AIAA 94-2376 PREDICTION OF TRANSITION LOCATION FOR A 2-D NAVIER-STOKES SOLVER FOR MULTI-ELEMENT AIRFOIL CONFIGURATIONS

Kazuhiro Kusunose and Hoa V. Cao Boeing Commercial Airplane Group Seattle, Washington

25th AIAA Fluid Dynamics Conference

June 20-23, 1994 / Colorado Springs, CO For permlsslon to copy or republlsh, contact the Amerlcan Institute of Aeronautlcs and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

..-. PREDICTION OF TRANSITION LOCATION FOR A 2-D NAVIER-STOKES

SOLVER FOR MULTI-ELEMENT AIRFOIL CONFIGURATIONS

Kazuhiro Kusunose’

Roeing Commercial Airplane

Abs t r ac t

A transition prediction method for 2-D Navier-Stokes flow analysis has been developed. Three different tran- sition mechanisms are identified for multi-element air- foil boundary layers: the growth of unsteady Tollmien- Sclilichting waves; laminar boundary layer separation; and transition due to turbulence contamination.

Laminar boundary layer properties are computed by solving the boundary layer integral equations with the pressure distribution of the Navier-Stokes solution as thc input and used in the prediction of transition due to Tollmien-Schlichting waves or laminar boundary

~, layer separation.

Contamination by the turbulence in an adjacent shear layer is simulated using the Spalart-Allmaras turbu- lence modcl. Thus, transition may occur upstream of the transition point associated with the first two mech- anisms.

Numerical results for two-element and three-element coilfigurations using the NASA Ames I N S 2 D in- compressible Navier-Stokes code with the Spalart- Allmaras turbulence model are compared with exper- imental data. Agreement is generally good.

1. Introduction

In the last several years, 2-D Navier-Stokes code c a pabilities for a multi-element airfoil configuration has been significantly improved. Two workshops have been conducted by NASA: the Ames ”Flow Physics Issues in a High-Lift System” (May, 1991) and the Lan- gley ”High-Lift Flow Physics Workshop” (May, 1993). Tlicse greatly stimulated the high-lift community to

‘Senior Specialist Engineer f Senior Specialist Engineer Copyriglit 01994 by The Boeing Company. Published by

tlic American Institute of Aeronautics and Astronautics, Inc. w i t h permission.

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and Hoa V. Caot

Group, Seattle, Washington

advance the application of Navier-Stokes methodology to multi-element airfoil analyses. During the Lang- ley workshop, most of the participating Navier-Stokes codes predicted flow of the McDonnell Douglas three- element airfoil [l] well compared to the experimental data.

With numerical-algorithm and turbulence-model de- velopments getting closer to their maturities, the fol- lowing two issues have become important in the quest t o improve the accuracy of 2-D Navier-Stokes codes. One is the grid quality, and the other is the predic- tion of boundary layer transition. I t should be noted that turbulence modeling and transition prediction are two distinctly different flow modeling problems, except when transition occurs through confluence, or ”con- tamination”.

A successful Navier-Stokes analysis for a multi-element airfoil configuration requires a correct rendition of the complex flow physics. This includes laminar boundary layer flow, boundary layer transition, and turbulcnt boundary layers flow; flow separation and reattach- ment; as well as the interaction between wakes and boundary layers. To achieve this it is necessary to pro- vide a physically adequate grid in the critical areas[2] and correct boundary layer transition modeling.

The necessity of having a free-transition capability in a viscous flow analysis code for a multi-element airfoil configuration has been clearly demonstrated in [2,3]. Depending on the specified transition location, a wide range of flow solutions can be obtained from fully at- tached to massively separated. Transition-prediction capability is the final touch for a Navier-Stokes code in order to predict the real flow around a multi-element airfoil. In this paper, we present a transition predic- tion method for 2-D multi-element airfoil flows using the NASA Ames INSZD incompressible Navier-Stokes code [4]. Predicted transition locations for two multi- element airfoil configurations (a two-element NLR- 7301 [5] and a three-element high lift airfoil [I]) are

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compared with experimental data

2. Trans i t ion Mechanisms

Our studies have indicated that three different tran- sitioii mechanisms need to be considered for the pre- diction of the boundary layer transition point for 2- D flows. The first mechanism is the growth of the uiisteady Tollmien-Schlichting waves in the laminar boundary layer,where the Reynolds number plays an important role. The second one is the laminar hound- ary layer separation, which is a weak function of the Reynolds number. The final one i s transition due to turbulence contamination,

The commonly used "e"" method, which assumes that the transition will occur when the disturbance am- plification ratio has grown to be en , can predict the transition onset point if transition is caused by the Tollmicn-Schlichting wave instability.

For a typical multi-element airfoil flow, a strong suc- tion pcak exists near the leading edge on the upper sur- face of most elements. In this case, a laminar bound- ary layer separation is most likely to occur a t a point downstrcam of the suction peak, and the separation causes transition. Since the transition process is pri- marily due to the bursting of the vortices born in the laminar separatcd shear layer, the transition is very fast.

In the case in which a contact between a laminar boundary layer and a turbulcnt wake occurs (it will happen frequently for a multi-element airfoil configu- ration), the transition due to the Tollmien- Schlichting wave or thc laminar boundary layer separation may be "bypassed" because of the oncoming turbulence con- tamination caused by the wake of the leading airfoil elenient.

For these reasons, it is necessary to check all three transition mechanisms to reliably predict transition for a multi-clement airfoil configuration.

2.1 Transition Due to Tollmien-Sclrlichting Wavc or L a m i n a r Scparation

For the case where turbulence contamination is ab- sent, the onset point of transition is assumed to oc- cur either a t the point where the unstable Tollmien- Schlichting wavc has grown by a factor e' (known as the e9 method) or at the point where laminar bound- ary layer separation is encountered. Laminar bound- ary layer separation i s declarcd when eitbcr the skin friction cf is negative, or the shape factor €I exceeds 3.5.

Since the e' method assumes that the transition occurs when the most unstable Tollmien-Schlichting wave in the boundary layer has grown by some factor, e', it is necessary to calculate the amplification factor of the most unstable Tollmien-Schlichting wave. To do so, the disturbance growth rates must be related to the local laminar boundary layer parameters.

Drela and Giles [6] , using the Falkner-Skan profile fam- ily, have solved the Orr-Sommerfeld equation for the spatial amplification rates of a range of shape param- eters and unstable frequencies. The logarithm of the maximum amplification ratio n is calculated by intc- grating the local amplification rate downstream from the point of instability, simply assuming that the slope of the maximum amplification rate dn/dRee is only a function of local calculated shape factor H and is given by the corresponding self-similarity solution.

-'

Here, Res is the Reynolds number based on the mo- mentum thickness. The slope d n / d R e s and the critical Reynolds number Rea0 are expressed hy the following formulas

dn - = 0.01[{2.4EI-3.7+ dRee

2.51anh(1.5H - 455)) ' + 0.25]0.5 ( 2 )

1.415 20.0 H - 1 H - 1

log,,Reso = (- - O.489)tanh(- - 12.9)

+- 3'295 + 0.44 H - l

2.2 'Ikansition by Contamination

This type of transition is naturally predicted by utilia- ing the Spalart-Allmaras turbulence model [7] or any other transport model. In the Spalart-Allniaras model, a source term is added at the transition onset point in ordcr to obtain transition. The source term will be nonzero only in a small domain of influence, which should not extend outside of the boundary layer.

The transition onset point is specified, as previ- ously discussed, either where the unstable Tollmien- Schlichting wave has grown by a factor e', or where laminar boundary layer separation is encountered, as- suming that the turbulence contamination level is neg- ligihle. However, when the calculated turbulence level

,-

in the incoming boundary layer exceeds the magni- tude of the source term at the specified transition on- set point, transition will naturally be triggered at a point upstream of the specified transition point.

Currently, in the Johnson-King [8] or Baldwin-Barth [9] t.urbulence models, the laminar boundary layers are distinguishcd from bhe turbulent boundary layers by ”ovcr-writing” the eddy viscosity with zero in a region upstrcam of the specified onset transition point. It is, in general, not an easy task especially for multi- block or unstructured grids. The over-writing will also prevent the contamination process.

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3. Calculat ion of Lamina r Boundary Layer Properties

A difficult problem for applying the transition predic- tion method to a Navier-Stokes solver is the treatment of the boundary layer parameters (momentum thick- ness, shape factor, skin friction, etc.) of the boundary layer near the transition point. Transition causes the boundary layer velocity profile to change from laminar to turbulent, with a largc impact on the value of the boundary layer parameters (especially for the shape factor).

If thc boundary layer parameters are calculated by us- ing the velocity profile from a Navier-Stokes solution, a meaningful new transition point cannot be found in thc region downstream of the current transition point [lo]. Laminar boundary layer quantities must be pro- vided even downstream of the current transition point in order to allow a downstream movement of predic- tion point during the iterative solution.

In the present method, the boundary layer momentum iutegrsl equation

-

(4) d0 0 dU, cf d s We d s 2 - + ( H + 2 ) - - = -

with two additional closure conditions (shear and shape factor correlations) are utilized to calculate the laminar boundary layer parameters. 8, 11, Ue and cf are the momentnm thickness, the shape factor, the outer-edge velocity, and the skin friction coefficient, rcspectively; s is the coordinate along the body sur- face.

Thwaites’ method [ll] has been applied to solve the system of the momentum integral equations in or- der to estimate the equivalent laminar properties. In Thwaites’ method, it is assumed that the boundary layer is laminar for a given surface pressure distribu- tion. The outer-edge velocity Ue distribution, which

.-

is required as an input t o the Thwaites’ method, is estimated by utilizing Bernoulli’s equation [lo] and the surface pressure distribution is obtained from the Navier-Stokes solver. Since the INS2D incompressible Navier-Stokes code is used in the present calculation, the actual Ue distribution is simply given by

(5)

With this method of determining the outer-edge ve- locity distribution, the current method works for both structured and unstructured grids. Note that Bernoulli’s equation fails after separation, but this is of little importance since the boundary layer calcula tion ends there.

4. Implementation of the Transition Model in the INSZD Code

The transition prediction model has been implemented in the NASA Ames INS2D incompressible Navier- Stokes code [4]. The coupling procedure used in the present method is simple. For a given surface pres- sure distribution calculated from the INS2D code, an outer-edge velocity distribution is obtained by utiliz- ing Bernoulli’s equation. Once the laminar bound- ary layer parameters are calculated along the body surface by solving the momentum integral equations, those parameters are used to compute the transition point by checking both the growth of the Tollmien- Schlichting waves and the laminar boundary layer sep- aration. If the incoming turbulence contamination is large enough, transition will be naturally triggered at a point upstream of the specified transition point by the Spalart-Allmaras turbulence model. The transi- tion points are, currently, updated every 10 iterations.

5. Results and Discussion

Numerical solutions using the INS2D code with the Spalart-Allmaras turbulence model are compared to experimental data for a two-element NLR-7301 air- foil [5] and three-element airfoil configuration[l], which was used for the NASA Langley’s High-Lift Flow Physics Workshop in 1993. A total of 6 blocks which contained about 70,000 grid points were used for the NLR-7301 airfoil. A total of 9 blocks with approx- imately 140,000 grid points were used for the three- element airfoil configuration. The method for grid gen- eration used in the current analysis is discussed in [Z].

5.1 Results for Two-Element NLR-7301 Airfoil Case

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Utilizing the currently developed free transition capa- bility, converged solutions for tw.0 different of angles of attack (6 and 13 degrees) are obtained for a Reynolds number of 2.5 million. The geometry of the NLR-7301 airfoil is shown in Figure 1. Figure 2 shows the com- parison of computed pressure distributions with test data [5]. Predicted transition onset points and skin friction coefficients are also compared to the experi- mental data in Figure 3 and 4 , respectively. In gen- eral, agreement between the calculations and test data is good.

Figure 1. Geometry of NLR-7301 2-element airfoil

e = 6.0.

I

0 = 13.1.

0 0 (1.2 0 4 0.6 0.8 1 0

x/c

Figure 2 . Pressure distributions for NLR-7301

The present code captures the separation bubblc which was observed in the experiment. The skin friction coef- ficient shows the numerical evidence of a laminar bub- ble near the suction peak of the main element. Cal- culated velocity profiles across the boundary layer at the bubble location are shown in Figure 5. For the

Figure 3. Transition points on NLR-7301

u - : Cdcula,iao

01 = 13.1' 0 : Expnmenr 0.m

0 021

0.01,

0.OlO

cF om5

om

0.w

dOi0

4.011

0.010 Y,,- .., -

0.w

0.m

CF 0 . m

4.030

*.o>, 0 0

x/c Figure 4. Skin friction on NLR-7301 W

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tested angles of attack (6 and 13 degrees), the flow is basically attached. These cases required about 300 it- erations to obtain a 5-order of magnitude reduction in flow residuals. Convergence history for total lift and the change in flow residual are shown in Figure 6.

-

\

Figure 5. Velocity profiles at the bubble location on NLR-7301

5.2 Resu l t s for Three -E lemen t Airfoil Case

Converged solutions are obtained with free-air condi- tion for Reynolds number 5 and 9 millions. The geome- try of the NASA Langley CFD challenge three-element airfoil [l] is shown in Figure 7. There is no significant flow separation on any of the three elements of the airfoil for the tested angles of attack (8, 16, and 21 degrees), and about 300 iterations are enough for an adequately converged solution (a 4-order of magnitude reduction in flow residuals). For INS2D, 300 iterations for 140,000 points grid requires about one hour of CPU time on the CRAY Y-MP. In general, more iterations will be required to achieve the same level of conver- gence if the size of the separated flow region increases.

Typical convergence histories for those tested angles of attack (8, 16, and 21 degrees) are shown in Figure 8. Computed pressure distributions are compared to the (uncorrected) experimental data [l] in Figure 9. The

-

cL!i , *m ~o

0.0

4 .5 0 103

Number of Iterations

Figure 6. Convergence history for NLR-7301

agreement is fair. Some discrepancies are seen on the upper surface of the flap element and the region near the trailing edge of the main element.

Figure 7. Geometry of NASA Langley CFD challenge 3-element airfoil

The discrepancies are basically caused by the exis- tence of tunnel walls in the experiment. When the actual tunnel walls are modeled in the computation, the agreement is much improved [12]. The effect of the tunnel walls on pressure distributions is shown in Figure 10.

Predicted transition points and transition zones from the experiment are plotted in Figure 11. Predicted transition points compare very well with the data, ex- cept those on the flap element. The transition zone on the upper surface of the flap, in the experiment, is detected a t the leading edge for all three angles of attack. Since the transition zone is located in the

5

I .J.-

CL

rm 2m xa ua 1.0

Number of Iterations

Figure 8 . Convergence history for CFD challenge airfoil

strong favorable pressure gradient region, it is difficult to explain the phenomenon without considering by- pass transition. The flow near the leading edge of the flap may be contaminated from some unknown causes, for instance, surface roughness (introduced by the hot film sensors) or effects of brackets between the main and the flap elements.

Although our numerical results could not predict by- pass transition on the flap element for those tested angles of attack, bypass transition has been obtained by reducing the angle-of-attack to 0 degrees. The cal- culated nonzero eddy viscosity regions including the transition points ate shown in Figure 12, for angles of attack of 16, 8, and 0 degrees. It can be seen that the size of the cove wake is enlarged when the angle of at- tack is reduced. At 0 degree angle-of-attack, both the slat and main element's cove wakes contaminate the boundary layer in the strong favorable pressure gra- dient region (near the leading edge) of the respective trailing elements.

I 4 . P 0.2 (1.6 1.0

X I C

Figure 9. Pressure distributions for CFD challenge airfoil

UPPER AND LOWER WALLS AT 2 CHORDS

SOUD UNE i. WITH TUNNEL WALLS DASHED LINES = FREE AIR SYMBOLSUNE = LTPT TEST DATA

RN;5.E+6 ALPHb16.2 DEG

CP

-0.2 0.2 0.6 1.0 V C

Figure 10. Wind tunnel effects for CFD challenge airfoil W

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Figure l l (a) . Transition points on CFD challenge airfoil

W

6. Conclusions

A transition prediction method for 2-D Navier-Stokes flow analysis has been developed. Three different tran- sition mechanisms, the growth of unsteady Tollmien- Schlichting waves, the laminar boundary layer sepa- ration, and contamination by another share layer, are equally important for multi-element airfoil configura- tions.

To allow a downstream movement of predicted tran- sition point during the iterative solution, the lami- nar boundary layer quantities must be provided even downstream of the transition point. The laminar prop- erties are calculated by solving the momentum inte- gral equation by applying the Thwaites' method to the pressure distribution provided by the Navier-Stokes so- lution.

For the case where the turbulence contamination level is negligible, the onset point of transition is assumed to occur either at the point where the unstable Tollmieu- Schlichting wave has grown by a factor e9 (known as the e9 method) or at the point where the laminar boundary layer separation is encountered. The con- tamination is naturally simulated by using the Spalart- Allmaras turbulence model.

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Figure l l (b) . Transition points on CFD challenge airfoil

Predicted transition points are compared with experi- mental data for two-element and three-element con- figurations using the INS2D incompressible Navier- Stokes code with the Spalart-Allmaras turbulence model. Generally, agreement is good. For the two- element airfoil configuration, the present code captures the separation bubble which was observed in the ex- periment.

Acknowlegements

We acknowledge Drs. J.D. Crouch, P.R. Spalart, N.J. Yu, W.H. Jou and K.K. Ishimitsu for many discussions and comments during the course of the model develop- ment. We also thank Drs. S.E. Rogers and D. Kwak of NASA Ames for providing the original version of INS2D code and their continuous supports.

References

[I] Chin, V.D., Peters, D.W., Spaid, F.W. and McGhee,R.J., "Flowfield Measurements About a Multielement Airfoil at High Reynolds Numbers', AIAA-93-3137, 1993.

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Figure ll(c). Transition points on CFD challenge airfoil

[2] Cao, H.V. and Kusunose, K., "Grid Generation and Navier-Stokes Analysis for Multi-Element Airfoils", AIAA-94-0748, Reno, Nevada, January 1994.

[3] Kusunose, K., Wigton, L. and Meridith, P., "A Rapidly Converging Viscous/Inviscid Coupling Code for Multi-Element Airfoil Configurations", AIAA-91-0177, Reno, Nevada, January 1991.

[4] Rogers, S.E., Wiltberger, N.L. and Kwak, D., "Efficient Simulation of Incompressible Viscous Flow Over Single and Multi-Element Airfoils", AIAA-92-0405,Reno,Nevada,January 1992.

[5] Van Den Berg, B., "Boundary Layer Measure- ments on a Two-Dimensional Wing with Flap", NLR T R 79009 U , 1979.

[GI Drela, M. and Giles, M.B., "Viscous-Inviscid Analysis of Transonic and Low Reynolds Num- ber Airfoils", AIAA Journal, Vo1.25, No.10, Oct, 1987.

[7] Spalart, P.R. and Allmaras, S.R., "A One- Equation Turbulence Model for Aerodynamic Flows", AIAA-92-0439, Reno, Nevada, January 1992.

Figure 12. Nonzero eddy viscosity regions and transition points

u [8] Johnson, D.A. and King, L.S., "A Mathemati-

cally Simple Turbulence Closure Model for At- tached and Separated Turbulent Boundary Lay- ers", AIAA Journal, Vo1.23, No.11, 1984.

[Q] Baldwin, B.S. and Barth,T.J., "A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows", AIAA-91-0610, Reno, Nevada, January 1991.

[lo] Radespiel, R., Graage, K. and Brodersen, O., "Transition Predictions Using Reynolds- Averaged Navier-Stokes and Linear Stability Analysis Methods", AIAA-91-1G41, Honolulu, Hawaii, June 1991.

[ll] Cebeci, T . and Bradshaw, P., Momentum Trans- fer in Boundary Layers, McGraw-Hill, New York, 1977.

[I21 Cao, H.V., Kusunose, K., Spalart, P.R., Ishim- itsu, K.K., Rogers, S.E. and McGhee, R.J., "Study of Wind Tunnel Interference for Multi- Element Airfoils Using a Navier-Stokes Code", AIAA-94-1933, Colorado Springs, Colorado, June 1994.

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