Multielement Airfoil Flows

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CALCULATION OF MULTIELEMENT AIRFOIL FLOWS,

INCLUDING FLAP WELLS

by

Tuncer Cebeci*, Eric Besnard** and Hsun H. Chen***

Aerospace Engineering Department

California State University, Long Beach

* Professor and Chair, AIAA Fellow**

Graduate Student***

Professor, AIAA Associate FellowCopyright 1996 by the American Institute of Aeronautics andAstronautics Inc. All rights reserved.

Abstract

A calculation method for multielement airfoils

based on an interactive boundary-layer approach

using an improved Cebeci-Smith eddy viscosity

formulation is described. Results are first presented

for single airfoils at low and moderate Reynolds

numbers in order to demonstrate the need to calculate

transition for accurate drag polar prediction and the

ability of the improved Cebeci-Smith turbulence

model to predict flows with extensive separation, and

therefore to predict maximum lift coefficient, (cl)max.

Results, in terms of pressure distributions and lift and

drag coefficients, are presented for a series of two-

element airfoils with flaps or slats. The method is

extended to the computation of configurations with

flap wells. Results show that the same accuracy can be

reached as for faired geometries. A slight

compressibility effect was accounted for by

introducing compressibility corrections to the Hess-

Smith panel method. Again, the importance of the

compressibility effects and the turbulence model on

stall, and the need to calculate the onset of transition,are demonstrated. Recommendations are then made

for the preferred approach to predicting the

aerodynamic performance of multielement airfoils for

use as a practical and efficient design tool.

1. Introduction

In recent years, there have been significant

accomplishments in computational fluid dynamics.

Whereas in the early 1960s calculations performed

for airfoil flows with panel methods and boundary-

layer methods were under development for simple

flows, today the calculations are being performed

routinely with Navier-Stokes methods not only for

airfoil flows but also for complex aircraft

configurations. Our capabilities in aerodynamic flows

have reached levels which were difficult to imagine in

the early 1960s. Progress has been so great and so

rapid that one may even say that computational fluid

dynamics has reached, or is very near to reaching, its

maturity.

Despite these advances, however, there is still

more to be done in developing design methods for

high lift configurations. The presence of high and

low Reynolds number flows on various components of

airfoils, including flap wells and significant regions of

flow separation near stall conditions as well as

possible merging of shear layers, offer significant

challenges to code developers. The required generality

and accuracy of the code and, equally important, its

efficiency as a design tool, add additional challenges.

The current development of design algorithms forhigh-lift devices follows two approaches. One

approach pursues the solution of the Navier-Stokes

equations with structured and unstructured grids.

Some of the Navier-Stokes methods used for this

purpose are based on the solutions of the

incompressible flow equations [1, 2] while others are

based on the solutions of the compressible flow

equations [3]. The incompressible form of the

equations employing the numerical procedure of

Rogers et al. [1] provides accurate results and allows

the calculations to be performed efficiently. The


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solution of the compressible equations, on the other

hand, takes a considerable amount of computer time;

their convergence rate is slower and, at this time, they

are in the research stage.

The other approach for developing methods for

high lift configurations is based on the interactive

boundary layer theory, which involves interaction

between inviscid and boundary-layer equations [4, 5].

In this approach the inviscid flow is often computed

by a panel method with or without compressibility

corrections, and the viscous flow is computed by a

compressible boundary-layer method. This approach

is very efficient but is not as general as the Navier-

Stokes approach for more complicated geometries.

Regardless of which approach is used to develop

a computational tool for high-lift configurations, it is

necessary to include the compressibility effect if the

calculation method employs incompressible flow

equations, and to calculate the onset of transition.Both can strongly influence the accuracy of the

calculation method. Experiments on single or

multielement airfoils show that the compressibility

effect, even at Mach numbers around 0.30, has a

pronounced effect on the maximum lift coefficient.

Similarly the predicted location of transition in the

calculation method is important in properly

identifying the effects of wind tunnel and flight

Reynolds numbers. Individual components of multi-

element airfoils at wind tunnel Reynolds numbers can

experience relatively lower Reynolds numbers than

the main airfoil. At chord Reynolds numbers less than

500,000, the components can have large separation

bubbles, with the onset of transition occurring inside

the separation bubble [6]. As a result, the behavior of

the flow can be significantly different from the

behavior of the flow on the main airfoil at higher

Reynolds numbers. Furthermore, the transition

location can drastically influence the drag coefficient,

and therefore determining the onset of transition is

crucial for predicting drag polars, which are of major

interest to aircraft designers.

Section 2 describes an interactive-boundary-layer

approach to the calculation of high lift configurations

in two-dimensional flows. It is applied to a variety ofhigh lift systems, including airfoils with flaps and

slats, and airfoils with and without flap wells. Results

are presented and discussed in Section 3. Finally, in

view of the present results, a preferred approach for

predicting the flow about multielement airfoils is

discussed in Section 4.

2. Calculation Method

2.1 Inviscid Method

The inviscid flow field is computed by the Hess

Smith panel method [7]. Three options were used in

the multielement panel method. As in a single airfoil

panel method, in one option the vorticity strength was

taken to be constant on all panels, and a single valuewas adjusted to satisfy the condition associated with

the specification of circulation. Multielement

configurations computed with the calculation method

employing the multielement panel method with

viscous corrections showed that, while the results

were in good agreement with experimental data for

conventional airfoils, the results for airfoils with thin

trailing edges and/or supercritical airfoils were not.

To improve the results, two additional options for the

vorticity distribution were incorporated into the panel

method. One option assumes that vorticity varies

quadratically with the surface distance and the second

option assumes that it varies quadratically in a smallregion near the trailing edge. In this latter option, to

be referred to as the partially parabolic vorticity

distribution option, the size of the region is to be

specified in the panel method. The compressibility

correction depends upon the linearized form of the

compressible velocity potential equation and is based

on the assumption of small perturbations and thin

airfoils. The correction used in the present panel

method is based on the Prandtl-Glauert formula as

described in [7].

2.2 Inverse Boundary Layer Method

Boundary layer equations

The compressible boundary layer equations

(mass, momentum and energy) for laminar and

turbulent flows are well known and, with the

algebraic eddy viscosity ( )m and turbulent Prandtlnumber (Pr )t formulation of Cebeci and Smith [7],

they can be expressed as follows:

x

uy

( ) ( )+ =v 0 (1)

uu

x

u

yu

du

dx y

u

ye ee

m+ = + +

v ( ) (2)

uH

x

H

y yk c

T

yu

u

ypm

tm+ = + + +

v (

Pr) ( ) (3)

where T is the temperature, H is the total enthalpy

given by

H c Tu

p= +2

2(4)

and

v v v= + ' ' (5)


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In the absence of mass transfer, the boundary

conditions for an adiabatic surface are

y = = =0 0 0, ,u v ,H

y= 0 (6a)

y u xe , ( )u , H He (6b)In the wake, where a dividing line at y = 0 is

required to separate the upper and lower parts of theinviscid flow, in the absence of normal pressure

gradient, the boundary conditions at y = 0 become

yu

= = =0 0 0, ,y

v

(7)

Solution procedure

The above equations are first expressed in

transformed coordinates. Two sets of transformed

coordinates are used, one for the direct problem when

the equations are solved for the prescribed pressure

distribution, and the other for the inverse problem

with the external velocity updated during theiterations. The Falkner-Skan transformation is used in

the direct mode and a modified version is used in the

inverse mode. Reference 7 presents and discusses the

transformed equations and their solution procedure in

detail. The airfoil is divided into upper and lower

surfaces. For each surface, the calculations start at the

stagnation point and proceed in the standard mode up

to a certain specified x-location. Then, the inverse

calculations begin from that location and continue up

to the far wake.

Interaction law

The boundary layer equations become singular at

flow separation and do not allow the calculation of

separated flows for a prescribed velocity distribution.

However, when the external velocity is treated as an

unknown, this difficulty can be overcome as discussed

in [7]. The external velocity is represented by

u x u x u xe eo

e( ) ( ) ( )= + (8)where ue

o is the inviscid velocity computed by the

panel method and ue is the perturbation velocity dueto viscous effects, which is given by the Hilbert

integral

u x dd

u dx

e

x

x

e

a

b

( ) ( . ).*=1 (9)

in the interaction region (a, b). Its evaluation is

described in detail in [7].

The flow calculations in the flap well region

which are similar to those for a backward facing step

require modifications. A large portion of the flow

separates immediately after the sudden change of the

airfoil geometry. The flow reattaches and gradually

recovers downstream in the flap well region or in the

wake. The calculation of such flows is difficult, and

potential theory is not adequate because of the

singularity that occurs at the geometry discontinuity

and the strong viscous effects in the separated region.

To compute such flows, the interaction law has to bemodified. First, a fairing is placed in the flap well

region, and the Hilbert integral formulation is used to

generate initial guesses for the displacement thickness

distribution. Then, the relaxation formula

( ) ( )* * + = +

1 1 1

u

u

ev

ei

(10)

where is actually the sum of the distance from thefairing to the flap well cove and the displacement

thickness measured from the fairing, is used in the

inverse method to replace the Hilbert integral

formulation of the external boundary condition. The

new edge boundary conditions are given by Eq. (6b)and Eq. (10), where uev and uei correspond the

external velocities computed by the boundary layer

and inviscid methods, respectively, and is arelaxation parameter. At the end of the flap well, the

solution procedure reverts to the Hilbert integral

approach.

Once the boundary layer development is

computed, the blowing velocity distribution is

determined from

vn ed

dxu= ( . )* (11)

and used as a boundary condition in the inviscid

method. This interactive procedure is repeated untilconvergence of the lift coefficient is achieved.

Turbulence model

An improved version of the algebraic eddy-

viscosity formulation of Cebeci-Smith is used here.

The eddy viscosity distribution across the boundary

layer is defined by two separate expressions,

m

m i tr c

m o e tr c

yy

A

u

yy y

u u y

=

=

=

( ) . exp . .

( ) ( )dy . .

0 4 1 0

2

0

y

(12)

where

=0 0168

1 5

..

F, A

w

=

26

1

, u

=

max

12

(13a)

where F is related to the ratio of the product of the

turbulence energy by normal stresses to that by shear

stress evaluated at the location where the shear stress

is maximum. As discussed in [7], it is given by


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Fu x

u y= 1

(14)

where the parameter is a function ofR u vt w= ( ' ')max , which, for w 0 , is

represented by

=+

+

61 2 2

10

110

R R

R

R

t t

t

t

t

t

( ).

.

R

R

(15)

Forw t 0, R is set equal to zero.Also, whereas in the original Cebeci-Smith eddy

viscosity formulation the intermittency expression was

valid only for zero pressure gradient flows, the

expression in the improved formulation is applicable

for flows with favorable and adverse pressure

gradients as well as zero pressure gradient flows. It is

based on Fiedler and Heads correlation [8] and is

given by( )

=

1

21

2erf

y Y(16)

where Y and are general intermittency parameterswith Y denoting the value of y where = 0.5 and denoting the standard deviation. For details, see [7].

The condition used to define yc is the continuity

of the eddy viscosity, so that m is defined by ( ) m ifrom the wall outward (inner region) until its value is

equal to that given for the outer region by ( )m o .The expression tr models the transition region

and is given by

tr trex

x

G x xdx

utr

=

1 exp ( ) (17)

Here x tr denotes the onset of transition and G is

defined by

GC

uR

e

x tr=

32

3

2

1 34

.

(18)

where C is 60 for attached flows and the transition

Reynolds number is u xx e trtr= . In the low

Reynolds number range from Rc = 2 10 toRc = 6 10 , the parameter C is given by

( )C Rxtr

2213 4 7323= log . (19)

The corresponding expressions for the eddy-

viscosity formulation in the wake are

[ ] m m w m t m w

x x= +

( ) ( ) ( ) .exp.e.

0

20(20)

where ( ) .e. m t is the eddy viscosity at the trailing edgecomputed from its value on the airfoil and ( )m w isthe eddy-viscosity in the far wake given by the larger

of

( ) . ( ).

min

m wl

e

y

u u dy= 0064 (21a)

and

( ) . ( ).min

m wu

ey

u u dy=

0 064 (21b)with ymin denoting the location where the velocity is

minimum.

For high Reynolds numbers, if the flow is

attached, the onset of transition is determined by

Michel's criterion as described, for example, in [7]. At

high angles of attack, the flow separates downstream

of the pressure peak before Michel's criterion can be

satisfied. Therefore, the onset of transition is chosen

to coincide with laminar separation.

At Reynolds numbers less than 106 , where large

separation bubbles may be present even at low angles

of attack, it is necessary to calculate the onset oftransition by the en-method as discussed in [7]. For

high lift configurations, flaps usually have low

Reynolds numbers. Fortunately, the pressure

distribution on the flap upper surface does not vary

greatly with angle of attack, and therefore the

computation of the transition location does not have

to be performed at each angle of attack.

3. Results and Discussions

3.1 Single Airfoils

Before we present a sample of results for high lift

configurations, it is useful to discuss the role of

transition by examining the results shown in Figs. 1

and 2 for two airfoils at high and low Reynolds

numbers. The results in Fig. 1 are for the NACA

0012 airfoil at a chord Reynolds number of 3 x 106.

As can be seen, there is a significant difference

between the calculated lift coefficients in which the

transition location was fixed near the stagnation point

for all angles of attack or computed. A similar

observation can be made for the drag coefficients.

Those calculated with the transition computed show amuch better agreement with experimental data than

those in which the transition was fixed.

Fig. 2 shows the results for the Eppler airfoil at a

chord Reynolds number of 200,000. Here, the location

of transition does not play an important role in

predicting lift coefficients (Fig. 2a). However, the

calculations with a fixed transition location can be

performed until = 13.5o. As before, the drag


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coefficients obtained with the transition location

with a fixed transition location (Fig. 2b).

(a)

0.0 5.0 10.0 15.0 20.0

0.0

0.5

1.0

1.5

2.0

Cl

(b)

0.0 0.5 1.0 1.5Cl

0.000

0.010

0.020

0.030

0.040

cd

measurements

transition computed

transition at the

leading edge

Fig. 1. Force coefficients for the NACA 0012 airfoil at

Rc = 3 x 106, (a) lift and (b) drag.

(a)

-2.0 3.0 8.0 13.0 18.0

0.0

0.5

1.0

1.5

measurements

transition calculated

transition at the leading edge

Cl

(b)

-5.0 0.0 5.0 10.0 15.0

0.00

0.02

0.04

0.06

0.08

0.10

cd

Fig. 2. Force coefficients for the Eppler airfoil at Rc =

200,000, (a) lift and (b) drag.

3.2 Two-element Airfoils without Flap Well

A sample of results is presented below for three

two-element configurations without flap wells.

Figs 3 and 4 show the results for the NLR 7301

supercritical airfoil/flap configuration. The

experimental data of Van den Berg and Oskam [9]

include pressure distributions at = 6o and 13.1o, alift curve and a drag polar. A flap of 32% chord was

used at a deflection angle of 20o and with a gap of

2.6% chord. The experimental freestream Mach

number was M = 0.185, and the chord Reynolds

number was 2.51 x 106. Fig. 3 shows a comparison

between measured and computed pressure

distributions at = 6.0o and = 13.1o, and Fig. 4shows a similar comparison for the lift coefficient and

drag polar. The inviscid flow calculations wereperformed using the partially parabolic vorticity

option, and the viscous flow calculations were

performed with the onset of transition location

calculated on the main element and flap. As can be

seen from the results in Fig. 4a, the calculated results,

including drag, agree well with experimental data.

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/c

-2.0

0.0

2.0

4.0

6.0

8.0

-C

MeasurementsCalculations


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(b)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/c

-2.5

0.0

2.5

5.0

7.5

10.0

12.5

-Cp

Fig. 3. Pressure distribution for the NLR 7301

supercritical airfoil/flap configuration for (a) = 6o

and

(b) = 13.1o.

(a)

0.0 5.0 10.0 15.0 20.01.5

2.0

2.5

3.0

3.5

cl

(b)

Measurements

Calculations

1.5 2.0 2.5 3.0 3.50.00

0.02

0.04

0.06

0.08

0.10

cd

clFig. 4. Force coefficients for the NLR 7301 airfoil/flap

configuration, (a) lift coefficient, and (b) drag polar.

Figures 5 to 9 show the results for two NASA

high lift configurations tested by Omar et al. [10] for

a freestream Mach number of M = 0.201 and chord

Reynolds number of 2.83 x 106. As in the NLR 7301

multielement configuration, the inviscid flow

calculations were performed using the partially

parabolic vorticity option. In addition the

compressibility corrections were made to the inviscid

flow using the Prandtl-Glauert formula.

Figure 5 shows the pressure distributions for the

airfoil/flap configuration at = 0.01o and 8.93o. It isseen that the calculated results are in good agreement

with experimental data, although there is some

discrepancy on the upper surface of the flap. This

discrepancy is due to two phenomena, the merging of

shear layers and the inaccurate wake center lineprediction, as shown in Figure 6. It is seen that in the

experiments, the merging occurs at the trailing edge

of the flap for both = 0o and = 8o, and that it isnot predicted by the present method. Also, in the

present method, the wake center line is assumed to be

the dividing streamline merging from the main

element trailing edge and is closer to the flap upper

surface than the measured one (thick line in Figure

6), which causes a greater predicted flow acceleration

on the flap upper surface than the actual acceleration.

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/c

-2.0

0.0

2.0

4.0

.

-Cp

Meas rements

Calc lations

(b)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/c

-4.0

0.0

4.0

8.0

12.0

16.0

20.0

-Cp

Critical Pressure Coefficient

Fig. 5. Pressure distribution for the NASA airfoil/flap

configuration at (a) = 0.01o and (b) = 8.93o.


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(a)

computed

easured

measured merging

wake center lines

(b)

measured merging

shear layer edges

Fig. 6. Comparison of measured and computed shear

layers for the NASA airfoil/flap configuration, (a) = 0o

and (b) = 8o.

Figure 7 shows the lift and drag coefficients for

the same configuration. The lift coefficient is slightly

over predicted due to the discrepancies on the flap

upper surface, and similarly the drag coefficient is

slightly under predicted. Also, the calculated

incompressible stall angle is rather different from the

calculated compressible one. The critical pressure

coefficient for M = 0.201 is indicated in Fig. 5b. At

= 8.93o, the measured stall angle, there exists asmall region of supersonic flow and a shock may

occur. Even though this shock cannot be predicted by

the panel method, the compressibility corrections that

are applicable at lower angles of attack provide a

significant improvement to the stall prediction.

(a)

-10 -5 0 5 10 15

0.0

1.0

2.0

3.0

Measurements

IncompressibleCompressible

cl

(b)

cd

cl0.0 1.0 2.0 3.0

0.00

0.05

0.10

0.15

Fig. 7. Force coefficients for the NASA airfoil/flap

configuration, (a) lift coefficient, and (b) drag polar.

Figures 8 and 9 show the results for a slat/airfoil

configuration. Figure 8 presents a comparison

between measured and calculated pressure

distributions at 26.88o. While the agreement is

excellent for the slat, the pressure peak on the main

airfoil is slightly under-predicted. Figure 9 shows the

lift and drag coefficient comparisons for the same

configuration. The agreement with data is excellent

up to stall for the lift coefficient and the drag polar is

predicted satisfactorily. The computed maximum lift

coefficient and stall angle are 2.88 and 27.5o,

respectively, and the measured ones are 2.90 and

26.9o. This gives a 0.7% error for the maximum lift

coefficient and an error of 2.2% for the stall angle.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

x/c

-2.0

2.0

6.0

10.0

14.0

18.0

-CpMeasurements

Calculations

Fig. 8. Pressure distribution for the NASA slat/airfoil

configuration at = 26.88o.


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(a)

10.0 15.0 20.0 25.0 30.0

0.0

1.0

2.0

3.0

Total

MainElement

Slat

cl

(b)

Measurements

Calc lations

0.0 1.0 2.0 3.00.000

0.025

0.050

0.075

0.100

0.125

0.150

cl

cd

Fig. 9. Force coefficients for the NASA slat/airfoil

configuration, (a) lift coefficient and (b) drag polar.

3.3 Two-element Airfoil with Flap Well

The method described in Section 2 was applied to

an airfoil/flap configuration tested by Omar et al. [10]

similar to that of the previous section, but comprising

a flap well cove to illustrate the applicability of the

method to airfoils with flap wells.As mentioned in Section 2, the calculations are

performed in several steps. First, a fairing is assumed

in the flap well region. Inviscid calculations are then

performed for this faired geometry and an external

velocity distribution is obtained. The boundary layer

equations are solved over the airfoil for this pressure

distribution with the Hilbert integral inverse

formulation of Section 2 so that an initial

displacement thickness in the flap well region can be

obtained. Then, in the flap well cove, the inverse

formulation employing the relaxation formula of Eq.

(10) is used starting with this initial displacement

thickness distribution added to the distance from theflap well cove to the fairing. After several iterations, a

new displacement thickness distribution in the flap

well cove is determined. The corresponding

distribution over the faired airfoil is calculated and

simulated into the inviscid method with the blowing

velocity. As in the cases without flap well, several

inviscid/viscous iterations are performed until

convergence.

Figure 10 presents the geometry in the flap well

region and the fairing used for the calculations.

Results in terms of computed displacement thickness

are shown for = -8.11o and = 8.59o. As expected,the recirculation region is much greater for = -8.11o

than for = 8.59o. Figure 11 presents the

corresponding skin friction coefficient variation in theflap well region and demonstrates the ability of the

method to compute large regions of separated flow.

(a)

fairingdisplacemen hickness

(b)

Fig. 10. Computed displacement thickness in the flap

well region for (a) = -8.11o and (b) = 8.59

o.

Figure 12 shows a comparison of measured and

computed pressure distributions in the neighborhood

of the flap well cove and on the flap for = -8.11o, = -0.02o and = 8.59o. Comparing measurementswith those for a smooth geometry (Fig. 5), it is seen

that the presence of the recirculating flow essentiallycauses an almost constant pressure distribution in the

flap well cove and increases the flow velocity in the

vicinity of the flap leading edge.

0.65 0.70 0.75 0.80 0.85

x/c

-0.005

0.000

0.005

0.010

Cf

= 8.11 = 8.59

main element

trailing edge

beginning of

flap well

Fig. 11. Computed skin friction coefficient in the flap

well region for = -8.11o and = 8.59

o.


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(a)

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

x/c

2.0

1.0

0.0

-1.0

-2.0

-3.0

Cp

measurementsith flap ell calc.ithout flap ell calc.

(b)

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

x/c

2.0

1.0

0.0

-1.0

-2.0

-3.0

Cp

(c)

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

x/c

2.0

1.0

0.0

-1.0

-2.0

-3.0

Cp

Fig. 12. Comparison of measured and computed

pressure distributions in the flap well neighborhood for

(a) = -8.11o, (b) = -0.02

oand (c) = 8.59

o.

Figure 12 shows that flap well results are better

than those obtained with the fairing which ignored

the presence of the flap well. On the main element,

while the agreement is satisfactory for = -8.11o, theregion of recirculating flow is under predicted for =

-0.02o

and = 8.59o

. This may be due to severalfactors, such as the inappropriateness of the

interaction law and of the turbulence model for such

recirculating flows. Work is in progress to improve

the results. However, the pressure distribution on the

flap upper surface, in particular close to the leading

edge, is satisfactory, which suggests that the

displacement thickness prediction is accurate near the

trailing edge of the main element.

Lift and drag coefficients were computed but are

not shown here since they are essentially the same as

for the case without flap well (Fig. 7).

4. Concluding Remarks

An interactive boundary-layer method with an

improved Cebeci-Smith eddy viscosity formulation is

used to calculate the aerodynamic performance

characteristics of high lift devices. The results show

good agreement with measurements for a variety of

multielement airfoils comprising slats, main element

and flaps. The method has also been extended to the

computation of airfoils with flap wells, and the results

show the same overall level of accuracy. In general,

the predictions are excellent for relatively low angles

of attack and very satisfactory up to stall. The study

shows that the onset of transition location plays a

significant role in predicting drag and that itscalculation must be a part of the computational

method. The study also shows that for some flows

merging of shear layers takes place and, since the

present method does not account for this merging, the

calculated results near stall conditions, while

satisfactory, are not as good as those at lower angles

of attack.

The stall prediction is satisfactory for most cases.

However, when the pressure peak on the main

element reaches values close to or greater than the

critical pressure coefficient, a truly compressible

inviscid method, such as a full potential or Euler

method, should be coupled with the presentcompressible boundary layer method in order to

compute stall with reliability. Work is in progress to

incorporate these capabilities, including the merging

of shear layers.

References

1. Rogers, S.E., Wiltberger, N.L. and Kwak, D.,

Efficient Simulation of Incompressible Viscous

Flow Over Single- and Multielement Airfoils,

AIAA Paper No. 92-0405, 1992.2. Rogers, S.E., Progress in High-Lift Aerodynamic

Calculations, AIAA Paper No. 93-0194, Jan.

1993.

3. Valarezo, W.O. and Mavriplis, D.J., Navier-

Stokes Applications to High-Lift Airfoil Analysis,

AIAA Paper No. 93-3534, Aug. 1993.

4. Cebeci, T., Calculation of Multielement Airfoils

and Wings at High Lift, AGARD Conference


10/10

AAIIAAAA 9966--00005566

10

Proceedings 515 on High-Lift Aerodynamics,

Paper No. 24, Banff, Alberta, Canada, Oct. 1992.

5. Cebeci, T., Besnard, E. and Messi, S., A

Stability/Transition-Interactive-Boundary-Layer

Approach to Multielement Wings at High Lift,

AIAA Paper No. 94-0292, Jan. 1994.

6. Cebeci T, Essential Ingredients of a Method forLow Reynolds-Number Airfoils, AIAA Journal,

Vol. 27, pp. 1680-1688, 1983.

7. Cebeci, T., An Engineering Approach to the

Calculation of Aerodynamic Flows, to be

published, 1996.

8. Fiedler, H. and Head, M.R., Intermittency

Measurements in the Turbulent Boundary Layer,

J. Fluid Mech., Vol. 25, Part 4, pp. 719-735, 1966.

9. B. Van den Berg and B. Oskam, Boundary Layer

Measurements on a Two-Dimensional Wing with

Flap and a Comparison with Calculations,

AGARD CP-271, Sept. 1979.

10.E. Omar, T. Zierten and A. Mahal, Two-Dimensional Wind Tunnel Tests of a NASA

Supercritical Airfoil with Various High Lift

Systems, NASA CR-2215, 1977.

Multielement Airfoil Flows

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