AIAA 93-0009 Investigation of a Model for Predicting Separation Control with Vortex Generators Dayton A. Griffin Dept. of Aeronautics and Astronautics University of Washington Seattle, WA

31 st Aerospace Sciences Meeting & Exhibit

January 11 -1 4, 1993 / Reno, NV For permission to copy or mpubllsh, contact the American institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

AIAA-93-0009

INVESTIGATION OF A MODEL FOR PREDICTING SEPARATION CONTROL WITH VORTEX GENERATORS

Dayton A. Griffin* Department of Aeronautics and Astronautics

University of Washington Seattle, Washington

Abstract The use of vortex generators for

separation control is a matter of current practical importance, and the effect of vortex generators on boundary-layer behavior is discussed. A new symmetry-plane model for predicting the effectiveness of counter-rotating vortex-generator installations in incompressible flow is introduced, and the underlying physics and assumptions of the model are described. Special attention is paid to the consistencv of the eauations used and the evaluation of thLtems w i t h . Calculations are reported which predict the effectiveness of vortex generator installations, and comparisons are made with existing experimental data. With a scaling factor introduced, good agreement is found between theory and experiment. Physical justif- --’ cations for a scaling factor are suggested, and additional planned verification experiments are discussed.

I. Introduction Vortex generators are often used for both

internal and external flows in order to prevent or delay separation. Typically small protruding wings, they produce longitudinal vortices which increase mixing between the external stream and the boundary layer. As fluid particles with high momentum from the free stream are mixed with the boundary layer, the mean streamwise momentum of the layer is increased, and the flow is able to withstand larger adverse-pressure gradients without separation.

The first reported use of vortex generators was in 1950, when small lifting surfaces were applied normal to wind-tunnel diffuser walls(’). The work of Pearcy(2) then contributed a large base of empirical data by cataloguing types of

* AIAA Student Member Copyright 8 1993 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

generators, features of the vortices, and effectiveness under certain conditions. More recently, a series of experiments have studied the fundamental turbulence structure in representative simplified flows (Mehta et al.(3) and Westphal et al.(4)), while numerical simulations have been made by Liandrat et al.C5), and by Sankaran and Russell(6), both using reduced sets of Navier- Stokes equations. Boundary layer theory cannot be used for the complete flow because of the normal gradients near the vortex cores.

Although vortex generators have been used extensively for more than forty years, the complexity of the resulting flow has eluded simple solution, and experimentation has been necessary for each new design. No models have emerged to predict the effectiveness of vortex generator installations in delaying separation, and to assist in the optimization of new installation designs.

This paper investigates a model(7) which is addressed to that need. The following section outlines it and develops the equations used. This is followed by a discussion of relative vortex motion, and the general calculation procedure is outlined. Results are compared to recent experiments, and areas for further improvement are suggested.

11. The model The interaction of vortex pairs with a

developing boundary layer is represented by the schematic on Fig. 1, which shows a planar aerodynamic surface in a flow with velocity U. A coordinate system is established with x initially in the free stream direction but lying along the surface, with y normal to the surface, and z in the spanwise direction. Velocity-vector components are u, v, and w respectively.

1

Fig. 1 Coordinate system for vortex boundary- layer interaction.

A pair of vortex generators is placed near the leading edge at x = 0. Typically small plates attached normal to the surface, the vortex generators are set at an angle of attack in the y = 0 plane, and produce a system of bound and trailing vortices. As the roll-up of the inviscid vortex sheet from a lifting wing produces a tight vortex a few chord lengths downstream of the tip@), and the chord of the generators is small compared to that of the surface, they have been represented on Fig. 1 by axial vortices with initial centers located at x = 0, y = h, and z = k d12. The vortex circulation, ? can be estimated by the geometry of the generators by lifting surface theory. Figure 1 shows a pair of counter-rotating vortices which produce a flow with symmetry about the z = 0 plane. A vortex generator installation would be made up of an array of such pairs, with the next set being centered at & D as indicated on Fig. 2.

\ Y t 8 w/o vortices

The vortices shown in Fig. 1 contribute an outward spanwise velocity component to the flow near the surface, and this thins the local boundary layer. Figure 2, modified from Ref. 2, shows contours of constant u/ul in the cross- plane at an arbitrary x-location, where u, is the u- velocity just external to the boundary layer. With the dashed line indicating the layer thickness, 6, in the absence of the generators, it can be seen that the generators reduce 6 over a significant portion of the span.

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While the overall interaction of pairs of strong counter-rotating vortices with a turbulent boundary layer is complex, considerable simplification is expected in the symmetry planes between the vortices. The vortex cores with their strong normal gradients are relatively far from these planes, and their effect is to add a cross flow velocity that accelerates away from the plane. The flows in the symmetry planes are thus modelled by boundary-layer equations with an applied cross flow determined by inviscid analysis. Beginning with the equations of motion for this flow, the incompressible continuity and momentum equations are, respectively

L au av aw ax ay az -+-+-=o

and

u-+v-+w-=u,au’+”, aU aU au a2u . (2) ax ay aZ ax ay

init ial vortex location f

k d - 4 I t-------D--------I

Fig. 2 Velocity contours at an x = constant cross-plane. Lr

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Integrating the momentum equation across the layer gives

aU aZ 0 ay ax 0

6 J au aU j( u,+ v- - u1 -)dy + j w- =

From continuity

and integrating by parts gives

Inserting these into the integrated momentum equation and evaluating in the z = 0 plane yields

Regrouping, and defining

and

m . 1

6 E - j ( UI - U)dY u1 0

leads to

dB I *aw du €I C -+-J-(ul -u)dy+(Z+H)--=L, (3) dx uI2 aZ dx u1 2

with the form factor H E 6'B. Equation (3) is the two dimensional, incompressible form of the Karman momentum integral relation, with an additional term due to the applied spanwise flow. .-.,

An additional relationship may be. obtained Eq. (1) across the by direct integration of

boundary layer. Beginning with

and recognizing that

and

therefore

1 s a u i a a6 u1 0 ax u1 ax - j d y = ---uu,(6- ti*)-- ax .

Similarly

a 6 aw a6 aZ az

aZ 0 0

-fw dy = j d y + w1 -

and

in the symmetry plane. Then noting that

the continuity equation becomes

or

where the coefficient of entrainment, CE. is (pul)" times the mass flux per unit area entering the top of the boundary layer.

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Equations (3) and (4) can be used to evaluate the properties of the boundary layer in a stepwise fashion. Assuming that the crossflow derivative can be found by inviscid analysis, the two equations together have five unknowns: 6, 6*, 8, Cf, and CE. In the typical manner of solving turbulent boundary layer problems, three empirically-based relations were selected to form a closed sed7).

III. Crossflow Derivative and Vortex Motion

The derivative in Eq. (3) is evaluated by inviscid, incompressible flow analysis, and the results are restated here for completeness (ie. Ref 7). First consider the pair of counter- rotating vortices of Fig. 1, together with their below-the-plane images necessary to insure the v = 0 boundary condition at y = 0. The complex potential for a point in the cross-flow plane is then found by superposition. Differentiation of this with respect to the complex distance yields the complex velocity

1 ( z -d / / )+ i (y -h )

1 1 (2- d / 2)+i(y+ h) - (z+ d / 2 ) + i(y+ h)

+

1 (z+ d / 2 ) + i (y - h )

+ (5)

Differentiating w and evaluating at z = 0 results in

where ZE d/2h and 7 Equation (6) gives the expression for the

crossflow derivative due to a single vortex pair at the initial point x = 0. However, the value of aw/& decreases significantly in the streamwise direction. This is because the vortices do not retain their original z, y position as they move downstream, but move under the influence of each other. The velocity components at a vortex core are given by

y h .

and

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The displacement of the core in the cross-plane a distance Ax downstream is found by multiplying the velocity at the core by M u , , and the process is repeated at each subsequent step using updated values for d and h.

The dynamics of an infinite array of counter-rotating vortices are calculated in a manner similar to that above, successively adding identical counter-rotating pairs with planes of symmetry located at z = i7 nD. The complete expressions for an infinite vortex array are available in Ref. 7.

I Equations (3) and (4) may be written in

difference form giving d(6-6*)/dx and dWdx at a particular station in terms of the previous x- station values of 6, 6'. 8, Cf, CE, u ~ , dulldx, and aw/az. From the problem specification, inviscid flow analysis can be used to find u1 and dulldx. As the integrand of the crossflow term in Eq. (3) involves the product (aw/&)(ul-u), both factors must be known as functions of y and their product integrated across the boundary layer. aw/&(y) is readily found in a form similar to Eq. (6). By assuming a power law velocity distribution for a turbulent boundary lay&), u/ul = (y/6)lln, the velocity profile can be solved as a function of H at each station and (UI-u) is then known.

Now Eqs. (3) and (4), along with the relations of closure, knowledge of the crossflow derivative and vortex paths, and the u(y) velocity profile can be used in a stepwise manner to solve for all the boundary layer values at each station, as long as the initial values at the vortex generators are known. A FORTRAN computer code was modified to implement the calculation procedure(7). In generating the results reported here, the initial boundary layer values for 6, 6*, and 0 were estimated by a combination of panel method codes(*), Thwaite's method for laminar boundary layers, and Head's method for turbulent boundary layers('0). Separation was assumed to occur at a form factor value of H = 1.8.

Several errors and inconsistencies were identified in the computational methods of Ref. 7

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and eliminated in performing the calculations reported here. Principle among these was a correction in the integrated continuity equation, and replacement of the simple "no-twist" assumption previously used to directly integrate

latter was to better deal with vortex cores that move deep into the layer. Improvements were also made in the estimation of initial boundary layer values at the vortex generators. Finally, an iterative scheme was implemented which made the computer results more reliable.

configurations with vortex generators had D = 2 cm, h = 0.15 cm, and generator chords of 0.65 cm. One line which originally appeared on Fig. 3 has been deleted here as it contained only one data point and was therefore determined

Note that for the same number of generators, both the experimental data and the symmetry-plane calculations show that the configuration with initial d = 0.35 cm is more effective than that with d = 0.5 cm. Also note that the two configurations with vortex generators

V. Discuss ion show extreme improvements in separation locations over those for the clean airfoil. For the airfoil with no vortex generators, separation was near 50% chord at 22' angle of attack. At the Same angle of attack, the vortex genera tor configurations with d = 0.5 cm and d = 0.35

moved the separation locations aft to near 80% and 95% chord, respectively.

- the three-dimensional effects in Eq. (3). The to be unreliable.

Calculated results from the revised model of Section IV were compared with data from previous experiments conducted at the University of Washington(7). The experiments used a rectangular wing with a 1 . 8 ~ span, 0.3m chord, and a NACA 0016 section profile. A trip strip was applied at approximately 12% chord. Vortex generators were cut from thin aluminum with an additional base that was folded at a right angle and epoxied on the wing at a 15' angle of attack and at 15% chord. The wing was tested in the Kirsten Wind Tunnel, which is a split-return closed-circuit tunnel with a 2.4 x 3.6m test section, a M = 0.3 capability, and a 0.4% turbulence level. The tests used flow visualization measurements to determine the separation locations for a range of angles of attack and a variety of vortex generator configurations. Most of the experiments used generators of height h = 0.15 cm so as to slightly protrude above the initial boundary layer.

Calculations were performed for the same geometries and configurations as the experiments, and it was determined that the symmetry-plane model was over-estimating the effectiveness of the vortex generators in delaying separation. A scaling factor was introduced to adjust the effect

--..,' a(o)

of the crossflow derivative term in Eq. (3). 05 Ob 0.7 Ob 09 1

Figure 3 is a plot of chordwise separation X/C location as a function of angle of attack. Note that separation locations are plotted for d c values between 0.5 and 1.0, while angles of attack are plotted from 0 to 30 degrees. The solid lines on the figure are the result of the calculations made using the revised procedure described in Section IV, with a scaling factor of 114 multiplying the crossflow term. The symbols indicate data obtained during the experiments described above, with the error bars indicating uncertainty levels.

one with no vortex generators. Both of the

Fig. 3 Separation location for NACA 0016 wing at 70dsec.

The fact that a scaling factor was needed to obtain the good agreement between calculations and experiment seen on Fig. 3 merits closer examination. The crossflow term is

-J Three generator configurations are represented: one with d = 0.5 cm, one with d = 0.35 cm, and

1 'aw u,2 aZ -I-(., - u)dy .

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A decrease in this term is equivalent to a decrease in the calculated effectiveness of the vortex generators. Note that (u1-u) was found by assuming a power-law velocity profile in a turbulent boundary layer, and aw/az was found by inviscid flow analysis. This combination of viscid and inviscid analysis in the same term was originally justified on the grounds that viscous effects on the vortex flow will occur over a length scale which is small compared to the scale of the generators, while the scale of the generators is the same as that of the boundary layer. While this is true, it appears that significant errors could result from neglecting the viscous effects on aw/az.

First consider turbulent boundary layer U / U ~ versus y16 profiles as shown on Fig. 4 for three different values of H, where H can be shown equal to (n+2)/n. H = 1.3 is the initial form factor at transition from laminar to turbulent flow, and would be a representative value at the vortex generators for an airfoil at low angle of attack. H = 1.5 is an intermediate value which might characterize the boundary layer at the generators for high angles of attack, and H = 1.8 is a limiting value which would represent a turbulent boundary layer approaching separation. The boundary layer profiles of Fig. 4 are relatively full, and it is seen that the value of (ul-u) is a maximum at y = 0, and decreases rapidly away from the surface.

0 0.2 0.4 0.6 0.8 1

u/u , Fig. 4 Boundary layer velocity profiles as a

function of H.

Next consider a representative plot obtained from Eq. (6) of non-dimensional versus y/h as shown on Fig. 5 for values of y/h ranging between 0 and 2. The range of values over which aw/az is integrated depends on the height of the vortex relative to the boundary layer, as the product (aw/az)(ul-u) is always integrated from y = 0 to y = 6. If the vortex height is twice that of the local boundary layer, awl& is integrated for values of y/h between 0 and 0.5. However, if the vortex is buried in the layer so that the vortex height is 112 that of the layer, awlaz is integrated for y/h values between 0 and 2. In any case, it is seen that the value of awl& is at a positive maximum at the wall, and decreases towards negative values above y h = 1.

L

2.0

1.5

Ylh

1 .o

0.5

0 -0.2 0 0.2 0.4 0.6

Fig. 5 Non-dimensional dw/dz for a counter- rotating vortex pair.

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Finally consider the value of the integrand in Eq. (3), which is the product (aw/az)(ul-u) shown plotted versus y16 on Fig. 6 for two extreme cases. The line for H = 1.3 and h/6 = 2 represents typical initial conditions at the vortex generators for an airfoil at low angle of attack, with the generator tips protruding to twice the height of the local boundary layer. The line for H = 1.8 and h/6 = 0.5 represents a case of impending separation, where the value of h/6

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indicates that the vortex paths are such that the vortices have become buried in the local boundary layer.

While the curves for the two extremes shown in Fig. 6 are markedly different in nature, both have positive maximums at y = 0. This is not surprising and will always be the case as the two factors, (aw/az) and (u1-u), both have positive maximums at y = 0. At this point it is apparent that the present model is overestimating the crossflow term of Eq. (3) by neglecting the viscous effects in the vortex flow. The no-slip condition will necessitate that w = 0 at the y = 0 surface for all values of z. Therefore, aw/& must approach zero at y = 0. Even if the viscous effects cause aw/& to approach zero over a region which is small compared to h and is close to the wall, Fig. 4 shows that the region over which aw/az is overestimated is the same region over which (u1-u) is the largest.

I

- Hz1.3 h/6=2.0

-- Hz1.8 h/6=0.5

0 0 .1 0.2 0.3 0.4

Integrand

Fig. 6 Non-dimensional integrand for cross-flow derivative term.

A more rigorous accounting of the viscous effects in the calculation of aw/az will thus lead to a decreased value of the crossflow term as found necessary on Fig. 3. It is also important to note that the crossflow term diminishes rapidly downstream of the vortex

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generators. The physical explanation for this is that as the vortices travel downstream, they move away from each other due to their mutual influence, and therefore awlaz decreases as well. The calculations of Fig. 5 were for fixed values of d and h, and thus the integrand of Fig. 6 does not properly display this important effect. The design of a vortex generator installation would address the integrated contribution of the crossflow term over the downstream direction, and this leads to optimal confgurnt i~ns(~) of D/d, Dlh, T/Uh, and h/6.

Further work on the symmetry-plane model is needed in order to develop its full potential, and this will require more detailed experiments. These will use a new research wind tunnel which is presently being installed at the University of Washington. The experiments will use a symmetric wing with a 3 ft span and chord of 1 ft mounted horizontally across the tunnel test section from wall to wall. The wing will be fitted with multiple pressure taps in the plane of symmetry between vortex generators, and separation locations will be measured both by flow visualization and by pressure measurements. The pressure measurements may also be used in place of the inviscid panel methods to determine the external flow. Results from the experiments will then be compared with the symmetry-plane model to check for agreement

VI. Conclusion Analytical methods for vortex generator

design have not existed due to the complexity of the flows. In this paper, a simple method for predicting the effectiveness of counter-rotating vortex generators has been revised and improved. The model is easy to use and appears to contain the interaction physics necessary for optimization of vortex generator installations. Further development guided by new experiments is planned in order to establish the model as a useful design tool.

Acknowledeements The assistance of Professor David A.

Russell of the University of Washington is gratefully acknowledged. The support of Dr. Bannister Farquhar of the Boeing Company and helpful consultations with Dr. Gerald Paynter are very much appreciated.

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References Taylor, H.D., "Summary Report on Vortex Generators", R-05280-9, 1950, United Aircraft Research Department.

Pearcy, H.H., "Shock Induced Separa- tion and its Prevention". in Lachmann. G.V., Boundarv Laver and Flow Control, Val. 2,1964, pp. 1166-1344.

Mehta, R.D., Shabka, I.M.M.A., Shibl, A. and Bradshaw, P., "Longitudinal Vortices Imbedded in Turbulent Boundary Layers", AIAA Paper 83-0378, 1983.

Westphal, R.K., Eaton, J.K. and Pauley, W.R., "Interaction Between a Vortex and a Turbulent Boundary Layer in a Stream- wise Pressure Gradient", Turbulent Shear Flows V, ed. Durst et al., Springer Verlag, 1987.

Liandrat, J., Aupoix, B. and Cousteix, J., "Calculation of Longitudinal Vortices Imbedded in a Turbulent Boundary Layer", Turbulent Shear Flows V, ed. Durst et al., Springer Verlag, 1987.

Sankaran, L. and Russell, D.A., "A Numerical Study of Longitudinal Vortex Interaction with a Boundary Layer", AIM Paper 90-1630, 1990.

Amaud, G.L. and Russell, D.A., "Symmetry Plane Model for Turbulent Flows with Vortex Generators", AlAA Paper 91-0723, 1991.

Kuethe, A.M. and Chow, C., Foundations of Aerodvnamics: Bases o f Aerodvnamic Design, 4th ed., John Wiley & Sons, New York, 1986, pp. 128-143.

Schlichting, H., Boun r T h 7th ed., McGraw-Hil:aEk%rk, &;% pp. 638.

Cebeci, T. and Bradshaw, P., Momentum ~~

Transfer in Boundary Lavers, Hemisphere, 1977, pp 108-198

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