An investigation on Le Foll's method used for blade optimization based on boundary layer concepts

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KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE

61

AN INVESTIGATION ON LE FOLL'S METHOD USED FOR BLADE OPTIMIZATION BASED ON BOUNDARY LAYER CONCEPTS

by

K. PAPAILIOU

RHODE-SAINT-GENESE, BELGIUM

AUGUS"T

1969

TECHNICAL NOTE 6l

AH INVESTIGATION ON LE FOLL'S METHOD USED FOR BLADE OPTIMIZATION BASED ON BOUNDARY LAYER CONCEPTS

(Paper presented at the XXth International Astronautical Congress)

by

K. PAPAILIOU

u ...

s ö* p p

rr

G w

velocity at the edges of the boundary layer surf ace length

momentum form factor

energy form factor

momentum thickness displacement thickness energy thickness

dissipation factor

skin friction coefficient

density

shear stress at the wall

statie pressure

parameter characterizing the equilibrium turbulent boundary layers Clauser's form factor

u

_...

Fl function of G F 2 function of G F3 function of G and Re ~ v

T turbulent eddy viscosity Ö boundary layer thickness y distance from the wall

Figure 1 - The curve n(G) as predicted by different theories. Figure 2 - The plane wG,w2

n

where Stratford's data appear.

Figure 3 - Eddy viscosity distribution for different equilibrium boundary 1ayers taken from Bradshaw (ref.

8).

Figure 4 - Co v L for laminar and turbulent boundary layers.

Figure 5 - Equilibrium turbulent boundary layers. Second experiment of Bradshaw (ref. 8).

Figure 6 - Bradshaw's experiment. Response of a retarded equilibrium tur-bulent boundary layer to the sudden removal of pressure gradient

(ref. 20).

Figure 7 - Bradshaw's experiment. The response of a constant pressure tur-bulent boundary l~yer to the sudden application of an adverse pressure gradient (ref. 21).

Figure 8 - Schubauer and K1ebanoff's experiment (ref. 22). Calculation of the momentum form factor distribution without curvature effects. Figure 9 - Schubauer and K1ebanoff's experiment. Calcu1ation of the momentum

thickness distribution without curvature effects.

Figure 10- Schubauer and K1ebanoff's experiment. Calcu1ations of the momentum thickness distribution with curvature and flow convergence effects. Figure 11- Schubauer and K1ebanoff's experiment. Calculation of the momentum

form factor with curvature and flow convergence effects • .

Figure 12- Experiments made on a C4-series blade (ref. 23). Figure 13- Experiment n° 6 of Stratford (ref. 10).

BLADE OPTIMIZATION BASED ON BOUNDARY LAYER CONCEPTS

1. INTRODUCTION

During the past five years we have developed at the von Karman Institute a procedure for the design of optimized compressor blades based on Le Foll's theory (ref. 1, 2) for boundary layers and A. Goldstein's (ref. 2, 3) inverse method for the calculation of potential flows. The pro-cedure briefly consists in selecting the kind of boundary layers we want to produce on the suction side of the blade and from them derive the velocity distribution which will produce these boundary layers with the aid of Le Foll's methode Using a pressure side velocity distribution which produces a laminar boundary layer over most of the pressure surface we can arrive at the blade shape with the aid of the inverse method of A. Goldstein. A special

proce-dure has been devised to deal automatically with the closure conditions

(ref. 2, 4). The treatment is, at the moment, limited to incompressible flow. A compressor blade of 45° turning back to axial direction has been designed according to the above procedure. This blade was tested in the V.K.I. cascade tunnel C-l. These experiments showed : the prediction of the location of transition was excellent the prediction of the velocity distribution prior to separation was very good . and,although unpredicted, a small region of se-paration was detected in the last 5% of the chord. This flow sese-paration re-sulted in higher losses than predicted and lower circulation than predicted.

An investigation of the causes of the initial difficulties with the method of blade design shows that the assumption of local similarity was not in fault. The investigation indicates that the discrepancy in the

losses could be explained on the basis of two phenomena : one of these is the flow convergence, which is always strongly present on a highly loaded blade.

-The secohd is the blade surface curvature, which affects greatly the turbu-lent boundary layer. A method has been formulated for including in Le Foll's theory, the effects of flow convergence and curvature.

In this paper we shall briefly explain Le FollIs method and its poss"ibilities when used as a design method, we shall describe the investi-gation undertaken to check Le FollIs method and finally we shall comment on the effects of curvature and flow convergence.

2. BRIEF DESCRIPTION OF LE FOLL IS METHOD

2.1 - The idea of Le Foll.

The aS8umption of local similarity (ref.

5, 4)

combined with the assumption of a one parameter family of velocity profiles permits the cal-culation of all the properties at a boundary layer cross section, once two properties are known. These two properties are a form factor and a charac-teristic length of the boundary layer. The problem of boundary layer cal-culation thus reduces to the problem of the determination of these two properties •

Any two integral moments of the boundary layer equation of mot ion can be used and we shall consider the two basic equations of Le FollIs methode a) The integral equation of momentum

where U2 d(eu:)

+

eH 12u ... dU ...

= C

f (H12,e)-; ds

e

=

momentum thickness 15

**

= energy thickness ó

*

=

displacement thickness momentum form factor

**

ö

=

~ energy form factor

(1)

Cf = skin friction coefficient C

_n

= dissipation factor

U~ = velocity at the edge of the boundary layer

s

₌

surface length

lt is possible to use the same equations and calculate the velo-city distribution U (s) starting from the boundary ~ layer properties, that

is starting from the distribution Hl2(6). We can complete this idea by

say-ing that, considersay-ing the plane which has as coordinates, a form factor and a Reynolds number based on a characteristic length of the boundary layer, any line on that p1ane represents a boundary 1ayer ; we can ca1culate the

velocity distribution corresponding to it using the equations (1) and (2). lf, additionally, the genera1 properties of the boundary layers can be

re-presented on that p1ane we can chose the boundary layer with the properties

we want and then calcu1ate the velocity distribution under which the chosen boundary layer will develop.

This is precisely Le Foll's idea : To introduce a plane having as

coordinates a form factor and a Reynolds number based on a characteristic length of the boundary layer on which all the unseparated laminar and tur-bulent boundary layers can be represented along with their general properties.

2.2 - The possibilities of Le Foll's theory.

The fo11owing properties of boundary layers are represented on Le F011's plane.

a) Schlichting's curve of neutral stability for laminar layers (ref. 6). b) Clauser's curve of turbulent stability (ref. 7). Le Foll has given the

first theoretical proof of its existence.

c) The upper limits of laminar and turbulent boundary layers.

Along with these properties the following processes have been established :

a) How to control a boundary layer in order to have it laminar, how to introduce instability and how to provoke transition accurately and quickly where the external turbulence level is given.

b) How to control a boundary layer for a range of Reynolds numbers of ope-rat ion.

c) How to control turbulent separation when the turbulence level of the external stream is higher than the one for which the design was realized. d) How to ~ptimize the deceleration of a turbulent boundary layer.

e) How to introduce the limit deceleration supported by a turbulent boun-dary layer.

f) How to optimize boundary layers used for the suction side of a blade. From these properties one can easily see that Le Follis method is a very powerful tooI for the design engineer. We might also state that it can be considered as an inverse method for the solution of the boundary layer problem.

3. THE TESTS MADE ON LE FOLLIS METHOD.

Le Foll's method is sufficiently accurate from the point of view of laminar layers. For turbulent layers, however, the fact that Le Foll used not weIl established data of equilibrium turbulent boundary layers and the hypothesis of local similarity, which is indispensable in the mathematical formulation of the problem, but is contested strongly in the light of today's knowiedge, demands a further investigation of Le Follls method, especially in view of the failure of our experimental data to match the theoretical pre-dictions.

3.1 - Inves·tigation made in respect to the calibration of the methode Le Follls calibration of the turbulent functions is based on data from the equilibrium turbulent boundary layers. At the time that he developed his theory he had as experimental guide in the domain of the equilibrium

turbulent boundary layers only Clauser's experiments. Reeently Bradshaw (ref. 8) realized two more experiments with the help of whieh it is possible to establish the true relationship between the parameter eharaeterizing the equilibrium turbulent boundary layers and defined below

where

n -

ö*

.2e.

- T ds

w

T is the wall shear stress

w

p is the statie pressure

and the form factor of Clauser G (see definition in ref. 7).

The available data along with the predietion

oi

some theories

(3)

are given in figure 1. We can see that we cannot be sure a priori about the extension of the

(n,

G)-curve passing from the experimenta1 data. It is possible, hcwever, to trace accurately the complete curve

(n,

G) on the basis of the available experimental data using a new plane having as coor-dinates wG and

(./n,

where w'=

I

_{Cf /2. In this new plane, given in figure 2,}

it is possible to represent the point corresponding to Stratford's experi-ment of a turbulent equilibrium boundary 1ayer with zero wall shear stress

(ref. 9, 10); On the same plane we have transferred the available data from the initial

(n,

G)-plane for different values of the skin friction coefficient (which amounts in different values of Re

_.

_ö*

). We can see in figure 2 that the curves G,

Jn

are ncw well established. Transferring back in the

(n,

G)-plane

points from any such curve we obtain a unique extension of the curven, G

passing through the experimenta1 data. A point to be noticed is that the

second experiment of Clauser seems doubtfu1, a fact already remarked by other workers. From this we can see that the theoretical curve taken by Le Foll is not in agreement with the experimental results.

We have decided then to try to calibrate once more the dissipation factor used in Le Foll's theory, because it depends on the form of the curve (TI, G). A new way was devised for its calibration based on the work of Mellor

(ref. 8). We have to note that the eddy viscosity is not constant (as can be seen from figure 3) in the outer part of the boundary layer but can be repre-sented by a unique curve. The development of this work is reported in detail in reference 12. Here we shall give the final expression found for the dissi-pation factor (4) where

_u""

(5) = -and F

2(G), F3(G, Re6

*)

are functions defined in ref.' 12 which is similar to the one given by Le Foll

U+2

""

and the one given by Rotte.

+2 U"" C ----= I D 2 = 1

+

FI (G)

+

U+

""

(6)

""

(7)

The numerical results of our calibration are compared with those of Le Fo11 in figure 4. Note that L is the form factor end X is the Iogarithm of the Reynolds number used by Le Foll. Much to our surprise the comparison was found to be very good.

3.2 - Investigation made in respect to the limit of deceleration end the separation criterion used by Le Foll.

The optimum deceleration as established by Le Foil, is composed by either equilibrium or highly non-equilibrium boundary Iayers. In fact in the final part of an optimized boundary layer for a suction side the limit dece-leration is employed. This part of highly non-equilibrium boundary layer cannot be calculated by methods using the hypothesis of local similarity.

A method was devised which is based on Stratford's (ref. 9 and 10) experimental results and analysis and the subsequent analysis of Townsend

(ref. 13). This method which can be regarded as an extension of Stratford's method can be employed to calculate the development of a highly non-equili-brium boundary layer, starting from any equilinon-equili-brium initial position. At the same time an extension of Stratford's criterion is provided. It is reported in det~il in reference 14. It can be used in the regions where Le Foll's theory might fail.

3.3 - Investigation made by applying Le Foll's method on available expe-rimental results and by comparing it with the calculation method of Bradshaw

(ref. 15).

Finally, Le Fol1's method has been compared with Bradshaw's method of calculation and a series of experimental turbulent boundary layers covering all the domain, from equilibrium to highly non-equilibrium boundary layers. We have used Bradshaw's method because it is supposed to be the best

avail-able today and so it may be used as a reference in the absence of experimental results.

The results of the calculation are given in figures 5 to 13. The comparison between the two methods and boundary 1ayers up to a certain

de-gree of severity is very good. Of course Bradshaw's method giv~superior

re-sults although the accuracy of Le Fo1l's method is judged quite sufficient for engineering purposes. It has to be pointed out, however, that the effects of flow convergence and the effects of blade curvature were the origin of deviations between Le Foll's theory predictions and experimental results. These effects could be taken account of with Bradshaw's method, which gave very good agreement with these experiments, once these effects were included

in the calculation.

Consider for instanee the classical experiment of Schubauer and Klebanoff. We can see in figures 8 to 11 that the two methods give the same results but fail to predict correctly the experimental values near the trailing edge. However, during the experiment of Schubauer and Klebanoff

there existed a certain convergence in the flow near .the trailing edge and the model had an important curvature there. These effects introduced in Bradshaw's method, the prediction near the trailing edge becomes very good.

Calculations were also made using Bradshaw's method and the ex-perimental velocity distribution of the designed blade with the aid of the optimization procedure established at the VKI, introducing now the flow convergence observed during the experiments and the curvature of the cal-culated blade.

The results showed an increase in

e

which accounted for the additionallosses observed experimentally and a separation was predicted near the trailing edge in accordance with the visualization done during the experiments.

In this way, we have showed that it was not the "local similarity" hypothesis that was responsible for the additional losses detected experi-mentally, but the combination of two effects : the flow convergence and the blade curvature which have not been taken into account in Le Foll's method.

4. SaME REMARKS ON THE EFFECTS OF BLADE CURVATURE AND FL~ CONVERGENCE. It has already been pointed out (ref. 16, 17) that the effect of curvature on laminar boundary layers is very small, the equation describing the flow remaining the same. However, the'centrifugal" forces created by the curvature have a direct and important effect on the turbulent properties. Although this effect was recognised some years ago (Thomson (ref. 18), for

instanee, took it into account in calculating his entraintment function), it is only very recently that a theory··with a concrete physical background was given by P. Bradshaw. From calculations we have performed up to ncw it

seems that this effect is acting indirectlyon the loss increase. lts in-fluence is predominant on the form factor H12' affecting very slightly the momentum thickness

e

(which is directly proportional to the losses). The tendency is to increase H

12 for a convex surface and to decrease it for a concave one so that, as the suction side of the blade is a convex surface,

the boundary layer becomes more critical. If separation is reached, then the losses increase disproportionally. In other words the deceleration

possible without separation is smaller for a boundary layer developing along a convex surface than for one developping along a flat surface.

The effect of convergence can be understood. physically by making the fOllowing remark. Consider figure 14 which presents the suction side of a blade and some of the flow streamlines on this surface. We can see that the losses created across any section AD, EF or CR are finally influencing the section BC. In other words the losses distributed over a width EF for instanee, shall be redistributed finally over a width BC. Consequently

measurements realized under this situation will not give the two-dimensional losses of the cascade but the losses including three dimensional effects.

The flow pattern showed in figure

1.

is very common in cascade measurements because of the presence of the corner vortices even if suction is applied on the wide wal~and the axial velocity ratio is equal to one. The reason is that the corner vortices cause alocal convergence of the flow

extending a small distance from the suctinn side. This is the reason why aspect ratio is very important when measurements are realized especially for highly loaded blades (the corner vortices creating highly convergent flow).

We have to add that once the flow leaves the blades, the rate of convergence of the streamlines is lower. The losses are redistributed accor-dingly and it is possible, if a divergent flow is established (as sometimes happens) to measure losses decreasing with distance from the trailing èdge in apparent contradiction with the fact that losses must be increasing be-cause of the mixing process taking place downstream of the blade trailing edge. We have to add finally that the convergence of the flow affects only the value of the momentum thickness leaving the value of the momentum fo~m

factor practically the same.

s. FINAL REMARKS AND CONCLUS laNS •

In the beginning of the present report we have shown that Le Follis method used as a "design" or "inverse" method for boundary layers is a very

useful tool and can be used to optimize boundary layers. There are, of course, better methods for the boundary layer calculation but these methods cannot be formulated in order to be used as "design" methods. However, the investigation presented showed that large discrepancies between Le Foll's theory and experimental results ahould be attributed to two effects : the effect of the blade curvature and the effect of the flow convergence which are not taken into account in Le Foll's methode

In reference 4 we have shown that the effect of flow convergence could be incorporated in Le Foli's method without major alteration of it. However, in order to incorporate the curvature effects a new calibration of

the dissipation factor is necessary. Although the mathematical formulation of this problem has been done, numerical results are not yet available. We

can finally say that in view of the fact that with Bradshaw's method and including the effects of curvature and convergence we have predicted the correct losses and in view of the agreement between Le Foll's and Bradshaw's theory without curvature and convergence effects it is hoped that these effects included, Le Foll's method will yield complete agreement with expe-riment.

1. LE FOLL "A Theory of Representation of the Properties of Boundary Layers on a P1ane." Proc. Seminar on Advanced Prob1ems in Turbomachi-nery, V.K.I. - March 29-30 (1965).

2. K. PAPAILIOU : "B1ade Optimization Based on Boundary Layer Concepts" -VKI CN 60 (1967).

3. A. GOLDSTEIN Unpub1ished 1ectures.

4. K. PAPAILIOU "Optimisation d'Aubages de Compresseurs à Forte Charge sur la Base des Théories de Couches Limites" •

Dissertation thesis. Liège University - June 1969.

5. J. ROTTA "Turbulent Boundary Layers in Incompressib1e Flow". Progress in Aeronautieal Sc. Vol. 2, pergamon Press (1962).

6. H. SClU..ICHTING : "Boundary Layer Theory" - pergamon Press (1960).

7. F. CLAUSER : "Turbulent Boundary Layers in Adverse Pressure Gradients". Int. Aero. Sc. 21, pp. 91-108 (1954).

8. P. BRADSHAW : "The Turbu1ence Structures of the Equilibrium Boundary Layer", ARC 27675 (1966).

9. B.S. STRATFORD : "The prediction of Separation of the Turbulent Boundary Layer" - Jn1 F1uid Mech., Vol. 5, pp. 1-16 (1959).

10. B.S. STRATFORD : "An Experimenta1 Flow with Zero Skin Friction throughout its Region of Pressure Rise" - Jnl F1uid Meeh., Vol. 5, Part I, pp. 17-35 (1939).

11. G. MELLOR : "The Effeets of Pressure Gradient on Turbulent Flow near a Smooth Wall". Jnl Fluid Mech., Vol. 24, part 2, pp. 225-274

(1966).

12. K. PAPAILIOU : "Calibration of the Dissipation Factor" - VKI IN 25 (1967). 13. A. TOWNSEND : "The Development of Turbulent Boundary Layers with

Neg1i-gib1e Wall Shear Stress" - Jnl Fluid Mech. Vol. 8, Part I, pp. 143-158 (1966).

14. K. PAPAllIOU : "An Extension to Stratford' s Theory for Highly Non-Equilibrium Turbulent Boundary Layers" - VKI IN 24 (1967).

Boundary Layer Deve10pment Using the Turbulent Energy Equation" NPL Aero Re 1269 (1968).

16. L.LEES, C.C. LIN (Editors) High Speed Aerodynamics and Jet Propulsion Lab. "Laminar Boundary Layers". Vol. V - Princeton Uno Press

(1959).

17. B.S. MASEY, B.R. CLAYTON : "Laminar Boundary Layers and their Separation from Curved Surfaces" - ASME J .B.E. p. 483 (discussion) , 1965. 18. B. THOMSPON : "The Ca1culation on Shape-factor Development in

Incom-pressible Turbulent Boundary Layers with or without Transpi-rat ion" - AGARDograph 97, May 1965.

19. P. BRADSHAW : "The Anal ogy be tween S treaml ine C urva ture and Buoyancy in Turbulent Shear Flow" - NPL. Aero. Re. 1231 (1967).

20. P. BRADSHAW,· D. FERRIS : "The Response of a Retarded Equilibrium Tur-bulent Boundary Layer to the Sudden Removal of Pressure Gradient" - NPL AERO RE 1145 (1965).

21. P. BRADSHAW : "The Response of a Constant Pressure Turbulent Boundary Layer to the Sudden App1ication of an Adverse Pressure Gradient" NPL AERO Re 1219 (1967).

22. G. SCHUBAUER, P. KLEBA~"'r'F : "Investigation of Separation of the Tur-bulent Boundary Layer" - NACA Re 1030 (1952).

23. J. GOSTELOW, A. LEWTOWICZ, M. SHAALAN : "Viscosity Effects on the Two-dimensional Flow in Cascades" - ARC CP No. 872 (1967).

n

I

16 J

L

Curve based on experimental result!:

/

I. Constant pressure

2. Bradshaw's first experiment

/

3. Clauser's first experiment

4 Bradshaw~ second experi ment

5 Clauser's sec.ond experiment

I+-Points tranfered from figure 3

14 12

J

V

10 8

IJ

"

..

Theor~ of le Foll 6

~/;/

//

4 rJ.:

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Downstream stability

l'"''

.'- Theor

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y

of Gibso n and Mellor

I

4 2 3

lil

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o

5

~v

THE

10

CURVE

15 20 25

n

(G) AS

PREDICTED

BY

30

35

DIFFERENT

I

_

.

Figure

1.

40

.G

45

THEORIES.

w~I~----~--~----~--_+----+_--_T~~~

·41

~ ~

I

The points corres»ponding to the

experi mentally studied equili bri u m

.3

r=

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boundary layers

I

I

.2

.J

•

o

I

I

I

I

0

.001

.002

.0)3

.004

-L-

Points to be

transfered in the

I

n

(G) - Plane

I

Figure

2.

lw'n

.005

.006

.007

.008

.009

.OJO

- - - Zero pressure gradient equilibrium boundary layer of Klebanoff

VT

I ---

Light adverse pressure gradient equilibrium boundary layer

of

Bradshaw

U

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ó'It

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Sev~re

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pre~surE?

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calcula

.

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Figure 3.

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,

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o

0.2

0.4

0,6

0,8

1.0

V/Ó.995

1.2

EDDV VISCOSITY DISTRIBUTION FOR DIFF ERENT EQUILI BRIUM BOUNDARV

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TAKEN

FROM

BRADSHAW '5

REPORT

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