseur de déplacement dans le cadre de la théorie du deuxième ordre.

Cette théorie est appliquée aux écoulements quasi bidi­

mensionnels autour d’ailes infinies en flèche. La couche

limite se développant sur des ellipsoïdes de révolution à

incidence est également calculée et comparée aux résultats

expérimentaux et aux calculs de premier ordre.

TABLE OF CONTENTS

ABSTRACT

LIST OF FIGURES LIST OF SYMBOLS

CHARTER 1 INTRODUCTION

CHARTER 2 GENERAL BACKGROUND OF SECONO-OROER BOUNDARY-LAYER THEORY

CHARTER 3 COOROINATE SYSTEMS

3.1 NON-DIMENSIONAL QUANTITIES ... 3-2 3.2 SURFACE-ORIENTED LOCALLY MONOCLINIC COORDINATE

SYSTEM... 3-3 3.3 THE SHIFTERS...3-4 3.4 RHYSICAL COORDINATES ... 3-10

CHARTER 4 SECOND-ORDER BOUNDARY LAYER EQUATIONS

4.1 GENERAL EQUATIONS ... 4-1 4.2 THE ORDER-OF-MAGNITUOE ANALYSIS ... 4-4 4.3 CLASSIFICATION OF THE BOUNDARY-LAYER EQUATIONS . . 4-8 4.3.1 Zero-Order Theory ... 4-8 4.3.2 First-Order Theory ... 4-8

^.3.3 Second-Order Theory ... 4-10

A.4 SECOND-ORDER GOVERNING EQUATIONS... 4-10 4.5 BOUNOARY CONDITIONS... ... 4-14 4.6 BOUNDARY-LAYER PARAMETERS ... 4-19

4.6.1 Shear Stress... 4-19 4.6.2 Displacement Thicknesses ... 4-20

4.6.2.1 Flux of a Quantity q across an Envelope S . 4-21 4.6.2.2 Mass- , Momentum- and Energy-Flow

Displacement Thicknesses ... 4-25 4.6.2.3 Equivalent Inviscid Sources Distribution . . 4-26

CHAPTER 5 APPLICATION OF SECOND-ORDER THEORY

5.1 GENERAL REMARKS... 5-1 5.2 INCOMPRESSIBLE BOUNDARY LAYERS ON INFINITE SWEPT

WINGS... 5-3 5.2.1 Leading Edge-Oriented Coordinates ... 5-4 5.2.2 Boundary-Layer Equations ... 5-7 5.2.3 The Attachment-Line Equations ... 5-10 5.2.4 Finite-Différence Solution of the

Boundary-Layer Equations ... 5-12 5.2.4.1 General Procedure ... 5-12 5.2.4.2 Discretization in the Normal Direction . . . 5-14 5.2.4.3 Différence Scheme for Quasi Two-Dimensional

Equations...5-16 5.2.4.4 Linearization of the Discretized Equations . 5-19

5.2.5 Results... 5-19 5.2.5.1 The Infinité swept Elliptic Cylinder .... 5-20

5.2.5.2 Comparison between First- and Second-Order

Boundary Layer Solutions ... 5-21 5.2.5.3 Velocity Profiles...5-21 5.2.5.4 Intégral Parameters... 5-23

5.3 THREE-DIMENSIONAL BOUNDARY LAYERS ON PROLATE

SPHEROIDS AT INCIDENCE ... 5-26 5.3.1 The Calculation Method... 5-27 5.3.1.1 The Governing Equations... 5-27 5.3. 1.2 The Numerical Method...5-28 5.3.2 Extension of the Method to Second-Order

Calculations ... 5-30 5.3.3 Three-Dimensional Boundary Layers on Prolate

Spheroids at Incidence ... 5-34

5 . 4 CONCLUDING REMARKS 5-39

CHARTER 6 CONCLUSIONS

REFERENCES

APPENDIX A BASIC 6E0METRIC RELATIONS

A. 1 CONVENTIONS ... A-1 A.2 BASE VECTORS IN LOCALLY MONOCLINIC

SURFACE-ORIENTED COORDINATES ... A-2 A.3 TRANSFORMATION OF A VECTOR... A-5 A.4 THE METRIC TENSOR... A-6 A.4.1 Covariant Metric Tensor... A-6 A.4.2 Contra variant Metric Tensor... A-7 A.4.3 Applications of the Metric Tensor... A-7 A.5 THE CURVATURE TENSOR OF THE SURFACE... . A-9- A.6 COVARIANT DERIVATIVES ... A-10

APPENDIX B DEMONSTRATION OF EQUATION (4.73)

FIGURES

LIST OF FIGURES

1. Surface-oriented locally monoclinic coordinates.

2. Section through the boundary layer.

3. Two-dimensional coordinates.

4. Surface éléments.

5. Connection of the velocity profile with the external flow.

G. Skin friction vector.

7. Control volume.

8. Leading-edge oriented orthogonal coordinates.

9. Flow near the attachement line of an infinité swept wing.

10. Block diagram of the intégration procedure.

11. Grid with variable normal spacing.

12. Crank-Nicholson molécule.

1 3

13. Cut of the swept elliptic cylinder in the (x ) plane.

14. Curvature coordinate b^^ on elliptic profiles in orthogonal coordinates.

R

15. Second-order velocity profiles, f = 0.125, a=5*, (p = 40*, Re=10

1 6 . First-order velocity profiles, f = 0.125, a = 5*, ip = 40*.

17. Second-order velocity profiles, f = 0.5, a=5*, (p = 40*, Re=10^.

18. First-order velocity profiles, f = 0.5, a=5*, tp = 40*.

19. Influence of Reynolds number on second-order calculation, f=1, a=0* , 4> = 0*.

20. Influence of Reynolds number on second-order calculation, f = 0.5, a=0* , (p = 0*.

21. Effect of incidence, f = 0.125, ip=0*.

22. Effect of sweep angle, f=0.125, a=5*.

23. The three-dimensional calculation method (from [28]).

24. Surface-oriented coordinates on prolate spheroids.

25. Longitudinal wall shear stress distribution, f=1/6, a=0*.

26. Circumferential distribution of the magnitude and the

direction of the wall shear stress, f=1/6, a=10*, x^=o,446.

27. Circumferential wall pressure distribution, f=1/G, a=10*, Re = 0.8x10^.

28. Limiting streamlines, f=1/6, a=10*, Re=0.8x10®,

Limiting streamlines, f=1/4, a=10*, Re=10®.

29 .

LIST OF SYHBOLS

1 .

ref a

aP

c P c

dl Ec e . .

“1 f 9 g . -J

Latin letters

1

déterminant of covariant surface metric tensor reference speed of Sound

covariant base vector of surface tangential coordinate covariant base vector of surface-normal coordinate coordinate of covariant surface metric tensor

coordinate of surface contravariant surface metric tensor

coordinate of covariant surface curvature tensor coordinate of mixed-variant surface curvature tensor chord length

/g

skin-friction coefficient

spécifie beat at constant pressure spécifie beat at constant volume lengtb element

Eckert number

unit vector of Cartesian reference coordinate System ellipse tbickness ratio, or ellipsoid fineness

déterminant of covariant off-surface metric tensor covariant base vector of off-surface points

coordinate of general covariant metric tensor

g a(3

h K a k L L

M M M

ref

P a m n 0(q) Pr P R Re r r r S S T t

V V

coordinate of covariant off-surface metric tensor in locally monoclinic coordinates

coordinate of general contravariant metric tensor coordinate of contravariant off-surface metric tensor

in locally monoclinic coordinates viscous stress tensor

grid spacing in the wall-normal direction principal surface curvatures

magnitude of largest local surface curvature reference length

curve in a plane x = constant 3

unit vector tangent to the curve L

déterminant of shifter of the first kind tensor Mach number

shifter of the first kind

géométrie progression term (^1) unit vector normal to the curve L order of magnitude of q

Prandtl number static pressure

1 //Re

Reynolds number

general position vector surface position vector radius of révolution

envelope of the control volume V surface enclosed by L at x =0 3

température

flow direction for three-dimensional calculation (equa tion 5.44)

velocity magnitude V=ly.|

control volume for flux calculation

\i velocity vector

V a contravariant surface-tangential velocity coordinate V 3 contravariant surface-normal velocity coordinate

X i general coordinate

Cartesian reference coordinate

Z . Greek Letters

a a

13

Ax 6

a

2a 3 a P

e. . 13k

■»

0

angle of incidence

contour angle of the wing profile in orthogonal coordinates

coordinates of covariant base vectors Christoffel symbol of first kind

Christoffel symbol of second kind ratio of spécifie beats

stepsize in the x*^-direction boundary-layer thickness

mass-flow displacement thickness

momentum-flow displacement thickness in x'^-direction energy-flow displacement thickness

Kronecker tensor truncation error

coordinate of covariant permutation tensor

transformed normal coordinate for the three-dimensional calculation method (section 5.3)

angle between surface coordinate lines

implicit weighting factor in the différence approximation of the normal dérivatives thermal conductivity

déterminant of the shifter of the second kind tensor

shifter of the second kind A. |J

Q O

T

a 3

4»

<l>

q Q i

viscosity coefficients

angle between the surface x^-coordinate line and the local skin-friction line

density

coefficient for the normal coordinate transformation in the three-dimensional calculation method

contravariant wall shear-stress coordinate wing backsweep angle

dissipation fonction

flux of a quantity q across the envelope S contra variant coordinate of vorticity vector

3 . Indices

3 . 1 Uoper and Lower Indices

i.j.k... =1,2,3 dénotés general tensor quantities

=1,2 dénotés surface-parallel tensor quantities 3 dénotés surfa ce-normal tensor quantities

3.2 Uoper Indices

quantity in Cartesian reference coordinate System

* physical quantity

3.3 Lower Indices

e external inviscid flow

oe refers to the boundary-layer outer edge

ref reference quantity

s refers to the attachment-line

“ freestream conditions

3.4 Other Svmbols

overbar: dimensional quantity

quantity stretched with the square root of the Reynolds number

quantity expressed in surface-coordinates (shifted) dot superscript: quantity at a point on the body

surface

— underbar: vector quantity î Z at X = 2 3

lj covariant dérivative

■i partial dérivative with respect to x^

CHAPTER 1

INTRODUCTION

In high Reynolds number attached flows, viscous effects are significant only in the immédiate neighbourhood of the walls. The viscous part of the flow can then be studied with boundary-layer theory, which allows the simplification of the Navier-Stokes équations by neglecting terms that are of lower order of magnitude than the leading terms.

Boundary-layer theory has been widely used for many years since Prandtl first proposed it. Numerical two- and three-dimensional calculation procedures are of practical interest for the aerodynamicist since they very accurately predict the skin-friction drag, wall beat flux or wall température, and, to a certain degree, the flow séparation on real configurations. Also many aspects of the transition from laminar to turbulent flow are treated in the frame of boundary layer theory.

At présent, several numerical methods and computer programs are available for the calculation of three- dimensional laminar and turbulent, incompressible and compressible boundary layers. The so-called inverse methods are being developed to compute two-dimensional flows with

1-1

small séparation régions. Other research programs deal with combinée! inviscid-viscous solutions, where the boundary layer and the inviscid-flow région are computed separately as usual, but coupled by means of the displacement surface or the équivalent inviscid source distribution.

There are, however, cases where these methods fail to predict the flow that is observed in experiments. This situation happens when the assumptions of the classical boundary-layer theory are violated, so that the complété Navier-Stokes équations hâve to be solved. The recent development of large computers allows the numerical solution of those équations for particular flow cases, and there is hope that complex configurations at high Reynolds numbers can be treated in the future, even if such calculations require very large amounts of computational effort.

Furthermore, there are questions with regard to turbulence modelling which need to be answered.

The limitations of boundary-layer theory can be overcome to a cetrain extent by considering second-order effects. The basic goal is thus to extend the domain of application of boundary layer theory by retaining contributions that are neglected in the classical theory.

The first approach of this extended theory was developed some twenty years ago for two-dimensional and axisymmetric problems [49]. Although it was demonstrated that higher-order effects can be included into boundary- layer theory, there was no attempt to study the influence of second-order terms in the computation of real flows. Since then, various authors hâve tried both successfully and unsuccessfully to incorporate second-order effects into first-order boundary-layer calculations, but these approaches were neither rigorous nor general, for they did not include ail secondary effects. In fact, no general three-dimensional second-order method has been developed until the présent work, the major contribution of which is that the problem is globally defined, from the governing équations to the boundary conditions, including the

1-2

détermination of intégral parameters, the limita of validity, and applications to realistic three-dimensional configurations.

The présent analysis is restricted to laminar flows, in order to eliminate uncertainties in the turbulence modelling because it would be difficult to détermine whether the turbulent boundary layers are mostly influenced by second- order effects or by the choice of a spécifie turbulence model. In priciple, however, turbulent boundary layer theory can also be extended to second-order. It is also important to note that the équations are formulated for thermally idéal gases. This assumption is valid for most of the aeronautical applications, but care must be taken when high Mach number flows are considered.

The présent thesis research was largely influenced by the investigations carried out in the field during the last two décades. Chapter 2 présents a review of these works, with emphasis on the different methods used for deriving the second-order boundary-layer équations. Second-order correction is divided as usual into external flow interaction and curvature effects. and the physical interprétation of each of them is given.

Since surface-normal dérivative terms play an important rôle in boundary-layer équations, it is important to express these in a coordinate System where these terms can be easily recognized. The best suited coordinates are the surface- oriented locally monoclinic coordinates which are defined in Chapter 3, while the necessary basic tensorial relations and the surface metric characteristics are given in Appendix A.

In this chapter are also defined the so-called Shifters which are very helpful for understanding second-order metric properties, and also allow saving of computer time and

storage for the numerical solution of the équations.

In Chapter 4, the three-dimensional second-order

boundary-layer équations are derived from an order-of-

magnitude analysis, and the domain of applications of

second-order theory is discussed in terms of the surface curvature. The final set of équations includes ail curvature effects and dis placement effects can be accounted

for if the proper surface velocity is used. Furthermore, it is demonstrated that two-dimensional second-order équations previously derived can be considered as particular case of the présent general theory.

The boundary conditions that hâve to be imposed at the outer edge constitute an important part of this chapter and are discussed extensively, since they do not evolve in as a straightforward a manner as in classical boundary-layer theory. Finally, the problem of three-dimensional mass- and momentum-flow thicknesses is discussed and the displacement

thickness équations are derived from flux balances.

Applications of the theory are presented in Chapter 5.

The quasi-two-dimensional flow around infinité swept wings is first investigated, for it is easier to numerically solve than fully three-dimensional problems. A parametric study of second-order effects is performed on elliptical shapes, the curvature and curvature gradient of which dépend on the thickness ratio. As a second application, the boundary- layer on prolate spheroids at incidence is solved with a computer-program adapted to second-order from the NLR (*) code .

These results are compared to those obtained both experimentally and numerically from classical theory. It is shown when curvature terms play a significant rôle, and how it confirma the theoretical analysis of the previous chapters.

Final conclusions are drawn in Chapter 6, together with proposais for future work on the subject, based on the expérience gained during the présent research.

* Nationaal Lucht- en Riumtevaart Laboratorium, the Netherlands

1-4

CHAPTER 2

GENERAL BACKGROUND OF SECONO-ORDER BOUNDARY-LAYER THEORY

Second-order boundary-layer theory was first proposed by Kaplun [19] and studied extensively by Van Dyke [A9], [50]. Their approach is based on the fact that the boundary-layer équations can be regarded as an asymptotic expansion of the Navier-Stokes équations for large Reynolds numbers, Re. A perturbation scheme is established for this purpose in which Re is the perturbation parameter. In such a singular perturbation problem, the asymptotic expansion of the solution can be separated into an outer expansion for the external non-viscous flow and an inner expansion applying to the boundary-layer flow (see [43],

[50]). Both asymptotic expansions are simultaneously constructed and matched in their overlap région of common validity, as stated by Lagerstrôm [25].

The first term of the outer expansion yields the Euler équations, from which the potential flow équations are derived if the oncoming flow is assumed irrotational. In the second approximation, viscous terms are still not présent, but the boundary conditions account for the displacement effect. In the inner expansion, the first term is precisely the classical boundary-layer solution. The

2-1

continuation of the perturbation process up to second order brings out supplementary terms that represent secondary effects [49].

This second-order problem consista of linear differential équations and can therefore be subdivided into a number of simpler problems, each of which has a clear physical interprétation. It is customary to distinguish between the effect of the surface curvature and the interaction with the external flow, the former being further subdivided into transverse and longitudinal curvature terms.

Concerning the interaction with the external flow, Van Dyke [49] recognized the displacement and the external

vorticity effects.

Displacement effect is the easiest second-order effect to understand but the most expensive to incorporate in an inviscid flow/boundary-layer calculation procedure. As fluid is slowed down by viscosity near the wall it is forced outward so that the effective body is thickened by the displacement thickness of the boundary layer. The inviscid flow is consequently altered and can be corrected by recomputing it past the enlarged body. Another technique is to use the wall-normal velocity on the géométrie body as the boundary condition reflecting the viscous influence on the outer inviscid flow [27], [34], [37]. Both methods involve an itérative procedure in which the outer flow is modified by the newly computed displacement thickness, and therefore require at least twice the boundary-layer calculation on the entire body.

The external vorticity effect is discussed in detail in the review paper by Van Dyke [51]. It has not been considered in the présent thesis, the oncoming flow being assumed irrotational, as is usually the case in low speed flows. This secondary effect can become significant in supersonic flow, where the presence of shocks may generate vorticity in the inviscid outer flow.

2-2

Curvature terms account for the most significant second-order effect in several flow cases of practical interest, as in fuselage-like bodies, wing leading edges and configurations where locally the boundary-layer thickness is not small compared to the smallest radius of curvature of the surface. Centrifugal forces then produce a pressure gradient across the boundary layer, expressed by the wall- normal momentum équation. Moreover, curvature contributes other terms to the second-order équations and modifies the boundary conditions. This was misunderstood by some authors who erroneously included this normal pressure gradient without correspondingly extending to second order either the governing équations or the boundary conditions. Van Dyke [51] reports several investigations of the separate effects of longitudinal and transverse curvature for simple two-dimensional configurations, showing that longitudinal curvature decreases skin-friction, whereas transverse

curvature increases it.

An alternative method used for deriving the second- order boundary-layer governing équations is the well-known order-of-magnitude analysis which Prandtl first applied in the classical theory [^3]. In this method, the Navier- Stokes équations are first expressed in dimensionless form by referring ail flow variables to the corresponding free- stream quantities while ail linear dimensions are normalized by a characteristic length. The order-of-magnitude of each term is then estimated and those smaller than a prescribed value are dropped. In first-order theory, terms of order unity only are retained. Second-order équations are obtained by keeping terms up to order which is precisely the perturbation parameter defined in the previous approach.

Davis et al. made use of the order-of-magnitude method for the formulation of second-order compressible [7] and incompressible [8] two-dimensional and axisymmetric laminar boundary layers. Similarly, Grundmann [11], [12], [13]

applied these équations to particular flows where surface curvature was présent.

2-3

Three-dimensional second-order boundary layers were studied by Kux [23], who first introduced the tensorial formulation and the use of the shifters, as defined in the next chapter. The équations are derived from the order-of- magnitude analysis and the domain of validity of second- order theory is determined, based on the surface curvature and curvature gradients (see also [42]). However, no attempt has been made to solve the équations obtained by Kux, and the three-dimensional outer boundary conditions hâve not been defined.

In the présent work, the order-of-magnitude analysis has been chosen for the détermination of the governing second-order équations. The reason is that this approach requires less mathematical background and is simpler to formulate than the perturbation method, therefore staying doser to the physics of the phenomena. Moreover, it allows the classification of the équations following the level of the approximation of the full Navier-Stokes équations, as shown by Grundmann [11], whereas such a hierarchy is difficult to establish from the other method. The problem of the outer boundary conditions is however more difficult to solve. because it is not a conséquence of the matching between inner and outer expansions, as in the perturbation method. A method for determining these outer boundary conditions is presented Chapter 4, yielding boundary conditions similar to those obtained by van Dyke [49].

From the above review, it appears that although the problem of second-order boundary layers has been widely investigated for two-dimens iona1 and axisymmetric flows, three-dimensional flows hâve only been partially approached.

In the présent thesis, use is made of these previous theoretical studies on the subject, and a three-dimensional second-order theory is presented which is general, complété and well suited to practical applications.

2-4

CHAPTER 3

COOROINATE SYSTEMS

The boundary-layer équations hâve been formulated vwith regard to numerous coordinate Systems. Although these may appear to differ from each other, they ail exhibit the same property that one coordinate is rectilinear and normal to the surface over w/hich the boundary layer is flowing, the remaining coordinate lines lying on this surface. This common property evolves from the fact that the velocity and température gradients in the direction normal to the wall are very large within the boundary layer and hâve to be separated from those parallel to the surface. Another reason for this choice is the simpler formulation of the boundary conditions.

Following Hirschel and Kordulla [15], use is made in the présent work of a general locally monoclinic, non- orthogonal, curved, surface-oriented coordinate System that covers ail the particular boundary-layer coordinate Systems used so far.

3-1

3.1 NON-DIMENSIONAL QUANTITIES

As mentioned in Chapter 1. use is made in the présent

of the order-of-magnitude analysis for deriving the second-order governing équations. It is then necessary to express ail quantities in dimensionless form by relating them to certain characteristic values, bearing the subscript (ref), and chosen in such a way that the resulting non- dimensional quantities do not exceed a prescribed order of magnitude [43]. For convenience, this normalizing process is defined at this point, so that ail quantities appearing in the following sections will hâve to be regarded as nondimensional, unless written with an overbar.

The characteristic quantities are:

'^ref’ reference velocity, usually taken as the free- stream velocity, if it satisfies the condition that the corresponding non-dimensionalized velocity

components do not exceed order 1, which is abbreviated as v“=o(1).

2) ^j-ef* ^ characteristic length, such that the base vectors coordinates and the dérivatives v“ are 0(1) in the

» P

domain of interest. The non-dimensional boundary layer thickness 5=6/L will then be used as a basis

ref

of comparison for the order-of-magnitude évaluation.

3) Density, pressure and température are non- dimensionalized using as reference values 5

ref

^ref'^ref '^'ref respectively. and Cp^.^^ are taken for non-dimensionalizing thermal conductivity and spécifie beat at constant pressure. These reference quantities are chosen so that the corresponding physical non-dimensional quantities do not exceed 0(1), a condition usually satisfied when taking the freestream quantities as reference.

^) reference viscosity coefficient. Assuming that

3-2

Stokes hypothesis:

3X + 2ij = 0

is valid , we conclude that both viscosity coefficient M and X are of the same order of magnitude, and can therefore be non-dimensionalized by the same reference coefficient p (Y )

ref ref^

3.2 SURFACE-ORIENTED LOCALLY MONOCLINIC COORDINATE SYSTEM

The surface-oriented locally monoclinic coordinates are depicted in fig.1 where the Cartesian coordinates are represented since they are used throughout as reference coordinates. The two-dimensional subspace x=o defines the surface of the body, on which the x - and x -coordinate

lines are lying. These can make any angle with each other, but the

3 -coordinate line is rectilinear and has to be locally at right angles to both of them, hence the qualifier monoclinic, borrowed from the terminology of crystallo-

graphers.

This coordinate System can be dealt with most efficiently by using tensorial methods. Tensorial formulation of the équations is therefore used as derived from the équations of fluid mechanics for general coordinate Systems developed by Robert and Grundmann [Al]. The same tensorial concepts are used for determining the metric properties of the body surface.

The basic geometrical relations needed for the governing équations are presented in Appendix A. The reader will notice that transformations from and into Cartesian reference coordinates are very simple, and that surface- oriented locally monoclinic coordinates are easily constructed on most of the common shapes, provided these présent no corners and are convex. On concave surfaces, the X -coordinate lines are converging and may eventually cross 3

3-3

each other. A limit therefore will be defined for the validity on concave surfaces in the following section (équation (3.21)).

3.3 THE SHIFTERS

From a geometrical point of view, the boundary layer is similar to a shell as considered in the theory of elasticity [9], except that the boundary-layer thickness is not known in advance as in the case of the elastic shell thickness. The mathematical tools already developed for Shell theory can nevertheless be applied to boundary-layer theory. In this context. Kux [23] and later Robert [42]

proposed the use of the so-called shifters, defined and discussed in the présent section.

Consider in figure 2 a section of the boundary layer by a plane perpendicular to the body surface x =0. This boundary surface is known and we aim at expressing the metric quantities (base vectors, metric and curvature tensor

coordinates, Christoffel symbols) associated with the field

12 3 3

point P(x ,x ,x ) in terms of the wall-normal coordinate x and the metric quantities at the corresponding surface point

• 12 3

P<x ,x ,x =0). Adopting the nomenclature of Flügge [9], kernel letters g and a will designate base vectors and metric tensor coordinates referred to points off and on the surface, respectively. A dot superscript on any quantity dénotés a-a ssociation.

From fig.2, we write:

(3.1)

Differentiating (3.1) leads to:

X. (3.2)

3-4

and finally, with équations (A.3a), (A.28) and (A.33):

g = - a

X

3 (3.3)

where = 6^ - . (3.4)

a a a

Equation (3.4) defines the shifter tensor of the first kind

«Y , which relates covariant base vectors at the points P and P .

A similar relation is obtained for the contravariant base vectors :

gP =

P a^

If “ (3.5)

defining the shifter tensor of the second kind A'

From (3.3) and (3.5), we hâve:

. g = A^ a = A^

“ = A*^

- e y E —a "

e a a e

and therefore, with équation (A.4):

which explicitly yields [42]:

(3.6)

a

P = [6^ + x^(b^

a a ^{Ô^)]/M}

where M= det(

.

(3.7)

Equations (3.4) and (3.5) are used to calculate the shifter coordinates as a fonction of the known surface curvature tensor terms and the normal coordinate. The shifters are then used to express the metric at P in terms of corresponding quantities at P, as demonstrated below.

3-5

From the définition of the metric tensor (A.14) and with équations (3.2) and (3.5), it follows that:

(3.8a)

(3.8b)

The relations for the Christoffel symbols, given in reference [9], are:

The physical significance of the shifters is difficult to emphasize in general non-orthogonal coordinates but the simple case illustrated below shows clearly the meaning of the shifters of the first kind in the géométrie field.

Consider a two-dimensional curved domain, in which only the X -coordinate is curvilinear, the x -coordinate being 1 2

1 3

rectxlinear and orthogonal to the (x ,x ) plane (fig.3).

The metric then simplifies so that the only non-vanishing shifters are:

Similarly, the metric tensor at the surface reduces to:

(3.9a)

(3.9b)

3(3 ^(3.9c)

M 3 . 1 0 )

3-6

so that équation (3.8a) leads to:

1 1

=

1 1

(3.11)

From équation (A.20). the length of the surface line element ds. of fig.3 is:

ds= /T^ ^ dx’ . (3.12)

and the length of the line element d^ at a fixed distance from the surface and having the same component dx^ as d^:

ds= /g^ ^ dx^ = ^ dx’ . (3.13)

where équation (3.11) has been used.

From équations (3.12) and (3.13), we obtain:

ds=M^^ds , (3.1Aa)

and similarly:

ds=A^^ds . (3.14b)

Thus in the case of a two-dimensional geometry, 1 3

M^(x =h) is the ratio of the lengths of an element at a distance h from the surface to the corresponding element on the surface. is therefore the measure of the dilatation of the physxcal x -coordinate due to the surface curvature.

From this conclusion, it is clear that M^>i on the convex part of the wall, and M^<1 when the x^-coordinate line is concave, as shown in fig.3. In this case, the x - coordinate lines are converging and intersect at x =h', 3 where M^^ = o. A limit for the validity of the présent coordinate System is then given by:

b’ < 1/Ô . (3.15)

3-7

where 5 is the boundary-layer thickness and therefore the outer limit of the domain of interest.

A more general analysis can be performed when considering the déterminant M of the shifters of the first kind. Fig.4 illustrâtes the case of double convex and double concave surfaces, although the relations that follow are also valid for convex-concave surfaces.

The physical surface element dS defined by the éléments dx ,dx is demonstrated [15] to be: 1 2

dS= J~Â dx^ dx^ . (3.16)

where a = det(a ) a(3' •

The corresponding surface element dS at a distance x =h from the wall is:

dS= dx^ dx^= M J~Â dx^ dx^ , (3.17)

and therefore:

dS=MdS . (3.18)

Equation (3.18) is the general form of (3.14a), valid for orthogonal and non-orthogonal coordinates, and shows that M is the ratio of the areas of a surface element at a distance X =h to the corresponding surface element at the wall. With 3 équations (3.4), (A.34) and (A.35) the déterminant M is written, after some manipulation:

l<2)x^ + 1 , (3.19)

expressing M as a second-order polynomial in x^^ t^e roots of which are the inverse of the surface principal curvatures:

x^ = 1 /K^ , x^ = 1 /K^ (3.20)

3-8

For double convex surfaces (K

<0). équation (3.19) bas no positive root and M>1 for any x >o, confirming that dS>dS, as can be seen in fig.4a. For double concave surfaces (Ki.KgiO). however, dS'<dS' (fig.Ab), and M<1 away from the Wall. The déterminant M therefore has positive roots and reaches zéro at a certain distance from the surface, leading to an indeterminate situation. This can be avoided by satisfying the inequality:

max() < 1/6 . . (3.21)

insuring that M can neither be vanishing nor become négative within the boundary layer for any possible value of the

surface curvature.

This conclusion, deduced from physical considérations, could also be obtained from équation (3.7), according to which a unique détermination of the shifters of the second kind is possible only for non-vanishing values of the déterminant M.

Another way to understand shifters is to consider them as relating the components of a vector expressed in g-base vectors and its corresponding components in a-base vectors :

T- - - T

j; - r g = r ' a

-a (3.22)

With (3.3), the équation above becomes

a = a

a —f — (3.23)

g

Oot-multiplying each side by a.^ , we finally obtain

î-P = r“ , (3.24)

and the inverse transformation is similarly demonstrated to be :

(3.25)

3-9

The relations developed in the présent section can be inserted into any set of équations. Of ail metric quantities, only the shifters still dépend on the normal

coordinate x , and significant computational effort can be 3 saved by their use. Furthermore, a better understanding of the effects of curvature is insured by keeping in mind the fondamental significance of the shifters, as described a bove.

3.4 PHYSICAL COORDINATES

In non-Cartesian frames, the base vectors are not necessarily dimensionless unit vectors. For this reason, tensor components - including vectors - do generally not hâve correct physical dimensions.

Let y. be any vector expressed in terms of the covariant base vectors:

V = v“ g + v^ . (3.26)

-a ^3

Each of the terms in the sum of the right-hand side has the same dimensions as the vector y., but the dimensions of each of its components v^ dépend on that of the corresponding base vector g^.

If we define the physical components of y, by v , it is * i shown in [41] that:

* i i

V /g-

( 11 ) ( )

(3.27)

Similarly, second-order tensors, t. obey the transformation:

t J~q . . . t

(Il ) (33)

13

(3.28)

+ No sommation for indices between parenthèses (sect. A.1)

3-10

CHAPTER 4

SECOND-ORDER BOUNDARY LAYER EQUATIONS

The second-order, three-dimensional, compressible, laminar boundary layer équations are derived by the order- of-magnitude analysis of the terms of the general équations for fluid mechanics in locally monoclinic surface-oriented coordinates. The geometrical relations developed in the previous chapter and in Appendix A are used to emphasize the rôle played by the curvature terms that appear in the resulting équations and for determining their limit of validity.

The discussion of the boundary conditions and the définition of the mass- and momentum-flow displacement thicknesses conclude this chapter, defining a complété second-order theory applicable to computation of realistic flows.

4 . 1 GENERAL EQUATIONS

The fondamental balance and constitutive équations

given by Robert and Grundmann [41] are used as a starting

point. In the locally monoclinic surface-oriented frame

compressible, laminar defined in Chapter 3. the steady,

équations in dimensionless form are written:

. Continuity équation:

(ov“)| + (

^)^ = 0 . (4.

a , 3

X -Momentum équations a

0(v^v“|p + v^v“l3)= P 1/Re(H“^|p + , (4

X -Momentum équation:

0(v“v^| + v^v^_ ) =

ex , J

-P - + 1/Re(H3“| 4-

,3 a , 3

( 4

. Energy équation

OC (

“

+ v^T ) - Ec(v“p + v^p )

:Ec/Re «fr +1 / ( PrRe )

T ) ^ + (

) ] , (4 .a .P .3.3

where the viscous stress tensor is:

,,aP

aP

-y

3 , , P, fi-y a

H = Xg (v 1 + V ) + M(g V + g'^^v | ) , (4

I » I I

3 , a-y 3 , a ■ »

H =M(g v|^ + v|^) , ( 4

= A I + (2

+X)

^

"Y I 3 (4

2

3 )

4 )

. 5a )

. 5b )

. 5c )

4-2

and the dissipation fonction «fr :

(4.6)

The shifting rules of section 3.3 are then introduced into the general équations (4.1)-(4.4), together with the relations developed for covariant dérivatives [9], so that only the metric terms at the wall still appear, the dependence on the normal coordinate being contained in the shifters. This is a lengthy but rather straightforward procedure which will not be detailed here.

In the non-dimensional formulation above, the familiar characteristic parameters appear [43], [47]:

expression relating the thermodynamic quantities and with relations for the coefficients p, Â,

and c . At moderate Mach numbers, air may be considered to obey the équation of State for perfect gases:

(4.7a)

Prandtl number: Pr= M c

ref pref ref (4.7b)

Eckert number: Ec= /ir T

ref'^'^pref 'ref (4.7c)

Equations (4.1)-(4.4) are complemented with an

P

P = Q T/ (^ (4.8)

where 'i- c /c = est is the ratio of spécifie beats

P V

4-3

and is the reference Mach number . 3^.^^ being the reference speed of sound.

The epecific beat at constant pressure, c^, is as.umea constant, its variation being significant only for large température différences.

The viscosity coefficients are related to each other by Stokes hypothesis already mentioned in section 3.1 and following which the molecular dissipation phenomena are negligible. The viscosity coefficient p is taken to be independent from the pressure but its variation with température has to be accounted for. The semi-empirical Sutherland formula [43], C47] or simplified relations [28], [43] are used for this purpose.

Finally, the thermal conductivity K is related to the viscosity coefficient p and the spécifie beat c through the Prandtl number, which can be set constant in the flow conditions of interest.

4 . 2 THE ORDER-OF-MAGNITUDE ANALYSIS

Before submitting the équations to a term-by-term order-of-magnitude évaluation, it is necessary to define suitable parameters characterizing the surface curvature in order to analyze the relative importance of the corresponding terms and to dérivé a hierarchy for the boundary layer équations.

Kux [23] has defined two parameters, the first characterizing the wall curvature and the second being a measure of the curvature gradients tangentially to the surface. The order of magnitude of these quantities is evaluated independently, and several sets of équations are determined according to them. Robert [42], and later Hirschel and Kordulla [15], followed this approach, although they limited the number of possible combinations. In the présent work however, use is made of one parameter only.

4-4

namely the measure of the surface curvature:

k = max( II , II ) (4.9)

This choice is justifiée! by the fact that the considération of a surface curvature gradient that is evaluated independently from the curvature is inconsistent with the évaluation of the other dérivatives.

In the order-of-magnitude analysis , the dérivatives of a quantity with respect to the x^-coordinates are assumed of ô the same order as the quantity itself:

0(v“ ) = 0(v“) . (4.10)

I P P

since x =o(1). The velocity and pressure dérivatives must be related to the wall curvature and its gradients, a separate évaluation of which therefore leads to the need of independently considering velocity and velocity gradients as well. This is not the case in the work of Kux and it is questionable whether the numerous resulting sets of équations might still be considered as being of genuine boundary-layer type.

The same conclusion is drawn from the fact that the order of magnitude of a quantity is by définition such that it is not exceeded in the entire domain. Even if locally the curvature and its gradients are not of the same order, one has to consider the highest values they can reach on the entire surface. In this context , a small curvature with a large gradient must give rise to a large curvature at some location, and conversely, large curvatures must be associated with large gradients somewhere on the surface.

Therefore, a different order of magnitude of the curvature and the curvature gradients has to be considered as a local phenomenon which cannot be referred to in the general order- of-magnitude analysis.

4-5

The next important step is the détermination of an upper bound for the curvature parameter k. In section 2.1, the reference length L has been introduced as a non-

ref

dimensionalizing quantity for ail linear dimensions. It is such that the base vectors’ components are 0(1) in the

a

domain of interest, so that, with équation (A.15):

g^p= 0(1), (4.11a)

and at the wall:

a 0(1) . (4.11b)

ap

This together with équation (3.8a) leads to:

m

“= 0(1), (4.12)

P

and finally, with the définition (3.4) of the shifter of the first kind:

b“ ^ 0(1) (4.13)

If inequality (4.13) is not satisfied, the shifters are 0(5k)>0(1) and curvature effects are too large for boundary layer équations to be valid. This brings out the fact that, instead of condition (3.21), the inequality:

k 1 /5 (4.14)

must be satisfied, since x =0(5) within the boundary layer. 3 This last condition is more restrictive because it brings limitations to convex surfaces while the géométrie limit of validity (3.21) only restricted the use of locally monoclinic surface-oriented coordinates on concave surfaces.

It should be noted that the case k=1/5 in (4.14) is valid only for convex surfaces. Although the condition (4.14) will be applied in the following to ail kinds of surfaces, it has to be regarded as a strict inequality for

4-G

concave walls.

A conséquence of the upper limit for the curvature defined by (4,13) is seen from équation (3.24):

0(v“) = 0(M%^) = 0(v“) . (4.15)

denoting that vector coordinates hâve the same order of magnitude when expressed in terms of the general or the

surface base vectors.

The other quantities related to the wall curvature are the dérivatives of the shifters. Their order of magnitude is :

3 0t

"

,3 ' -'’

= •

A last remark is necessary concerning the normal velocity component, v^.

pointed out by Robert [42], its order of magnitude cannot be fixed beforehand. It is therefore inferred from the continuity équation and found to be 0(6), as in the classical theory [43], [47].

From the non-dimensionalizing process of section 3.1 and the above complementary analysis of the curvature, the terms of équations (4.1)-(4,4) can be compared to each other in order to define the governing boundary layer équations.

Since the typical quantities are the boundary-layer

-

1/2

thickness 5, which is 0(Re ) for laminar boundary layers, and the wall curvature parameter k, it follows that a hierarchy of the governing équations is determined from their comparison. Several combinations are possible, and a classification very similar to that of Hirschel and Kordulla [15] is presented here which covers the cases of practical interest.

4-7

4.3 CLASSIFICATION OF THE BOUNDARY-LAYER EQUATIONS

4.3.1 Zero-Order Theory

O(k) << 0(1)

For slightly curved surfaces, the surface curvature terms can be completely neglected , even in the surface metric. The problem then reduces to the flow on a fiat plate with a pressure gradient parallel to the wall. This représenta a good approximation for flows on wings [14],

[15], except in the leading edge région , where the curvature may reach values so large compared to the boundary

layer thickness that even first-order theory is not valid.

There are only a few cases where zero-order theory predicts the flow with sufficient accuracy, and it does not apply to fuselage-like bodies for which at least the transverse curvature has to be taken into account.

4.3.2 First-Order Theory

0(k) 4 0(1)

This corresponds to the classical boundary-layer theory, where it is assumed that:

0( 1 /k) >> 0(5)

that is, the local radius of curvature of the surface is very large compared to the boundary layer thickness. As a conséquence, the effect of dxlatation of the x -coordinate 3 lines observed in section 3.3 can be neglected in the boundary layer, and the metric quantities on any surface X =cst are approximated by those at the wall, hence the 3 dénomination "a-metric".

4-8

Since the dependence of the metric on x is neglected, 3 we Write :

g w a r « r

ap ' P'Y

where the dot superscript dénotés that the quantity is evaluated at the wall, and therefore, with (3.24):

extensively in the book by Hirschel and Kordulla [15], and applications to particular flow cases can be found in the literature, see e.g. [13], [16], [17], [39], [45]. When expressed in a cartesian coordinates System, these équations

can be easily seen to reduce to the familiar boundary layer équations of [43].

The tensorial formulation is avoided by most of the authors who still make use of the formulation in physical coordinates (see section 3.4). Blottner [3] considered three-dimensional boundary layer équations in surface- oriented orthogonal coordinates. Similar équations were applied by Cebeci et al. [6] to the flow around prolate spheroids at incidence. Schneider [44] considered the turbulent boundary layer équations on ellipsoids using external streamline coordinates. Lindhout et al. [28] made use of the physical formulation in general surface-oriented coordinates in their numerical calculation method.

V a

The general first-order équations are studied

^.3.3 Second-Order Theory

0(1) < 0(k) ^ 0(1/6)

If the curvature is not small compared to t)ie boundary layer thickness, the variation of the metric when going away from the wall is no longer negligible. Use is made of the so-called ”g-metric". where the dependence of the tensor and the vector components on the wall-normal coordinate is considered. Consequently , some terms in the Navier-Stokes équations that are neglected in the first-order theory must be retained, as discussed in the next section.

t, . 4 SECOND-ORDER GOVERNING EQUATIONS

The tedious but straightforward order-of-magnitude évaluation of the terms in the Navier-Stokes équations is carried out with the above second-order conditions and finally yields:

. Continuity équation:

(/aMpy*^) + /a(Mgv^) 0 (4.16a)

which can be further developed as:

0 (4.16b)

and confirms that v =0(6). 3

X -Momentum équations: a

-2A 3 =

(4.17)

4-10

X -Momentum équation

M® i_ "Y «

Vô^'' '' = - 1 /

. 3 (4.18)

Energy équation:

QC (v T +v T -Ec.(v P +v P.,)

P , a ,3 .a ,3

= Ec/Re uv“ +1/(PrRe)[KT ]

aPo^,3,3 ,3,3 (4.19)

The équation of state (4.8) and the expression relating the viscosity coefficient p to the température remain unchanged since they only contain scalar quantities, unaffected by the coordinate System used.

The naturel stretching of the wall-normal coordinate with the square root of the Reynolds number, applied to

laminar flow, is defined by:

K^= , (4.20)

= v^/Ri . (4.21)

Introducing this transformation into (4.16)-(4.19 ) , we obtain the "stretched" laminar boundary-layer équations:

. Continuity équation:

( /aMev” )

,a J~k ( M

) , 3

(4.22)

X -Momentum équations:

eCv*^pV^ +v'*~v^ +A^

5 +A^Apr5 - 2 R A^bf v *^v ^ ]

.0 .3 ^0 5-

50

• a.0 yô , a . 5 .6, y a ,

P.P "\^5^^'',3^ (4.23)

X -Momentum équation

R M°b, v^v“ = a 5^ -1/c P,3 ^(4.2*)

. Energy équation:

>w

ec (v“ï +v^T -) -Ec(v“p +v^p ~) = K

**5

Qt

J

= '■<\3',3 ■ '‘-2='

where R = 1//Te .

The stretching of the x -coordinate might hâve been 2 directly applied to the metric by defining a surface-normal base vector of magnitude:

1^3! = 193! = 1//Re

instead of a unit vector (see sect. A.2). The présent method is preferred since it séparâtes the geometrical properties of the surface and the physical properties of the laminar boundary layer.

It should be noted that the normal coordinate stretching has not been applied to the curvature tensor coordinates and the shifters appearing in équations (4.22)- (4.25). These are expressed in terms of the physical - coordinates because this formulation emphasizes the important property that the second-order stretched équations

4-12

are still dépendent on the Reynolds number, although this is not the case in first-order theory where the normal coordinate stretching éliminâtes the Reynolds number from

the governing équations.

The most évident change brought by the second-order considérations evolves from the x -momentum équation (4.18) 3 or (4.25), showing that the pressure cannot be assumed constant in the boundary layer. Integrating (4.18) yields a pressure increase of the order 5k across the boundary layer,

that can be neglected only in the frame of first-order theory.

The centrifugal term appearing in équation (4.18) cannot be included without simultaneously considering the other curvature effects, as was the case in some of the earliest higher-order investigations [51]. The terms underlined in the governing équations (4.16)-(4.19) are of the order 5k and therefore vanish for small curvatures. For this reason, they are referred to as the "second-order terms". However, it should be emphasized that the other terms in the équations are influenced by the wall curvature through the shifters and the décomposition of the velocity into general contra variant coordinates, the corresponding base vectors depending on the wall-normal coordinate and the curvature, see équation (3.3).

When the conditions of applicability of first-order theory are satisfied, the second-order équations above are easily demonstrated to reduce to the classical équations as formulated by Hirschel and Kordulla [15]. First-order theory can therefore be considered as a limiting case of the more general second-order theory, valid when the Reynolds number tends to large values for a given surface curvature, or when the curvature is decreasing for a fixed Re. This will be confirmed in Chapter 5 when considering the flow on

infinité swept elliptical cylinders.

4-1 3

For two-dimensional and axisymmetric flows, équations {4 , 1 S )-(4. 19 ) are similar to those obtained by Van Dyke [49], [51] from the perturbation theory, and to the

équations of Davis, Whitehead and Wornom [8] derived by the order-of-magnitude analysis.

The inadequacy of the curvature parameters chosen by Kux [23] was discussed in the previous section.

Consequently, the équations proposed by Kux do not compare with the présent équations, except for the combination corresponding to the case we identified as the actual second-order condition. Nevertheles s, Kux mentions the term

3 3 3

3V in the lefthand side of the x -momentum équation, although it is 0(5), and therefore negligible compared to the other terms.

Robert [42] excluded in his analysis the case k=0(1/5) and obtained three-dimensional équations where most of the spécifie second-order terms are missing, see also [15]. The X -momentum équation is however similar to équation (4.18), 3 in order to keep the typical second-order property of the non-vani shing normal pressure gradient.

From the comparisons with other works, it is concluded that the set of équations (4 . 1 6 )-(4 . 1 9 ) constitutes a suitable basis of second-order theory, applicable to three- dimensional configurations of practical interest.

Furthermore, the limits of validity of this theory are clearly established. As a last step, the boundary

conditions that hâve to be prescribed on both sides of the boundary layer are defined in the next section.

4 . 5 BOUNDARY CONDITIONS

At the Wall, the usual no-slip conditions apply:

V ( X a 1

(4.26)

4-14

v^(x\x^,x^ = 0) 3,1 2 , (4.27)

= V ( X , X )

w

where v is the surface suction velocity, which has to . . 3

satxsfy the condition v =0(5) in order to remain in the w

frame of boundary layer assumptions. Finally, the température profile is set by either an imposed wall

température or a prescribed first dérivative of the température in the wall-normal direction.

The outer boundary conditions are not so straightforward to obtain. The familier condition that the velocity at the boundary layer outer edge is equal to the inviscid velocity at the wall:

v(x=5)=v(x=0)=v (4.28)

e e

is valid in first-order theory, but it cannot be applied when considering second-order effects.

The outer boundary conditions are derived from the so- called matching principle, stating that the viscous profile merges smoothly with the inviscid flow field at the boundary layer outer edge. When second-order effects play a significant rôle in the viscous part of the flow, it is clear that they also must influence the external flow, and therefore affect the boundary conditions.

This phenomenon is clearly illustrated in the case of incompressible flows. The external flow then obeys Bernoulli's équation, which in non-dimensional form is:

'^e * ^ ’ (4.29)

3

where V^=

v'^v^

e e e

Differentiating (4.29) with respect to x^ leads to:

V V e e , 3 e,3

(4.30)

4-15