AN EDDY VISCOSITY FOR THREE-DIMENSIONAL BOUNDARY-LAYER FLOWS

Ridha r bid* NASA Langley Research Center

Hampton, VA 23665-5225

Abstract

This paper proposes an isotropic eddy viscosity model for three-dimensional boundary- layer flows which is an improved version of the Cebeci-Smith model. Both turbulence models are tested by comparison with two experiments. The proposed model is found to perform much better than the Cebeci-Smith model. This improvement is due to a reduction of the eddy viscosity coffecient.

Introduction

Advances in the field of computational fluid dynamics have allowed designers to make increasingly more accurate predictions of three- dimensional boundary-layer flows. However, many obstacles limit the effectiveness of engineering computational fluid dynamics. In particular, the turbulent shear stresses are difficult to model.

Most calculations employ turbulence models constructed for two-dimensional flows and applied in three-dimensional situations. A critical survey of models available to predict three- dimensional boundary-layer flows and their performance has been published by bid.' These studies have shown that the eddy viscosity models give poor predictions for flows with strong crossflow. These models are not able to reproduce the three-dimensional effects on the turbulence structure. In fact, the three-dimensional boundary-layer ex eriments reviewed by ~ohnston' and Van Den Berg3'show that the direction of the shear stress vector lags behind that of the mean velocity gradient vector, therefore invalidating the assumption of isotropic eddy viscosity. Also a significant decrease in shear stress magnitude is observed if compared to that of an equivalent two-dimensional boundary layer.

~ r a d s h a w ~ has suggested that good predictions could be expected only from calculations using models based on the Reynolds stress transport equations. However, the models have not et been found to perform well in practice,' thus, necessitating the use of simple isotropic eddy viscosity models. In fact, bid^ has shown that an isot~opic eddy viscosity which leads to a reduction of the shear stress magnitude can provide better results even for flows with strong crossflow.

The purpose of this paper is to propose an isotropic eddy viscosity model which significantly improves the Cebeci-Smith model The experiments of Van "en Serg and Elsenaar' and Oechow and Felsh8 are used to evaluate the proposed model. The equations of the boundary layer are solved by using an implicit finite difference method.

*NRC Fellow, Theoretical Aerodynamics Branch, TAD. Member AIAA

Turbulence Modeling

For an eddy viscosity model approach, it is assumed that:

In the Cebeci-Smith formulation9, the eddy viscosity is defined by two formulas: one in the inner region of the boundary layer, and another in the outer region. The inner eddy viscosity is given by:

where

In the outer region, v is defined by t

where

y is the Klebanoff intermittency correction k factor.

V is the magnitude of the local mean velocity vector defined by

2 112 v = (u2 + w )

Ve is the magnitude of the external velocity vector.

An exponential function is used to effect smooth transition between the inner and outer eddy viscosity distributions.

Note that this 3-D Cebeci-Smith model (CS model) does not reduce to their classical 2-D model for separated flows.

This paper is declared a work of the U. S. Government and is not sub- ject to copyright protection in the United States.

F A 7

Several comparisons between experiments and boundary-layer calculations have shown that the CS model gives reasonable predictions for attached flows, but the predictions deteriorate when separation is approached. l Measurements indicate that the eddy viscosity levels in the outer region of the boundary layer are much smaller than those predicted by the CS model. This is partly due to the outer length scale Av which was found to be nearly equal to the streamwise displacement thickness, 61'

To reflect more the three-dimensionality of the flow, a new outer length scale for the outer eddy viscosity is proposed

where

6 u 6 = I (1 -2) dy, streamwise displacement 1 0 'e thickness

6 'c 6 = I - dy, crosswise displacement 2 o v

thickness

U , U components of the mean velocity vector a?ongCand normal to the external streamlines.

The new length scale reflects the experimental fact that the streamwise component of the turbulent shear stress vector is reduced as the crossflow increases. For separated flows where 6 is of the same order as 61, the length scale gzven above needs to be modified as follows:

Note that the new model reduces to the standard CS model for two-dimensional flows.

Numerical Method

The equations for incompressible three- dimensional turbulent boundary-layer flows were integrated in a cartesian frame of reference

The equations of the boundary layer are solved by using an implicit finite difference method which can proceed either in direct or inverse mode. The streamwise derivatives are evaluated using a second outer upwind scheme, the transverse derivatives by a first outer upwind scheme, and the normal derivatives by a centered scheme. The turbulent boundary layer is described here with 60 points in the normal direction. For more details about numerical method, see Ref. 1.

Discussion and Results

The first case considered is the experiment of Van Den Berg and ~ l s e n a a r ~ which simulated the incompressible flow over an infinite 35' swepi wing at a free-stream Reynolds number of 2.42~10 per meter. The equations of the boundary layer are solved in the inverse mode by using the measured longitudinal and transverse displacement thicknesses. The inverse mode is used to avoid numerical difficulties when crossing separation line.

Fig. 1 shows the configuration of the wall streamlines (also call skin friction lines) and the external streamlines obtained by using the CS model. This figure clearly shows the strong skewness of the velocity in the boundary-layer region when the separation line is approached. In this case, the separation line is observed when the wall streamlines become parallel to the leading edge (located between stations 8 and 9). The total skin friction at separation line does not fall to zero as in two-dimensional flow, but reaches a distinct minimum.

The measured pressure coefficient distribution and those predicted by the proposed turbulence model and the CS model are compared in Fig. 2. The present model gives results in much better agreement with the experiment than those predicfed by the CS model. In Figs. 3 and 4, the mean velocity profiles predicted by the two turbulence models are presented in a cartesian frame linked to the external streamlines. The agreement with each other and with measurements is good up to station 5. Further downstream, the profiles predicted by the proposed turbulence model are in much better qualitative agreement with the experiment than are those predicted by CS model. Also, one sees a noticeable improvement in the prediction of the turbulent shear stresses, especially the crosswise component (See Figs. 5 and 6). The good results are attributable to the reduction of the eddy viscosity levels (See Fig. 7 ) . However, some discrepancies with the experiment still exists. The crosswise component of the velocity is still not well predicted in the separated region. This is due to an overestima- tion of the crosswise component of the turbulent shear stress vector. Also the eddy viscosity in the inner part of the boundary layer is over- predicted. To reproduce these effects, a more sophisticated turbulence model than the proposed turbulence model is needed.

The second case considered is the experiment of Dechow and ~ e l s h . ~ They investigated a three- dimensional turbulent boundary layer induced by a cylinder standing on a flat plate (See Fig. 8). The free-stream Reynolds number was 1.45~10~ per meter. The equations of the boundary layer are integrated in the direct mode by using the wall pressure distribution. The calculations were started at x = 0, and the initial velocity profiles were obtained from smoothing the measured velocity profiles. The starting conditions for the spanwise integration were obtained from solving the plane-of-symmetry equations.

The two turbulence models give the same good prediction of the mean velocity profiles up to station 5 (See Ref. 1). The negligible effect of the turbulence model on their results comes from

the fact that, for this flow, the pressure forces dominated over shear stresses, especially in the highly three-dimensional regions.

However, the calculation method using CS method overestimates the turbulent shear stresses, especially the crosswise component (See Fig. 9). This is due to an overprediction of the eddy viscosity. The proposed turbulence model leads to a substantial improvement in the agreement with the experimental shear stresses; the model is able to reproduce the decrease of eddy viscosity levels (See Fig. 10). Nevertheless, the agreement with experiment cannot be considered entirely satisfactory. In particular, the prediction of the inner layer profile of the streamwise component of the turbulent shear stress vector starts to deteriorate downstream of station 5 (in regions of flow having larger skewing angles). In fact, at stations 5 and 6, the measured mixing length distributions could no longer be represented by L = 0.40~. To improve the agreement, a complex model which requires more input information is needed as shown by Pierce and McAllister. lo

Conclusions

An isotropic eddy viscosity model which is based on experimental data, is proposed for three- dimensional turbulent boundary-layer flows. The experiments, which simulate incompressible flows over an infinite swept wing and in front of a cylinder mounted on a flat plate, are used to evaluate the proposed model.

For attached flows, the proposed model is found to perform much better than the CS model, especially in regions of flow with large skewing. The improvement is attributable to the reduction of eddy viscosity levels. This effect of the three-dimensionality is more important to take into account in calculations than the lag of the turbulent shear stress direction.

For separated flows, the present model seems to predict only the mean flow properties with reasonable accuracy. The prediction of the turbulent shear stresses is less satisfactory downstream of the separation line. Similarity laws are no longer valid in this part of the flow. Nevertheless, many comparisons with different experiments are needed before drawing valid conclusions.

References

1. Abid, R. : Turbulence Modeling for Three Dimensional Boundary Layer Flows. NASA TP (to appear in 1988).

2. Johnston, J. P: Experimental Studies in Three Dimensional Turbulence Boundary Layers, Stanford University, Stanford, CA. Report M?-34, 1976.

4. Bradshaw, P. : Complex Turbulent Flows, Trans. ASME I, J. Fluids Engrg., 97 (1975).

5. Cousteir, J.; Aupoix, B.; Falhas, G.: Synthese de resultats theorigues et experimentaux sur les couches limites et sillages turbulents tridimensionnels, ONERA N.T. 1980-4 (1980).

6. Abid, R.: Extension of the Johnson-King Turbulence Model to the 3-D Flows. AIAA Paper 88-0223, 1988.

7. Van Den Berg, B.; and Elsenaar, A.: Measurements in a Three Dimensional Incompressible Turbulent Boundary Layer in an Adverse Pressure Gradient under Infinite Swept Wing Conditions, NLR-TR-72092U, 1972.

8. Dechow, R.; and Felsh, K. 0 . : Measurements of the Mean Velocity and of the Reynolds Stress Tensor in a Three Dimensional Turbulence Boundary Layer Induced by a Cylinder Standing on a Flat Wall, Proceedings from Symposium on Turbulent Shear Flows, Penn State University Press, Vol. 1, 1977.

9. Cebeci, T.: Problems and Opportunities with Three Dimensional Boundary Layers, AGARD Report No. 719, 1984.

10. Pierce, F. J.; McAllister, J. E.: Near-Wall Similarity in a Pressure-Driven Three- Dimensional Turbulent Boundary Layer, Trans. ASME I, J. Fluids Engrg. 105: pp. 257-262, 1983.

3. Van Den Berg, B.: Some Notes on Three Dimensional Turbulent Boundary Layer Data and Turbulence Modeling, NLR-MP-82007U, 1983.

- External streaml~nes ..-... Sbn frlctlon lines

Leading edge

five Free stream d~rection

.4 Present model

0 Experiment C e .2

calculation

Fig . 1 General s t r u c t u r e of t h e flow. F ig . 2 Wall p r e s s u r e c o e f f i c i e n t .

8 0 10 Present model

- Present model --- CS model + Experiment

+ Experiment 60

Y(mm)

4 0

20

0 0 0 .2 1.0 0

0 0 0 0 0 0 0 . 2 . 4 . 6

U/"

Streamwise v e l o c i t y p r o f i l e s . F ig . 4 Crosswise v e l o c i t y p r o f i l e s .

' O r station 7 60r station 8 Y(mm) Present model

CS model

- Present mol

Fig . 5 Turbulent shea r s t r e s s p r o f i l e s .

0 2 0 "r Station 7

Present model

2 0

Fig . 6 Turbulent s h e a r s t r e s s p r o f i l e s .

0 2 0 r Station 8

Present model ,--, \ \

--- t CS model - Present model \

I, t - '\

+ Experiment Ve8l --- CS model

.020 r Station 9

\ \ + Experiment \

.020 r Station 10

Fig. 7 Turbulent eddy v i s c o s i t y p r o f i l e s .

.015

t - VA

.010

1 1 1 1 1 1 1 1 1 1 1 I I

0 15 3 0 4 5 60 7 5 0 15 3 0 4 5 6 0 7 5 Y(mm) Y(mm)

- /--_

.015 /

/ .

\ - / Present model 2 -

I \ --- I \ CS model ve81

I , + Experiment

- . r - - - _ . - - / /

Present model , \ ---

/ CS model /

/ \\, + Experiment - I \

Extemal streamline

Fig. 8 ~echow/Felsh experiment.

.05 r Station 5

.6 - .6 -

C --- CS model

Station 5 -

I + Experiment

Statlon 6

- /--kt

020 "r Station 5

Present model - - Present model 2

+ Experiment --- CS model L v 2 + Experiment 2 e

1 .

O 5 station 6

-.05

102 / -Resent model

-.I5 , --- , CS model + Experiment

Fig. 9 Turbulent shear stress profiles.

020 r Station 6

Fig. 10 Turbulent eddy viscosity profiles.