Investigations of three-dimensional incompressible turbulent boundary layers

INVESTIGATION OF THREE-DIMEMSIOML INCOMPRESSIBLE

TUHBULEMT BOUNDARY LAYERS

PROEFSCHRIFT

ter verkrijging van de graad van doctor in

de technische wetenschappen aan de Technische

Hogeschool te Delft, op gezag van de rector

magnificus prof.dr.ir. H. van Bekkum voor

een commissie aangewezen door het college

van dekanen, te verdedigen op woensdag

28 april 1976 te I4.OO uur

door

BERMRD VAN DEN BERG

werktuigbouwkundig ingenieur

geboren te Tjiandjoer, Indonesië

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR ^

ACKNOWLEDGEMENTS

I wish to express my gratitude to the director of

the National Aerospace Laboratory NLR for the

opportunity to carry out the investigations

described here and for the permission to publish

the results in this form. I want to thank all

people who contributed in one way or another in

carrying out the experimental program. My colleague

Bram Elsenaar should be mentioned in particular in

this respect. The successful completion of the

measurements is for a large part due to him.

Finally I wish to mention the stimulating

co-opera-tion with Piet Wesseling and Jan Lindhout.

L i s t of symbols

₇

1 INTRODUCTION ' 9

2 GENERAL BEHAVIOUR OF BOUNDARY LAYERS

2.1 Two-dimensional boundary layers

2.1.1 Two-dimensional laminar 9

boundary layers

2.1.2 Two-dimensional turbulent 10

boundary layers

2.2 Three-dimensional boundary layers

2.2.1 Three-dimensional laminar I4

boundary layers

2.2.2 Three-dimensional turbulent I7

boundary layers

3 DERIVATION OF EXTENDED VERSIONS OF THE

LAW OP THE WALL

3.1 General comments 21

3.2 Two-dimensional law of the wall

3.2.1 Effect of pressure gradient 23

in two-dimensional flow

3.2.2 Effect of pressure gradient 25

and inertial forces in

two-dimensional flow

3.3 Three-dimensional law of the wall

3.3.1 Effect of pressure gradient 26

in three-dimensional flow

3.3.2 Effect of pressure gradient 30

and inertial forces in

three-dimensional flow

^ DESCRIPTION OF A THREE-DIMENSIONAL

BOUNDARY LAYER EXPERIMENT

4.1 Model design

4.1.1 Basis of design 32

4.1.2 Pilot model investigations 33

4.1.3 Description of the final model 37

4.2 Experimental methods

4.2.1 Velocity measurements 38

4.2.2 Wall shear stress measurements 40

4.2.3 Calibration of wall pitot tubes 44

4.3 Experimental results

4.3.1 External flow 47

4.3.2 Boundary layer flow 50

4.3.3 Velocity profile analysis 53

^

COMPARISON OP THEORY AND EXPERIMENT

5.1 The extended law of the wall 54

5.2 Boundary layer calculation methods 57

6 CONCLUDING REMARKS 6O

X REFERENCES 6I

SUMMARY IN DUTCH 65

SUMMARY

A theoretical study is made of the flow near

the wall in a three-dimensional turbulent boundary

layer. An extended law of the wall is derived,

which predicts the variation of the magnitude and

direction of the velocity close to the wall.

Fur-thermore an experimental investigation is

describ-ed. Measurements were performed in a

three-dimen-sional turbulent boundary layer, developing under

well defined conditions. The e:-.periment provides a

test case for three-dimensional turbulent boundary

layer calculation methods and actually has been

used for this purpose. The predictions of the

ex-tended law of the wall have been compared with

these experimental results.

-6-LIST OF SYMBOLS

Sym'bols which are used only locally are defined in the text, A B = b.a

C

"C'^C

h , etc 1' 2' ^ K , etc. X

i

m n P P' Ap Re , etc.

constant in law of the wall constant in law of the wall wall shear stress coefficient,

^f ^

\/i <

s t a t i c p r e s s u r e c o e f f i c i e n t , S ^ ^ P - P r e f ) / ! P"refj ^'f ^^ ^^^ ^^-periment C = (p-p-j^)/- pUe^ diameter of pipe f u n c t i o n , r e p r e s e n t i n g t h e v e l o c i t y d i s -t r i b u -t i o n near -the wall according -t o -t h e simple law of the w a l l , see e q , ( 2 , 7 ) f u n c t i o n s , defined by e q . ( 3 . 2 6 ) and (3.30) r e s p e c t i v e l y

height of wall p i t o t tube Lame c o e f f i c i e n t s

i n t e g r a l s , defined by e q , ( 3 . 9 ) ) (3.14) and (3.42) r e s p e c t i v e l y

Von Karman constant

geodesic c u r v a t u r e of the curve X = c o n s t a n t , e t c .

mixing l e n g t h exponent i n U oc x

surface c o - o r d i n a t e normal t o the flow d i r e c t i o n a t t h e boundary l a y e r edge, p a r t of t h e s t r e a m l i n e c o - o r d i n a t e system mean s t a t i c p r e s s u r e

f l u c t u a t i n g p a r t of s t a t i c p r e s s u r e pressure d i f f e r e n c e , more s p e c i f i c t h e pressure d i f f e r e n c e measured with a wall p i t o t tube

d i s t a n c e from pipe a x i s

Reynolds number based on ó, e t c . , Re, u , e t c . x ' u,- , e t c . ^x U U , e t c . X U ó//' e ' surface c o - o r d i n a t e i n t h e d i r e c t i o n of the flow a t t h e boundary l a y e r edge, p a r t of t h e s t r e a m l i n e c o - o r d i n a t e system f l u c t u a t i n g v e l o c i t y magnitude f l u c t u a t i n g v e l o c i t y component i n t h e d i r e c t i o n of t h e x - c o - o r d i n a t e , e t c . s k i n f r i c t i o n v e l o c i t y magnitude, s k i n f r i c t i o n v e l o c i t y component i n the d i r e c t i o n of the x - c o - o r d i n a t e , e t c . mean v e l o c i t y magnitude dimensionless v e l o c i t y , U = U/u-^

mean v e l o c i t y component in the d i r e c t i o n of the x - c o - o r d i n a t e , e t c .

surface c o - o r d i n a t e i n g e n e r a l

995

distance from the leading edge of the test plate, measured parallel to the tunnel axis (see e.g. i''ig. 24) distance from the surface

dimensionless wall distance, y = yu-r/w wall distance of the first grid point for the numerical boundary layer calcula-tions in chapter 5

surface co-ordinate in general

distance from a' lino on the test plate surface parallel to the tunnel axis (see e.g. Fig. 24)

pressure gradient parameter in two-dimen-sional flows, o ^ (i/pu^)(dp/dx)

parameter for the pressure gradient nor-mal to the wall shear stress direction, a^ = (r/pu^)(ap/öC)

parameter for the pressure gradient in wall shear-jStress direction,

a^ = (r/pu^)(3p/d5)

wall shear stress gradient parameter in two-dimensional flows, (3 •= (r/u"!^) (du^/dx) parameter for the gradient of the wall shear stress angle, p - (I'/u^) (ö'p /of)

parameter for the gradient of the wall shea-r stress magnitude,

P^ = (r/u2)(9uT/3fJ

measure for the boundary layer thickness in general

wall distance where U/u = 0.995

-7-nn 3 ns 3 sn 3 SS

\

displacement thickness in two-dimensional flows, defined by eq.(2.17)

crosswise displacement thickness, defined by eq.(2.38)

streamwise displacement thickness, defined by eq.(2.38)

surface co-ordinate normal to the local wall shear stress direction, part of a Cartesian co-ordinate system

angle between the pressure gradient vec-tor and the £-axis, positive clockwise momentum thickness in two-dimensional flows, defined by eq.(2.l8)

momentum thicknesses in three-dimensional flows, defined by oq. (2.38)

pressure drop coefficient for pipe flow angle of sweep

dynamic viscosity kinematic viscosity

surface co-ordinate in the direction of the local wall shear stress, part of a Cartesian co-ordinate system

pressure gradient parameter, ;;= (6/T^)(dp/dx)

density

shear stress magnitude

dimensionless shear stress, T = r/y ' w laminar shear stress, e.g. T« = jiOU /Sy turbulent shear stress,

e.g. -pu u X y

T , etc. shear stress component in the direction of the x-co-ordinate, etc,

(p angle between the velocity vector and the X-axis, positive clockwise Subscripts

e value at the edge of the boundary ]ayor w value at the wall

1, etc. value at measuring station 1, etc. (see Fig,32)

30 free stream value isymboll denotes absolute value

symbol denotes time mean value symbol denotes vector quantity X proportional to

1 IHTRODUCTIOK

In fluids of small viscosity the influence of viscosity is in general mainly felt in a thin layer close to the rigid boundaries: the boundary layer. In this thin layer the fluid velocity decreases rapidly from a certain outer value to zero at the wall. The flow in the boundary layer may be laminar or turbulent. In the latter case, which occurs most frequently in practice, the fluid motion consists of irregular fluctuations superimposed on a mean velocity field. These irregular fluctuations, which are called turbulence, occur due to the fact that the well-ordered, i.e. the laminar solution appears to be unstable for sufficiently small viscosity. Although the turbulent fluid motion is random,

statistically distinct average values can be dis-cerned. Little is understood as yet about turbu-lence, notwithstanding extensive research during nearly a century. The theory of turbulent flows consequently lacks solid foundations, so that ex-perimental investigations and physical intuition have to play an important role in this field of research,

The fluid motion in boundary layers is govern-ed by shear forces, in addition to the pressure forces. In laminar boundary layers the shear stress follows from the velocity field accordingly to a simple law due to Newton, In turbulent boundary layers an additional apparent shear stress occurs as a result of the turbulent motion. This "turbu-lent shear stress" is usually much larger than the laminar shear stress, so that it is essential for the calculation of the turbulent boundary layer development to know the magnitude of the turbulent shear stress. Various semi-empirical relations for the turbulent shear stress have been employed in turbulent boundary layer calculation methods. The experimental knowledge about two-dimensional tur-bulent boundary layers is quite extensive, so that the empirical assumptions used are reasonably well established. The state of the art is much less satisfactory with regard to three-dimensional boun-dary layers. As most bounboun-dary layers happen to be three-dimensional, more work in this area is badly needed from the practical point of view.

During the last decade a number of calculation methods for three-dimensional turbulent boundary layers has been presented in the literature. Such a method was also developed at the National Aero-space Laboratory NLR. In all these calculation methods existing empirical relations for the tur-bulent shear stress in two-dimensional boundary layers were generalized to three dimensions, for instance by assuming a direction for the turbulent shear stress in addition to the magnitude. Because of the small amount of experimental data available on three-dimensional turbulent boundary layers, these generalizations are far from well founded, so that the accuracy of the calculation methods is questionable. Therefore an experimental investiga-tion was started at NLR with the specific aim to provide a good check for three-dimensional turbu-lent boundary laye^' calculation methods. A special test set-up was designed to facilitate comparisons between theory and experiment. The experiment will be described in this report,

Special attention will be paid here to the flow close to the wall. This is an important region, since much of the velocity decrease to zero at the wall

oc-curs in the wall region. The velocity distribution in

that region is described by a so-called law of the wall. An extended law of the wall will be derived here, which describes the variation of the magnitude and direction of the velocity near the wall in a three-dimensional turbulent boundary layer. Use is made of this three-dimensional law of the wall in the three-dimensional turbulent boundary layer cal-culation method of NLR.

This report will start with a short review of the state of the art in chapter 2. Particular-ly section 2.1, which deals with the two-dimen-sional boundary layers, contains mainly material many readers may be familiar with. In chapter 3

the three-dimensional extended law of the wall will be derived. Chapter 4 describes the three-dimensional turbulent boundary layer experiment. A comparison between the measured velocity varia-tion close to the wall and the predicvaria-tions by the new three-dimensional law of the wall will be made in chapter 5. I" this chapter the experimental data will also be compared with a number of three-dimensional turbulent boundary layer calculation methods, including the one developed at NLR.

2 GENERAL BEHAVIOUR OP BOUNDARY LAYERS 2.1 Two-dimensional boundary layers

2,1,1 Tw£-dimensi_ona]^ l_aminar boundary_la;y£r£ The motion of incompressible viscous fluids is governed by the Navier-Stokes equations of mo-tion and the continuity equamo-tion. In two-dimensio-nal flows the continuity equation reads in a Cartesian co-ordinate system x,y:

du au

3x 9y (2.1)

where U-^ and Uy are the velocity components in the direction of the x- and y-co-ordinate respec-tively. We are considering here boundary layers, i.e. thin shear layers adjacent to walls, in which the velocity variation with distance from the wall is much larger than the velocity variation with distance parallel to the surface. In that case the Navier-Stokes equations in a steady flow simplify to Prandtl's boundary layer equation:

au

au

pU - r ^ + pU --^ X ax y ay dx

a^u

3y (2.2)

where x and y are the co-ordinates parallel and normal to the wall respectively. The symbols p and |i denote the density and dynamic viscosity of the fluid, Newton's law for the shear stress reduces in the boundary layer approximation to

^= ji(aU / 9 x ) , The pressure p may be taken constant across the boundary layer within the approximation considered here. In this approximation terms of order {^b/X) have been neglected, where 6 is the boundary layer thickness and X is a characteristic

length parallel to the wall. Thin boundary layers only exist in fluids of small viscosity. The rela-tive magnitude of the viscosity may be exp-essed in the Reynolds number Re„ = UX/i;, where U and X are a characteristic velocity and length and v = ^x/fx

For the boundary layer approximation to be valid, the Reynolds number must be large.

-The left-hand side of equation (2,2) contains the inertial terms. The right-hand side represents the effect of the pressure force and the viscous shear stress. IJhen the pressure gradient dp/dx is positive, the flow deceleration near the wall may become substantial. The shear stresses will help the flow near the wall to overcome the pressure increase. At the wall the pressure gradient must be balanced completely by the shear stress gradient

9r/öy. When the pressure gradient is sufficiently large, reverse flow will occur. Consequently the fluid originating from upstream has to leave the surface. This important phenomenon is called boun-dary layer separation. At the separation point the wall shear stress T^ = (i(dU^/ay) Q = 0, Downstream of the separation point the thickness of the viscous region increases rapidly and the viscous part of the flow ceases to be a thin shear layer, so that equa-tion (2.2) is not valid there.

When the pressure distribution p(.xj along the wall is given and also a velocity profile l^xi^J ^'^

an initial section, equation (2,l) and (2.2) can be solved, with the no-slip condition at the wall as boundary condition: y = 0 —»• U = U = 0, The velocity at the edge of the boundary layer and the pressure are related by:

pu,

dU e e dx d£ dx (2,3)

This follows directly from equation (2,2), when noting that outside the boundary layer the viscous term is negligible and that U <K U ,

Solutions of the equations have been obtained for several oases. The Blasius solution for a boun-dary layer along a flat plate at constant pressure is well known. The Blasius boundary layer belongs to the so-called similar boundary layers, i,e, boun-dary layers with velocity profiles that are similar at all streamwise positions and differ only by sea]e iactors for U^ and y. Similar boundary layers may also occur in cases with a pressure variation p(x) and a corresponding velocity variation U (x) in the outer flow. The equations defining the simi-lar solutions of equations (2,l) and (2.2) were derived by Falkner and Skan m 1930 (Ref.l), Numerical solutions were given later by Hartree

(Ref,2). It appears that similar solutions may be obtained when the outer flow velocity varies as Ug ex X™, Some typical velocity profiles for a number of values of m are given in figure 1. The case m = 1 represents a boundary layer in a large negative pressure gradient, as occurs in the neigh-bourhood of a stagnation point on a two-dimensional body. The Blasius solution is obtained for m = 0. Negative m-values correspond to positive pressure gradients. Boundary layer separation occurs for m = -0.09.

The solutions just discussed are exact solu-tions, Various methods exist to obtain approximate solutions of the laminar boundary layer equations (e.g. Ref, 3, 4 ) . After fast computers have become available, numerical procedures, using finite dif-ference methods for instance, have been used to obtain accurate solutions of the equations (see e.g. Ref.5). Generally speaking the calculation of two-dimensional laminar boundary layers does not constitute a real problem at present.

Laminar boundary layers appear to occur only at rather small Reynolds numbers. For Reynolds num-bers exceeding Re^ = order (lO^) the laminar

solu-1.0 995 0.8 0.6 0.4 0.2 m = - C

/

/

r

SEPARATIO m = - 0 . 0 9

/

/"

\j p p 0 F 11 F

/ /

i

l\

~y/j

/

/

/

/

/ STAGNATION F L A T P L A T E m = 0 1 Fi nw

Figure 1;

Typical velocity profiles for two-dimensional

similar laminai' boundary layer,s.

tion can be shown to be unstable. The exact Reynolds number, at which the laminar boundary layer becomes unstable, depends on the pressure gradient. In practice the boundary layer becomes turbulent some distance downstream of the instability point. Tur-bulent boundary layers will be discussed in the next section.

2.1.2 Tw£-d_imensi_onal^ t_urbule_nt_ bound_ary_lay£rs^ In turbulent flows the fluid velocity can be divided into a mean velocity, U^, U^, and a random fluctuating velocity, u^j, u , u^, and the pressure in a mean pressure, p, and fluctuating part, p'. We will consider here flows, which are stationary in the mean, hence the time mean_valu£S of_the fluctuating parts will be zero: u-^ ^ Uy = u^ = p' ^ = 0. The resultant momentary velocity, (Q^+u^) etc.,

-10-and pressure, (p+p'J, can be substituted in the unsteady Navier-Stokes equations of motion and in the continuity equation. Time averaging of latter equation leads to the conclusion that the mean velocity and the fluctuating velocity must fulfil the continuity equation. The equation of moan motion for turbilent boundary layers is found

to become after averaging (see e.g. Ref.7, 9 ) :

HU

au

X X dx + pU

au

X y 3y d£ dx

au

u

X .y

a^u

sy

(2.4)

As usual, a term a/ax(u'^-u2) has been neglect-ed in the above formula. This Is permissible with-in the boundary layer approximation swith-ince the term has the order (6/X), but it should be noted that this is one order of magnitude larger than al"! other neglected terms Uti the boundary layer equa-tions.

Equation (2.4) shows that a "turbulent shear stress" t.(- = -pUjj-Uy is created by the fluctuating motion of the fluid. Generally the turbulent shear

stress is much larger than the viscous shear stress,

'^l = n6U^/ay, exc3pt in the so-called viscous sub-layer, which is a very thin region adjacent to the wall, where the turbulent motions are suppressed. Since the resistance to fluid deformation is much smaller in the viscous sublayer than in the turbu-lent part, very sabstantial velocity increments with distance from the wall occur in the viscous sublayer. Consequently turbulent boundary layer velocity profiles are characterized by a large velocity increase near the wall. This is illustrat-ed in figure 2, where some typical velocity pro-files in turbulent boundary layers have been plotted. The velocity profiles for an outer flow velocity variation Ug cc x™ are shown and may be compared with the similar solutions for laminar boundary layers, plotted in figure 1. For turbu-lent boundary layers exact similarity is not pos-sible in general, but nearly similar velocity profiles may occur for an outer flow velocity varying as Ug oc x'". These nearly similar turbulent boundary layers are commonly called equilibrium or self-preserving boundary layers. The precise defin-ition of self-presei'vation will be given later in this section.

At the outer edge of the boundary layer a sharp boundary between turbulent and non-turbulent fluid exists, as illustrated in figure 3. The shape of this boundary is very irregular and changes continuously with time. The shape is determined to a large extent by the motion of the large eddies ill the turbulent part of the layer. The amount of turbulent fluid is increasing in downstream direc-tion by a process, which is called entrainment, I'.ost production of turbulence occurs near the walls

,;ust outside the viscous sublayer. The processes there have been found to be roughly periodic, Actually the turbulence production at the edge of

Ine sublayer occurs distinctly in bursts. It seems that many processes in turbulent boundary layers have a highly intermittent character (see e.g. Ref.99).

Unlike the laminar shear stress, the turbulent shear stress -pu^^Uy is not simply related to the mean velocity field. An equation for the turbulent

snear stress can be obtained from the Navier-Stokes equations after some suitable manipulations (see

0.4 1

\y

SEPARATION P m = -0.23

/

/ F L A T P / FLOW, r

^

/

/

T l = - 0 . 2 1 ^ LATE /

n=0

/ \ y

]

J

/I

/

/

/

/ / /

/ / /

/

/

/ 1

^ ^ m . 1

F'igure 2 : Typical velocity profiles for two-diraensii s e l f - p r e s e r v i n g turbulent boundary layers ( from -'' "• " " ' " ^ ^

for two-dimensional s e l f - p r e s e r v i n g turbulent bo

ref. 6 , Re ^ * = 1()4 ).

R e f , 7 ) . The equation for pu^-Uy r e a d s :

a(-u u ) pu _{ax "^ y} a(-u u ) ay x - y pu.

au

au X / X au y 3y 3y 9y (p+pu T - | i ( u V' X u +u y y

F ^

-+ ^ - ^ ) -+ 3x ( 2 . 5 ) S i m i l a r e q u a t i o n s may be w r i t t e n for t h e o t h e r com-ponents of t h e t u r b u l e n t s t r e s s t e n s o r . These equa-t i o n s do noequa-t l e a d , however, equa-t o a knowledge of equa-t h e magnitude of the t u r b u l e n t s t r e s s e s , since t h e e q u a t i o n s contain new unknowns: the t r i p l e products

-MOMENTARY POSITION OF

T U R B U L E N T - N O N - T U R B U L E N T INTERFACE

'«wms.^^^v^vvmv^vvvvv^^iw^^^^

ss^

Figure 3 : Sketch of turbulent boundary laver.

of the turbulent fluctuations. In principle equa-tions can be derived for the triple products of the turbulent fluctuations, which, however, con-tain quadruple products, etc. It appears that by averaging the Navier-Stokes equations so much in-formation is lost, that a theoretical determination of the turbulent shear stress is not possible along these lines. The equations for the turbulent shear stresses can be helpful to a certain extent in un-derstanding the turbulent processes. The best known turbulence equation is that for the total turbulent kinetic energy per unit mass

— n ^ — (u + u + u ). This equation reads (see 2 2 X y z R e f , 7 , 9 ) :

a(i ;?) a(i ^)

pu —I +pu —4, =

X ax y 9y

au

-pu u -— X y 3y

a_

9y , ( p + _pu + u ( u r u + u r u + u r ' X X y y z ( 2 , 6 ) An inspection of equation (2,6) learns that the

left-hand side of the equation represents the trans-port of turbulent energy by the mean flow (adveotion), The first term at the right can be interpreted as the work done by the mean flow against the turbulent shear stress. Therefore it is generally called the production term. The second term at the right can be understood as a diffusion term. The last term on the right may be shown to comprise mainly the dis-sipation of turbulent energy into heat by viscosi-ty, Although ultimately the dissipation of turbu-lence occurs by viscous forces, the rate of dissi-pation has been found to be independent of viscosi-ty, This may be explained by assuming the energy transfer between eddies to be a cascade process

(Ref,-7). The larger eddies, which contain most of the energy, exchange energy only with smaller eddies of a neighbouring size, which exchange ener-gy with still smaller eddies and thus finally with eddies which are so small that dissipation by vis-cous forces occurs. The fact that viscosity does not play a role in the rate of dissipation can be explained now, by assuming that the rate of energy transfer is determined by the first stages along this chain of processes,

An important feature of turbulence is the es-sential role that large eddies appear to play in the processes. It was just mentioned that the dis-sipation rate is controlled by the larger eddies. The shape of the turbulent - non-turbulent inter-face at the boundary layer edge is determined in

the main by the large eddies, so that the rate of entrainment of non-turbulent fluid will depend on these large eddy motions. It is likely that also the diffusion of turbulence is caused at least partly by transport of turbulence by large eddies. The fact that large eddies play such an important role, makes it very difficult to generalize tur-bulence properties, since the properties of the large eddies are not determined by local flow conditions, such as the velocity gradient, but they will depend on the characteristics of an extensive part of the flow and these characteristics will differ from case to case.

We will now discuss some generally accepted properties of the velocity profiles in turbulent boundary layers. First attention will be focussed on the velocity distribution close to the wall, which is described by the so-called law of the wall. In a thin region adjacent to the wall, the total shear stress (ti + T.) often may be assumed approximately constant and equal to the wall shear stress T . Then the fluid velocity near the wall

w

is completely determined by the magnitude of the wall shear stress or skin friction, T" , the dis-tance from the wall, y, and the fluid properties, p and I' , It is convenient to introduce the skin friction velocity, which is defined as u_ = ('^w/p) • Dimensional reasoning leads to the conclusion, that we may write:

U f(y^) (2.7)

where U = U /LL, and y = yu^/r , When the turbulent stresses are negligible, i,e, well within the viscous sublayer, the wall shear stress f,^ = = ndU^/ay. Consequently in this region the velocity distribution will be:

U (2.8)

Outside the viscous sublayer the direct effect of

viscosity i s negligible. Consequently the velocity

increase outside the sublayer depends only on u ,

y and p. Dimensional arguments then show that

3Ux/3y = iv/'^yi where k i s a constant. Integration

gives:

U l { l n y ^ + Al (2.9)

where A is an integration constant. The existence of a region near the wall with a logarithmic veloc-ity distribution has been confirmed experimentally numerous times. It appears from experiments that k 5: 0.4 and A 5; 2. The constant k is generally called the von Karman constant. Close to the wall, say for 5 < y""" < 30, a gradual transition occurs from the velocity distribution given by equation (2.8) to that by equation (2.9).

In a similar way some statements can be made about the velocity distribution in the outer part of a turbulent boundary layer, now excluding the part close to the wall. The pressure gradient has a large effect on the velocity aistribution in the outer part of the boundary layer. The severity of the pressure gradient is appropriately expressed by the parameter:

/ / = ^ § ^ (2.10) T djc ^ '

w

•12-This parameter represents the ratio of the pressure forces to that of the shear forces. Boundary layers for which // = constant, are called self-preserving boundary layers. It can be demonstrated (see e.g. Ref, 6) that the requirement // = constant can be met approximately if the outer flow velocity varies as Ugocx™, Boundary layer velocity profiles for such flows were shown in figure 2,

It has been found that when viewing the velo-city relative to the velovelo-city at the edge of the boundary layer, i,e, when considering the velocity defect (Ug-U^), the /elocity distribution in self-preserving boundary layers nearly only depends on the wall shear stress and depends very little on the actual magnitude of the velocities relative to the wall. This so-called velocity defect law may be written:

U -U e X

ï(yA,;; (2.11)

It appears that generally there is a region in the boundary layer where equation (2.7) and (2.1l) are both valid. It is easy to prove that in this case the velocity disti'ibution must be logarithmic in the overlap region, which is another possibility to derive equation (2.9).

Equation (2.1l) only holds in self-preserving boundary layers. But also the velocity profiles of non self-preserving boundary layers are usually found to collaps with one of the members of the profile family of the defect law, when an appro-priate value for II is chosen and not the actual value. Equation (2,7) and (2.1l) may be regarded as constituting a two parameter velocity profile family, with the parameter Uj./Ug and a profile shape parameter replacing the pressure gradient parai,ieter//. The profile family of Coles (Ref. 8) is based on the law of the wall and the velocity defect law and has proved to be very valuable in practice. It may be written:

U U T

_ x _ £ . i ( l n / . A ) . C s i

e ' n ( j ' ^ y A )

(2.12)

where C is to be considered the profile shape pa-rameter, It is easy to see that for small y/ó equation (2.12) reduces to the law of the wall as given by equation (2.9). To derive the velocity defect law, we must first establish U /u_, which

' e t may be done by substituting y = 6 m equation

(2.12):

— = i ln(—^) + A + C

u,- k ^ 1- (2.13)

Substracting equation (2.12) from (2.13) gives the velocity defect:

U -U

C I (è-'yA) - è in(y/ö) (2,14)

Equation (2.13) may also be written

+ C (2,15)

This equation actually relates the skin friction U T I ÖU u,^

-^ = i ln(-^) + In(rf) + A

u- k ^ 1- ' ^U '

' e

velocity ratio VL./U„, or the wall shear stress coefficient C^. = TJ ^ pu| = 2(u.j./Ug)2, to a Rey-nolds number and a profile shape parameter. Such a relation is called a skin friction law.

A large number of calculation methods for two-dimensional turbulent boundary layers exists, Several calculation methods use the boundary layer momentum-integral equation, which is obtained by integration of equation (2,4) across the boundary layer thickness. With some algebra (see e.g. Ref. ll)

one finds, using the continuity equation: dO ( 2 0 + 6") '^"e '[w_

dx U dx ,,2 e pU

e

(2.16)

where 6 and 0 are the boundary layer displacement thickness and momentum thickness, which are defin-ed by: U ƒ (1 - ^ ) dy 0 6 U U / # (1 - # ) dy J e e (2,17) (2.18)

The ratio of the magnitude of the displacement thickness 6 to that of the momentum thickness 0 depends on the shape of the boundary layer veloci-ty profile. This ratio H = 6** /O is generally call-ed the shape factor.

For a given pressure distribution p(x), i.e, for given Ug(x), equation (2.l6) relates the mo-mentum thickness 0, the shape factor H and the wall shear stress coefficient Cf. When a boundary layer velocity profile family as that discussed earlier is supposed to describe all existing ve-locity profiles, a second relation betv.-een Cf, H and 0 (actually Reg) can be- obtained. One additional equation for Cf, G, H is still needed to solve

the problem. Several auxiliary equations are em-ployed in various existing calculation methods. At present the so-called entrainment equation, intro-duced by Head (Ref,lO), is often used as the auxil-iary equation. The entrainment velocity, which de-termines the rate of increase of the amount of tur-bulent fluid in downstream direction, may be written:

entr, dx u/y d_

dx U {è-è^ e (2.19)

Head assumed an empirical relation between V + /U and the shape factor H, Thus a new equation is obtained, which enables the calculation of the boun-dary layer development. Later more sophisticated entrainment equations have been employed,

The important advantage of calculation methods using the momentum integral equation is the moderate amount of computation required due to the elimina-tion of one of the independent variables by an in-tegration of the equation of motion over the boun-dary layer thickness. The advent of modern fast computers has reduced the importance of small amounts of computation. Consequently calculation methods that solve the equation of motion itself, making assumptions about the local turbulent shear stress,

-13-have received increased attention lately. Since in these cases the whole boundary layer flow field is computed, these calculation methods are called field methods.

One possible way to describe the turbulent shear stress is by introducing the so-called mixing length relation:

h

=

Pf

au

öy

au

X 3y (2.20)

where f is the mixing length. Near the wall, but outside the viscous sublayer, the mixing length generally is supposed to be given by f = ky where k is the von Karman constant. When t^ = ^w' equa-tion (2,20) with f= ky is compatible with the wall law equation (2.9), valid outside the viscous sub-layer. Within the sublayer a damping function has to be assumed that reduces the mixing length i fas-ter to zero than f = ky, in order to describe the suppression of the turbulent shear stress there, In the outer part of the boundary layer the mixing length is often supposed to be a constant fraction of the boundary layer thickness 6, e.g. t/ó » 0 , 1 . Now equations (2.l) and (2,4) can be solved numeric-ally and thus the boundary layer flow may be calcu-lated,

To get a correspondence with the laminar boun-dary layer equations an eddy viscosity is sometimes introduced:

pi-dU

X

e 6y (2.21)

The variation of the eddy viscosity in the wall region is generally chosen so that a close corres-pondence exists with the assumed mixing length variation in that region. In the outer part of the boundary layer the usual assumption is a constant value for Ugó^/i-g, The differences between the mixing length and eddy viscosity approaches appear to be small in practice.

A principal objection against the approaches just discussed is that the turbulent shear stress is completely determined by the conditions at the position X considered, while it is known that the turbulent shear stress is actually dependent on the history, i.e, on upstream conditions. The his-tory effect may be incorporated by adapting a dif-ferential equation for the turbulent shear stress, Various such differential equations have been pro-posed, The differential equations are most often modelled wi-^h the aid of turbulence equations, like equation (2,5) and (2.6), In view of its later use, one of the proposed semi-empirical turbulent shear stress equations will be discussed more extensively here. This is Bradshaw's equation (Ref,12), which is based on the turbulent energy equation (2,6). A shear.stress equation is obtained by putting

T^ = a{— pu?), where a is an empirical (positive) constant. This means that the turbulent shear stress is assumed to be positive everywhere, Bradshaw's shear stress equation reads:

u ar

dr.

u

ar,

au ,

r+

ax a ay t ay ay W t^ 1

3/2

.K

( 2 , 2 2 )

The left-hand side of the equation represents the adveotion of turbulent shear stress, i,e. the his-tory effect. The first term on the right-hand side is a production term and the last one a dissipation term. These two terms are the dominant terms in the equation. The expression given for the dissipation term is purely empirical. The dissipation length L, or actually L/ó, is a function of y/ó.

The fact that L is proportional to the boundary layer thickness ó, expresses the general belief, discussed earlier, that the rate of dissipation of turbulence is determined by the large eddies, which may be supposed to be proportional to 6. When only the dominant terms are taken into account, equation (2.22) reduces to the mixing length rela-tion, equation (2,20), The variation of L across the boundary layer has been taken by Bradshaw roughly similar to the mixing length variation de-scribed earlier,

The second term on the right-hand side of equa-tion (2,22) has not yet been discussed. This term represents the diffusion of tur>)ulence. A particu-lar feature is that Bradshaw assumes the diffusion of shear stress to be mainly bulk diffusion instead of gradient diffusion, as is often assumed. The empirical quantity V^ may be called the diffusion velocity. This velocity, the variation of which across the boundary layer is prescribed, is assum-ed to depend on the turbulent shear stress level at the position x considered. An important role is played by this term near the edge of the boundary layer. In fact the diffusion term in equation (2,22) determines the rate of entrainment.

In 1968 a conference has been held at Stanford (Ref.13), where a number of calculation methods for two-dimensional incompressible xurbulent boundary layers has been compared with selected boundary layer experiments. It was found that calculation methods employing the momentum integral equation and a well chosen auxiliary equation were not less accurate than methods that compute the whole velo-city field with the aid of a turbulent shear stress equation. Yet, field methods are generally consider-ed to have a greater potential for the future, pa7-ticularly as regards extensions to more complicated shear layers, such as three-dimensional boundary layers,

2,2 Three-dimensional boundar.y la.yers

2. 2.1 Thre£-d_imensi_ona]^ l_aminar boundary_laye_r£ In a boundary layer beneath a three-dimension-al externthree-dimension-al flow not only the magnitude of the ve-locity changes rapidly over the boundary layer thickness, but also the direction of the velocity. A physical explanation of this may be given as

follows. Consider a boundary layer over a plane surface as shown in figure 4. The streamlines at the boundary layer edge are supposed to be curved, The centrifugal forces on the fluid elements are balanced there by the pressure gradients. Inside the boundary layer the velocity decreases. However, the pressure gradient remains the same, so that inside the boundary layer the streamlines must be more highly curved to maintain the balance. This leads to a significant rotation of the velocity vector in the boundary layer,

In the above considerations we have neglected the effect of the shear forces which may partly

-14-STREAMLINES AT EDGE OF BOUNDARY LAYER

5 \

WALL STREAMLINES

STREAMLINES IN A THREE-DIMENSIONAL BOUNDARY LAYER

DIRECTION OF

EXTERNAL STREAMLINE

DIRECTION OF WALL STREAMLINE

flow around a t h r e e - d i m e n s i o n a l body or a r b i t r a r y shape, a c u r v i l i n e a r c o - o r d i n a t e system h a s t o be used.

The t h r e e - d i m e n s i o n a l boundary l a y e r e q u a t i o n s i n an orthogonal c u r v i l i n e a r c o - o r d i n a t e system may be found i n r e f e r e n c e I 4 , e , g . . In such a c o o r d i n -a t e system the length ds of -a l i n e element i s given

i^y

( d s ) ^ = h^(dx)^ + (dy)^ + h j d z ) ^ ( 2 . 2 3 )

The so-called Lame coefficients h^^. and h^ are func-tions of X and z. The Lame coefficient for the co-ordinate normal to the wall has been put h = 1, which is permissible within the boundary layer

approximation. The Lame coefficients h-^ and h^ are

related to the geodesic curvatures K^ and K^ of the curves x = constant and z = constant respectively:

ah

_-1

ah

X h h ax

X z h h az X z

( 2 , 2 4 )

For a Cartesian co-ordinate system h-j^ 1 and

K^ _{Kz = 0.}

The continuity equation in an orthogonal curvi-linear co-ordinate system reads:

h h X z a(h u ) Z X ax au 3y a(h u ) ^ X z h h X z ( 2 , 2 5 )

b. PERSPECTIVE PLOT OF A VELOCITY PROFILE IN A THREE -DIMENSIONAL BOUNDARY LAYER

Figure 4 : T h r e e - d i m e n s i o n a l lx3und;iry layer flow. balance the p r e s s u r e f o r c e s . At the wall the cen-t r i f u g a l force v a n i s h e s and cen-t h e p r e s s u r e g r a d i e n cen-t must be balanced completely by shear f o r c e s . The l i m i t of a s t r e a m l i n e a s t h e surface i s approached i s c a l l e d a w a l l s t r e a m l i n e . This l i n e i s every-where tangent t o t h e d i r e c t i o n of t h e l o c a l wall shear s t r e s s . Mostly, but not always, t h e maximum d e v i a t i o n from t h e outer flow d i r e c t i o n occurs a t t h e w a l l .

I t i s u s u a l t o r e s o l v e t h e v e l o c i t y v e c t o r of a t h r e e - d i m e n s i o n a l v e l o c i t y p r o f i l e i n streamwise and crosswise components. The streamwise v e l o c i t y Ug i s the component p a r a l l e l t o t h e wall i n t h e o u t e r flow d i r e c t i o n . The crosswise v e l o c i t y U^ i s t h e component p a r a l l e l t o t h e wall normal t o Ug, The crosswise v e l o c i t y i s zero a t t h e wall and a l s o a t the edge of t h e boundary l a y e r ,

We w i l l now d i s c u s s the governing e q u a t i o n s for t h r e e - d i m e n s i o n a l laminar boundary l a y e r s . The d i f f e r e n c e with t h e e q u a t i o n s for two-dimensional laminar bovindary l a y e r s , given i n s e c t i o n 2 . 1 . 1 , i s t h a t t h e v e l o c i t y component U^ i s not z e r o , This l e a d s t o extended v e r s i o n s of the e q u a t i o n s

(2,1) and (2.2) and t o one a d d i t i o n a l boundary layer e q u a t i o n , r e p r e s e n t i n g the e q u a t i o n of motion i n z - d i r e c t i o n , A mathematical complication with threedimensional boundary l a y e r s i s t h a t g e n e r a l -ly i t i s not p o s s i b l e t o choose a C a r t e s i a n co-or-dinate system, since t h e x- and z-axes a r e supposed t o be s i t u a t e d on t h e surface of t h e t h r e e d i -mensional body c o n s i d e r e d . Only when t h e surface of t h i s body i s d e v e l o p a b l e , a C a r t e s i a n c o o r d i n -a t e system i s c o n c e i v -a b l e . For t h e bound-ary l -a y e r

The laminar boundary l a y e r e q u a t i o n s of motion be-come i n t h i s c o - o r d i n a t e system:

au

u au au u au

X X „ X _ Z X h ax y 3j h az -K pU U +K pU = Z X Z X z h ax ^ X

a^u

3y ( 2 . 2 6 )

u au

_au

P T~ "^— +PU -^— ^ h^ ax "^ y ay

- u au

+ ,. - ^ ^ -K pU U +K pU^ = h az X X z z X z 1 a 3^U z ay

At t h e boundary l a y e r edge equation (2.26) and (2.27) reduce t o : U

_au

_{u au}

e e e e „ p — i T — i +p r - ^ — - i -K pU U +K pU'^ h 3x h az z e e x e X z X z 2 ^ 1 a£ h ax X

u au u au

P H ^ ^ r - ^ +P ^ T T ^ -K PU„ U„ +K PU^

( 2 , 2 8 ) h ax X h az z X e e z e X z X _j_ a£ h az z ( 2 , 2 9 )

-15-These are the Euler equations for inviscid flows, The equations are "two-dimensional", since the pressure has been assumed constant along a normal to the surface in the boundary layer approximation, The inviscid flow is not really two-dimensional, as the continuity equation for three-dimensional flows has to be employed,

Many orthogonal curvilinear co-ordinate sys-tems are possible on a given body, A frequent choice is the "streamline" co-ordinate system, for which the curves z = constant coincide with the projec-tions on the surface of the streamlines of the ex-ternal flow. Equation (2.28) and (2.29) become very simple then. Introducing the streamline co-ordinates s, n, gives: U 3U , . e e 1 dp P h as " " h as K pU^ n e h an (2.30) (2.31)

where Ug is the magnitude of the velocity at the boundary layer edge.

A particular case of three-dimensional laminar boundary layer flow, which is relatively easy to solve, is that on an infinite swept wing. In this case a Cartesian co-ordinate system is possible. We choose the x- and z-axis normal and parallel to the wing leading edge respectively. On an infinite swept wing all z-derivatives are zero. The equations in x-direotion then happen to reduce to the two-dimen-sional equations (see e.g. Ref.ll). Thus it is pos-sible to obtain three-dimensional similar solutions when Ug^ oc x™ , corresponding to the two-dimen-sional similar solutions given in figure 1. In figure 5 "the corresponding cross-flow velot-'ty profiles are plotted (taken from unpublished NLR work, see also Ref,15).

Figure 5 shows that the maximum crosswise velo-city in the cases considered occurs at about one third of the boundary layer thickness. The case m = 1 represents a stagnation line flow on an in-finite swept wing, i,e. a flow with a large negat-ive ohordwise pressure gradient. For m = -0.07 a positive chordwise pressure gradient exists lead-ing to a cross-flow in opposite direction. For m = -0.09 a rather special flow situation occurs. Then the wall shear stress component T^ = 0 , which means that the wall streamlines are straight lines parallel to the leading edge of the infinite swept wing. In analogy with the corresponding two-dimen-sional flow, this three-dimentwo-dimen-sional boundary layer .flow should be called just separated everywhere.

The definition of separation is a problem in three-dimensional flows and departs rather much from the existing notions about separation in two-dimensional flows. In two-two-dimensional flow a separ-ation line indicates the start of a region with reverse flow. The expression "reverse flow" is not well defined in the three-dimensional case, however, One of the proposed definitions in three-dimen-sional flows is: the separation line is an envelope of wall streamlines or, on an infinite swept wing, an asymptote of wall streamlines, A definition, which is satisfactory in all conditions, has not yet been found. For a further discussion of this subject one is referred to reference 16 for instance,

More complex shapes of the cross-flow velocity profile may occur than those given in figure 5.

0.8 0.2

/

/

'm =

\

\

/

/

1

\

^

/

/

/

\

i

w

_\

\

\

\

1

\

\

^

\

\

m . -0.07

/

A

m =

1

/

^

\

-0.09 \

/

/

-0.15 +0.15 Ue SIN2V)e

Figure 5 : Typical c r o s s - f l o w velocity profiles for similar t h r e e - d i m e n s i o n a l laminar boundary l a y e r s on an infinite swept wing. Same velocity profiles in x —direction and same

thickness 5 a s in fig. 1.

When t h e c u r v a t u r e of t h e e x t e r n a l s t r e a m l i n e chang-e s s i g n , c r o s s flow i n two d i r chang-e c t i o n s w i l l occur, a s shown i n f i g u r e 6, At t h e i n f l e c t i o n point of t h e e x t e r n a l s t r e a m l i n e n o t h i n g s p e c i a l can be n o t e d , but f u r t h e r downstream t h e crosswise v e l o c i t i e s n e a r t h e w a l l and i n t h e o u t e r p a r t of t h e boundary l a y e r a r e seen t o be i n o p p o s i t e d i r e c t i o n . This b e -h a v i o u r i s a consequence of t -h e f a c t t -h a t t -h e f l u i d near t h e w a l l a d a p t s i t s e l f much f a s t e r t o t h e chang-i n g c u r v a t u r e t h a n t h e f l u chang-i d away from t h e w a l l , s i n c e t h e i n e r t i a l e f f e c t s a r e l a r g e r f u r t h e r from t h e w a l l . The o c c u r r e n c e of t h i s t y p e of crosswise v e l o c i t y p r o f i l e s c e r t a i n l y has p r a c t i c a l importance,

since e x t e r n a l s t r e a m l i n e s often have i n f l e c t i o n p o i n t s .

Various c a l c u l a t i o n methods f o r t h r e e d i m e n s i o n a l laminar boundary l a y e r s have been d e v e l o p ed, Formerly a number of approximate methods, r e -sembling t h e two-dimensional c a l c u l a t i o n methods employing t h e momentum i n t e g r a l e q u a t i o n , have been proposed ( e . g . Ref. 17 - 1 9 ) . Later e f f o r t has been d i r e c t e d c h i e f l y to the numerical s o l u

-Figure 6 : Typical cross-flow velocity profiles for a laminar boundary layer in case of an external streamline with inflection point ( from ref, 14 ). tion of the full equations with the aid of the

fast computers, that became available (e.g. Ref. 20 - 23). The main problem is to reduce the com-puting times to acceptable levels.

2, 2,2 Thre£-dimensi^ona]^ turbule_nt_ boundary_layer£ We consider here turbulent flows, which are stationary in the mean. The velocity and pressure will be divided in a mean part and a fluctuating part: (U^+u^), (Uy+Uy), (U^+Ug), (p-t-p')) where the time mean of the turbulent velocities and pressure is zero. The resultant momentary velocities and pressure may be substituted in the unsteady Navier-Stokes equations of motion and the continuity equa-tion, Time averaging leads to the conclusion that equation (2,25) also holds for the mean velocity field in turbulent flows. Equation (2,26) and (2.27) become for turbulent flows:

u au au u au

p r^ - ^ -t-pU — i +p -^ - ^ -K pU U +K pU = ^ h ax y 3y ^ h az z^ X z x'^ z X z

a^u

•1 a 3u U 1 Sp X y — —*- -o ^ +li — — -h ax P ay '^ ^ 2 X '^ ay (2.32)

u su au u au „

p r^ — ^ +pU — 2 - +p - ^ — ^ -K pU U -i-K pU = ^ h ax y ay *^ h az v"^ v ^ ^r v X X z z X 2

a u

- - au u 1 dp z .Y , z '^ ay (2.33)

In these equations extra terms of the order (ö/x) have been neglected. The extra terms, which have been retained, may be regarded as representing the contribution of a "turbulent shear stress vector" with components T^ = -pu^Uy and T-^ = -pU2Uy ,

As in two-dimensional boundary layers, the magnitude of the turbulent shear stress is much larger than that of the viscous shear stress every-where, except in a thin layer adjacent to the wall: the viscous sublayer. In the viscous sublayer the fluid offers much less resistance to deformation than in the turbulent part of the boundary layer, Consequently substantial changes in magnitude and direction of the velocity vector occur in this thin layer. The large increase of the magnitude of the streamwise velocity near the wall was discussed already in the section 2,1.2. Some typical cross-wise velocity profiles of three-dimensional turbu-lent boundary layers are shown in figure 7. It is evident that considerable variations in the cross-wise velocity occur close to the wall. In contrast with laminar cross-flow profiles the maximum cross flow velocities generally occur very close to the wall in turbulent flows,

Analysis of measured velocities in three-di-mensional turbulent boundary layers has shown that generally the streamwise velocity profile shapes fit reasonably well in the known profile families of two-dimensional boundary layers (see e,g. Ref, 57 or 102), The crosswise velocity profiles con-stitute a more difficult problem. Many attempts have been made to describe the cross flow velocity profiles by analytical formulas (Refs, 24-31). Nearly all suggested cross flow profile family formulas do not include the possibility of cross flow in opposite directions. Yet, when the extern-al streamline has an inflection point, such cross flow profiles do occur in turbulent boundary layers as well as in laminar ones (see figure 6 ) .

We will discuss here only those few cross-flow profile families that have found wide application. In the first place Mager's formula (Ref,24) should be mentioned:

"s(^ ) "tg ('P,,-'P„) (2,34) where ('P„-'Dg) = the angle between the external streamline and the wall streamline. This formula is one of the first suggestions made for the cross flow profile and seems to be not worse than many of those proposed later,

Secondly, attention has to be drawn to Johnston's

-17-1.0 Y ^995 0.4 0.2

i

w

_\

_\

\ \ STATION

\

\ \ \ 5 — ^ \ \

1

\ . s7x~r*

\ ^

V

\

y

)

\

"

~

STATION 7

1

^

- \

^

\ ) 0.1 0.2

Figure 7: Typical cross-flow velocity profiles for

three-dimensional turbulent bouudaiy layers, R e ^ * ~

srl —2 X 104, Own measurements, station

5 and 7 ( see chapter 4 ),

p r o p o s a l ( R e f , 2 5 ) , based on t h e p o l a r p l o t of t h e v e l o c i t y v e c t o r , Johnston found t h a t i n a p o l a r p l o t t h e t i p s of t h e v e l o c i t y v e c t o r l i e very n e a r l y on t h e two l e g s of a t r i a n g l e , a s shown i n f i g u r e 8, According to J o h n s t o n t h e apex of t h e t r i a n g l e i s always s i t u a t e d j u s t o u t s i d e the v i s -cous s u b l a y e r . A c t u a l l y he assumed t h a t at t h e apex t h e v e l o c i t y magnitude follows from ö / u - = U = 1 4 . When a l s o u_ and ('O^-'Og) a r e g i v e n , t h e c r o s s f l o w v e l o c i t y p r o f i l e i s d e f i n e d . The c r o s s -flow v e l o c i t y i n t h e o u t e r p a r t of t h e boundary l a y e r , as defined by the t r i a n g l e r i g h t l e g , can be shown t o b e :

(U^-U^) tg (-,^-.J

14 Ur cos (ra -'p

_{' w e}

J

(2.35)

approximately a triangular shape indeed in many experiments. This would mean that there are two collateral regions: one near the__wal^l and, when looking at the velocity defect (Ug-U), one in the outer part of the boundary layer,

It is important to note that the collateral region near the wall, i,e, the left leg of the triangle in figure 8, corresponds with a very small part of the total boundary layer thickness, since the velocity increases very fast with distance from the wall in that region. This means that the rate of rotation of the velocity vector with distance must be very large indeed to notice such in a polar plot. Moreover, the accuracy of the measurements generally is poor close to the wall, so that on the whole not much significance should be attached to the fact that the polar plots suggest a collateral region near the wall. A more detailed look on the variation of the velocity vector near the wall learns that a substantial rotation of the velocity occurs there. As a matter of fact, the variation of the magnitude and direction of the velocity close to the wall is one of the subjects, that will be investigated extensively later. This will be done in chapter 3, where extensions of the two-di-mensional law of the wall to three-ditwo-di-mensional flows will be deduced.

More significance seems to have to be attached to the result that the velocity defect is collat-eral in the outer part of the boundary layer. It has been shown theoretically (Ref.32) that when an initially two-dimensional boundary layer is sub-jected to large pressure gradients, so that the influence of the shear stresses is negligible, a collateral velocity defect should result as a first approximation. Since in many experiments a two-di-mentional boundary layer is subjected suddenly to a large pressure gradient, as will appear from a discussion on this topic later in this chapter, the straight right leg of the triangle in figure 8 may also be fortuitous and mainly connected to the par-ticular test set-ups mostly employed up to now,

A number of calculation methods for three-di-mensional turbulent boundary layers have been de-veloped. As with two-dimensional boundary layers, one can distinguish between momentum integral methods and field methods. Momentum integral methods use the momentum integral equations, which are obtained by integration of equation (2.32) and

0.5

^ 9e'

r ^

L

0.5

The assumption made about t h e p o s i t i o n of t h e apex of t h e t r i a n g l e has not been supported by l a t e r measurements, but t h e p o l a r p l o t was found t o have

Figure 8 : Polar plot of the velocity vector in a three

-dimensional turbulent boundary layer

according to Jolmston.

•18-(2.33) across the boundary layer thickness (see e.g, Ref.14), In streamline co-ordinates, s, n, the momentum integral equations read:

, ao , 3 0 ( 2 0 +0") 1 ss _i^ sn ss s h as h an h U _{s e} au e as

au

(0 +G ) + _ J i V i l ^ ^ _K (6 -G )-K (Q +0 ) an s ss nn n sn ns h U n e pu (2,36) .,60 , SO 2 0 au 2 0 au 1 ns _1 nn ns e nn e h as h Bn h' U as "^ h U an * s n s e n e

calculation method (Ref,10), which employs the en-trainment equation as the auxiliary equation (see-section 2,1,2), The method of Michel et al CRef,29) also uses the entrainment equation, but with a different three-dimensional velocity profile family.

Instead of using the momentum integral equa-tions, the equations of motion (2,32) and (2,33), together with the continuity equation (2.25), can be solved directly when assumptions are made about the turbulent shear stress itself. The generaliza-tion of these so-called field methods from two to three dimensions requires less additional assump-tions than in the case of integral methods. General-ly the magnitude of the turbulent shear stress is taken the same as that in a two-dimensional flow with the same variation of the velocity magnitude, For instance for the mixing length relation, equa-tion (2,20), this leads to:

-2 K G +K (0 -G +6*) s ns n ss nn s

pu;

(2.37) au 2 au 2

l^tl

-

^^'

(#)

^

(^)

(2.39) In these equations the following thickness

para-meters have been used: 6 U 6 > | ( 1 - ^ ) d y 0 6 U 6 - - | ^ d y n o 6 U U

«ss^f(i-ir)ir^y

J e e o U U sn I ^ U ' J '' J e e 6 U U

7 1

dy Ü u nn J y2 o e dy (2.38)

The quantities 6g and 0gg are equivalent to the displacement thickness and the momentum thickness in two-dimensional flows. There is here also a crosswise "displacement" thickness 6* and there are three additional momentum thicknesses. One relation exists between the various thicknesses: ^n"''®sn~®ns ^ ^' which has been used to eliminate 6* from equation (a36) and (2.37).

Calculation methods based on the momentum in-tegral equations assume velocity profile families, so that the various boundary layer thickness para-meters are related to each other. These methods are generally extensions of existing two-dimensional

calculation methods. Roughly speaking, the flow in streamwise direction is calculated in these jnethods as if the flow was two-dimensional, apart from the fact that the streamwise momentum integral equation (2.36) is different from the two-dimensional one (2.16). The same velocity profile family, the same skin friction law for the streamwise component ^Wg/pUg, and the same or a very similar auxiliary equation are generally used. When a simple crosswise velocity profile family, like that of Mager (equa-tion (2.34))) is assumed, the cross flow is determ-ined by the cross-wise momentum integral equation (2.37).

Several momentum integral calculation methods for three-dimensional turbulent boundary layers have been proposed in the past (Ref. 29, 33-38), One of the more recent methods in use is that due to P,D. Smith (Ref,36), which is essentially an ex-tension of Head's two-dimensional boundary layer

The main new assumption then is related to the di-rection of the turbulent shear stress vector. The most simple and often used assumption is that the direction of the turbulent shear stress coincides with the direction of the maximum rate of deforma-tion, i,e,

a u / a y

X _ X

U ^ au/ay

(2.40)

This is the same direction as that of the viscous shear stress in laminar flow. Equation (2.39) and (2,40) may also be written:

au 2 au 2 p2 / X\ / z -3y 3y 1/2 ,f2 au 2 au 2 ^ay ^ ^ay ^ 1/2 au X ay au z ay ( 2 , 4 1 ) ( 2 . 4 2 )

When the variation of the mixing length across the boundary layer is known (e.g, taken identical to that used in two-dimensional boundary layer calculation methods) the governing equations can be solved numerically. A number of three-dimension-al boundary layer cthree-dimension-alculation methods has been de-veloped on this basis, either by using the mixing length relation itself, or by a related approach introducing an eddy viscosity as in equation (2,2l) (Ref.39-41, 95, 96).

It has been mentioned already in section 2.1.2 that more advanced turbulent shear stress equations have been applied for two-dimensional boundary layer calculations than those discussed above. In these calculations differential equations for the shear stress are employed, thus taking into account the upstream history of the turbulence. The semi-empirical differential equation for the turbulent shear stress proposed by Bradshaw et al was dis-cussed in some detail in that section. Extensions of this equation to three dimensions have been pro-posed by Bradshaw himself (Ref.42) and

-19-ly by Wesseling ( R e f . 4 3 ) . The extension to three dimensions is not evident, since Bradshaw's two-dimensional turbulent shear stress equation w a s deduced from the turbulent energy equation, which is a scalar equation. Y e t , both workers arrived at the same resultant equations for the turbulent shear stress in three-dimensional boundary layers:

U ^^t

X X

^h

U

%

+U

h ax ,y ay h az

X •' "^ z

(K U -K u )r,

_{X z z X t}

au

_{a K t | ^/^ '^'^x}

(2.43)

u 3^t

— 4U

h^ ax y ay

3^t U 9^1 ^ + r^ -T^ +(K U -K U )T, _{h Bz z X X z t}

au

•tl 3y

— (V r

3y ^ d 't

Ktl

)-(^)

1/2 h

(2.44)

The equations may be regarded as a vector equation for t^. Introducing the substantial derivative o/Dt, equation (2.43) and (2,44) may be condensed to:

!5

Dt

_au

3y

a_

3y

^^t)-0

1/2

(2,45)

The left-hand side represents the adveotion of the

shear stress vector T-^^.'^he first term on the right-hand side produces new turbulent shear stress in the direction of the maximum rate of deformation au/ay. The next term represents, the diffusion of T(. with a diffusion velocity Vjj^, The last term of the equation stands for the dissipation of exis-ting turbulent shear stress. The production and dissipation terms are the dominant terms in the equations. It is easy to show, When only the d o -minant terms are taken into account, that equation

(2.43) and (2.44) reduce to the mixing length r e -lation in three-dimensional boundary layers, equa-tion (2.41) and ( 2 . 4 2 ) . In that case the shear stress is in the direction of the maximum rate of deformation. Actually the shear stress direction according to equation (2,43) and (2,44) will differ

EXTERNAL STREAMLINE

INITIAL CONDITIONS REQUIRED OVER THIS WIDTH TO CALCULATE B.L. FLOW AT P

Figure 9: Typical regions of influence and dependence

of a point P in a three-dimensional boundarj'

1 aver.

somewhat from that direction due to the contribu-tion of the adveccontribu-tion and diffusion terms.

When the same empirical input for the constant a, the diffusion v e l o c i t y V ^ and the dissipation length L is used as for the calculation of two-di-mensional boundary layers (Ref,12), the equations ( 2 . 2 5 ) , ( 2 , 3 2 ) , ( 2 . 3 3 ) , (2.43) and (2.44) can be solved numerically. T h i s h a s been done by Bradshaw for the threedimensional boundary layer on an i n -finite swept wing (Ref.42) and at NLR by Wesseling and Lindhout for the general three-dimensional boundary layer ( R e f , 4 4 ) , Nash (Ref,45) has develop-ed a slightly different calculation method, with the same shear stress m a g n i t u d e , but with the shear stress taken in the direction of the maximum rate of deformation, which i s somewhat different in general from the d i r e c t i o n resulting from equation

(2,43) and ( 2 . 4 4 ) .

For the calculation of threedimensional b o u n -dary layers it is important to note the existence of a "region of influence" of a given point P, i.e, the region downstream of P, which is affected b y the flow conditions at P. This w a s studied first b y Raetz (Ref,20) for three-dimensional laminar boun-dary layers, but h i s analysis remains applicable i n the turbulent case. In b o t h cases information is transported downstream along streamlines and also along lines normal t o the surface. This means that the projection of the region of influence on the surface coincides with the projection of all stream-lines going through the normal on the surface of P, This is illustrated i n figure 9i where it has b e e n assumed that the external streamline is the least

curved and the wall streamline the most. One m a y also define a region of dependence, extending u p -stream of a point P, From this it can be deduced over which width the initial conditions of a three-dimensional boundary layer have to b e known to b e able to calculate the boundary layer flow at P

(see figure 9 ) .

We will now briefly review the available e x -perimental d a t a on three-dimensional incompressible turbulent boundary l a y e r s . Actually the number of three-dimensional b o u n d a r y layer experiments, that have been carried o u t , i s not really small (Refs. 31, 4 6 - 6 2 , 1 0 1 ) . Most of them, however, do not yield more than some shapes of skewed three-dimen-sional velocity p r o f i l e s . Particularly nearly all experiments are not v e r y well suited for a check of three-dimensional calculation m e t h o d s . Y e t , there is a great need for such checks, since tur-bulent boundary layer calculation methods contain much empiricism,

In many of the earlier tests, comparisons with calculations are impossible, since the initial con-ditions of the b o u n d a r y layer were not measured over a sufficiently large width in view of the r e levant region of d e p e n d e n c e . This occurs, for i n -stance, w h e n on a swept wing of finite aspect r a t i o , as shown in figure 1 0 , boundary layer measurements are carried out at one spanwise location only, There is also quite a number of test set-ups, i n which a three-dimensional boundary layer flow is created very abruptly. A typical example of such a test set-up is a cylinder mounted o n a flat plate (see figure 1 0 ) , the boundary layer along the plate having a thickness comparable with the cylinder radius. Here the b o u n d a r y layer develops from a two-dimensional zero-pressure gradient layer to a three-dimensional separation within a distance of