AI AA-88-0280 Experimental and Analytical Analysis of Grid Fin Configurations R. A. Brooks, Rockwell International Missile Systems, Duluth, GA; and J. E. Burkhalter, Auburn University, AL

AlAA 26th Aerospace Sciences Meeting January 11-14, 19881Ren0, Nevada

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EXPEKIMENTAL AND ANALYTICAL ANALYSIS OF GRID FIN CONFIGURATIONS

ROSS A. Brookst Rockwell International Missile Systems

Duluth, Georgia

John E. Burkhalter* Auburn University, Alabama

Abstract _- A compressible vortex lattice formulation has

been developed and utilized in predicting the aerodynamic coefficients for arbitrary grid fin configurations. To validate the theory, several representative grid fin designs were tested in a low speed wind tunnel. Results indicated that for most grid fin designs, the normal force and root chord bending moments are nonlinear in such a manner that it appears that upper and lower lifting surfaces tend to stall at different angles of attack. Hinge moments were small throughout the alpha range and drag coefficients are not necessarily parabolic. For most configurations, the cxnerimental data axreed quite well with the theory .

b

- C

C

CmC,2

RCBM 'm

C "y

c" CN

d<

M'

%

N' - r

- R

6

Nomenclature

F i n span

Reference length

Local chord length

Hinge moment coefficient about hinge line.

Root chord bending moment coefficient

Pitching moment coefficient about the leading edge of the m o t chord (y a x i s )

Sectional normal force coefficient

Total normal force coefficient

Differential length of a vortex filament

Pitching moment per unit span

Pitching moment about the leading edge of the root chord

Normal force per unit span

Position vector for the incompressible vortex lattice formulation

Position vector for the compressible vortex lattice formulation

Distance between trailing legs

TMember of the Technical Staff; Member, AIAA. *Associate Professor, Aerospace Engineering; Member, AIM.

- s,s Reference area

x,y,z Cartesian coordinates

Distance from the leading edge of the root (y axis) to the fin center of pressure.

xcP

x Distance from the leading edge of the LE root chord (y a x i s ) to the quarter

chord line of a subpanel

x distance from the y axis of the leading edge of leg 3 of the vortex filament

XI

(I Angle of attack

6 Mach flow parameter,

r Vortex filament strength

Angle of inclination of a subpanel with respect to the positive y axis

Introduction

OP i,

An unusual concept in missile fin design has led to a theoretical analysis and a series of subsonic wind tunnel tests on grid fin configurations. The fin design appropriately termed a "grid" fin is an all moveable fin which consists of (usually) high aspect ratio members of constant chord arranged in a grid-work pattern. The fin outer perimeter may be square, rectangrrlar, octogonal, or virtually any shape consisting of straight members connected end to end, while the inner section of the fin may consist of any arbitrary srid pattern. The internal fin density may range anywhere from an open cavity to a dense honeycomb arrangement. Obviously because of this flexibility, drag tailoring is easily obtainable. Additionally, these designs prove advantageous where control surfaces are span limited or situations where small actuator requirements are a high priority. A schematic of several general grid fin configurations is given in Fig. 1.

These fins are mounted transverse to the longitudinal a x i s of the missile so that flow passes through the grid members. Angle of attack of the fin is achieved by a simple rotation of the fin about its horizontal a x i s in much the same way a conventional fin control surface is deflected.

The theoretical aerodynamic analysis of these

W grid fins logically lends itself to a vortex lattice formulation. Primary coefficients of interest are normal force, root chord bending moment, and most importantly, the hinge moments. If potential flow assumptions are made, vortex

Fig. I Schematic of Several Grid Fin Configurations

lattice theory is quite adequate except for compressibility corrections.

General vortex lattice theory was first developed by Falknerl in 1943 and the descriptive phrase "vortex lattice" was first used by that investigator in several 1946 publication^.^,^ The advent of high speed digital computing machines made vortex lattice methods the first powerful tool for three-dimensional potential flow analysis. Investigators such as Rubbert,' H e d m ~ ~ n , ~ and RelotserkovskiiC extended Falkner's method in the early 1960s and adapted the method to computers. The vortex lattice method was applied to increasingly more complex configurations such as multi-planes, nanplanar wings, and entire aircraft lifting surfaces with intricate geometries, resulting in modern computerized methods such as those developed by Feifel.7 and Margason and Lamar. 8

Theory

Standard vortex lattice theory is centered around the Biot-Savart Law and is a constant pressure paneling method suited to thin lifting surfaces of moderate to high aspect ratios. For the present work, a compressible vortex lattice formulation has been developed. In the analysis, thc fin configuration is divided into a network of subpanels spaced uniformly (or nonunifarmly) on each fin member in the spanwise and chordwise direction (Fig. 2). A lifting vortex of unknown strength is located along the 114-chord line of each subpanel as prescribed originally by Pistolesiq as the optimum location for spanwise

vortices and later validated by ByrdlO for chordwise vortices. A pair of trailing vortices is shed along the panel edges parallel to the uniform flow, downstream to infinity and forms a so-called horseshoe vortex. The boundary condition of no flow through the surface is satisfied at the 314-chord lateral center of each subpanel and thus completes Pistolesi's 114-314 theorem. Imposition of the boundary condition results in a system of equations which can be solved to yield the strengths of the vortices. Once the vortex strengths are known, the fin sectional aerodynamic characteristics may be determined by appropriate summations in the chordwise direction. Finally, the total fin aerodynamic characteristics are computed by standard numerical integration techniques in the spanwise direction.

Fig. 2 Distribution of Vortices on a Typical Grid Fin.

Vortex lattice equations may be derived directly from the Biot-Savart equation which defines the velocity induced at a field point by a vortex filament. In vector form, this equation is

where r is the filament strength, and is the vector from a field point (x,y,z) to a differential

length of the vortex filament, da. developed to date (Refs. 5 , 7, and 8) consist of an incompressible formulation with the standard Prandtl-Glauert correction. However, a compres- sible downwash expression is highly desirable and may in fact be written as

Most methods

where

2

and is the vector from a field point (x,y,z) to

a differential length of the vortex filament, de. Validation of Eq. (2) was obtained by comparing compressible vortex lattice solutions to compressible lifting surface solutions which are discussed in Refs. 11 and 12. Comparison of the compressible formulation as presented in Eq. ( 2 ) was made with several other approaches including the standard incompressible form in Eq. (I), and other compressible lifting surface formulations for swept and unswept wings with and without taper at Hach numbers ranging up to 0.6. The compressible formulation, Eq. ( 6 ) , was in very good agreement with the compressible lifting surface solutions which as demonstrated in Refs. I1 and 12 agree quite w e l l with experimental data.

B

For the present case , Eq. (2) is applied to a single Swept horseshoe vortex, as depicted in Fig. 3, so that the induced velocity from each leg is

+ ( x - y tan@)kl and

Equations similar to these may also be derived for cases where dihedral is to be considered and where the trailing legs are not necessarily straight but may in fact bend and (generally) trail off in the free stream direction.

The boundary condition of no flow through the fin at control points can be written in matrix notation as

[ A I 7 = [ B l ( 6 )

The [AI matrix is known as the influence coeffi- cient matrix and is composed of the downwash terms

, Y

J

W

Fig. 3 Coordinates f o r a Single lbrseshoe VOT~CX.

given by Eqs. (4a), (4b), and (4c) without the unknown rs. The I B ] matrix is governed by the angle of attack and any other known upwash or downwash terms and for the present case is simply

B = - (sin di ; i = I , nc ( 7 ) i

where nc is the total number of grid fin control points. Once the vortex strengths are found by matrix algebra, the aerodynamic characteristics of the lifting surface may be computed through utilization of the Kutta-Joukowsky theorem for the Sectional properties followed by numerical integration in the spanwise direction for total coefficients.

For the sectional normal force, the Kutta-Joukowsky theorem yields

which can be written in summation notation as

where j denotes a particular spanwise station, and the right-hand side of the equation represents twice the summation of the vorticity over nc chordwise subpanels. The total normal force i s obtained by integrating over the span so that

where n = y/(b/2).

In a manner similar to the sectional normal force coefficient derivation, the sectional pitching moment coefficient about the leading edge of the root chord (y-axis) can be shown to be

U

3

where j denotes a particular spanwise station and - where 5 is the nondimensionalized distance from the where C is the chord length, and C is the reference y axis to the quarter chord line of the appropriate length. - subpanel given by

The grid fin center of pressure location is given by the nondimensionalieed distance from the

LE (12) leading edge of the y axis 5 =- x--x

C

(19) XCP The total pitching moment about the y axis is then - =

given hy C -

The root chord bending moment coefficient is given by

evaluation of Eqs. (IO), (13), and ( 1 4 ) requires that the sectional properties given by Eqs. ( 9 ) and (11) be known at various spanwise stations. The sectional properties for a grid fin are obtained by making imaginary vertical slices at predetermined spanwise stations. The strengths of a l l vortices cut by a single imaginary plane are summed to give the sectional property value at that particular Spanwise station. For the cases when a particular vortex is inclined with respect to the positive y axis, an appropriate component of the vorticity strength is summed. Summing a "component" of the vorticity strength is justified when the panel is inclined to a given axis. The normal force per unit span in the direction normal to the fin horizontal axis is then

-

where ap is the angle of inclination of the panel with respect to the fin horizontal axis. For a grid fin, Eq. ( 9 ) then becomes

where iivss I s the number of vortices cut by the imaginary plane associated with the jth spanwise station. Similarly, Eq. (11) becomes

nc - 1 2ri) COS ep) (17)

i=I

Note that the Spanwise stations must be selected so that no station f a l l s on a trailing leg.

The hinge moment coefficient is computed about the fin axis which passes through the chord midpoint and is given by

v

Verification of the theory was demonstrated by comparison with experimental wind tunnel data. Data were obtained for eight distinct grid fin geometries. Four models consisted of a 3-inch by 3-inch square outer shell and the remaining four models had B 3-inch by 6-inch rectangular outer shell ( s e e Fin. 6 ) . A l l fin models had R I-inch constant chord length and all fin grid elements were approximately 0.025-inch thick. The fin models were tested at a dynamic pressure of approximately 3 inches of water, which corresponded to a speed of about 120 feet per second. Farce and moment coefficients were based on a reference length of 3 inches and a reference area of 7 - 0 1 square inches. The average Reynolds number based on the chord length of 1.0 inch was 165,000.

Wind tunnel test data f o r the grid fins demonstrated several interesting trends. A s would be expected, horizontal slat configurations tend to provide higher normal force characteristics than open geometries, although structural limitations may reduce the practical effectiveness of such designs. These configurations would he suited to span-limited situations where more slats (lifting surfaces) could be added to a particular fin to attain a desired normal force. Complex grid-like inner fin arrangements provided increases in normal force over open geometries and a comparatively larger axial force and may be suited to applications requiring high structural integrity, well behaved lift characteristics, and high drag.

Results presented in Figures 5 and 6 indicate typical characteristics of grid fin configurations. For both cases, the square and the rectangular configurations, there is a significant increase in normal force at higher angles of attack as additional slats are added. A s expected, the double slat fin produced the largest normal forces while the open fin produced the smallest normal forces. However, the single slat and cross fins. at least over the linear range, generated approximately the same normal forces even though the cross fin had a larger lifting surface area. The 3-inch by 6-inch grid fins exhibited similar trends. It is possible that fins of the cross configuration produce less normal force than is dictated by the size of the lifting area because of interference between the fin members. Thus the cross configuration fin of both sizes produce roughly the same normal force as the single slat fin, but greater axial farces.

4

r- 11 OPEN

100-

z 0

050-

-0 50

Fig. 4 Grid Shapes for Wind Tunnel Tests; Chord Length = 1.0 Inch.

O0O;-/, , , , ,

The internal arrangement of a particular grid fin appears to also influence the stall point. Generally, the more complex an internal arrangement is, the higher the stall angle of attack. As a matter of fact, for the more complex geometries, the cross and double slat, the stall does not occur below 18 degrees angle of attack. Hinge moments (CMH) for all cases are very small and the root chord bending moments (CMRCBM) exhibit properties similar to the normal force behavior.

Agreement between theory and experimental data was quite goad for all configurations with typical results shown in Figures 7 and 8. It is interesting to note in both these cases that the slope of the Cralpha curves tend to increase up tc approximately 8.0 degrees to such an extent that the theory underpredicts the data slightly in this region which is atypical for vortex lattice formulations. From Fig. 8 in particular, it appears that the upper "slat" stalls before the lower slat which produces a distinct "bump" in the aerodynamic coefficients between 4.0 and 10.0 degrees angle of attack. Past 10 to 12 degrees, there seems to be a second "stall," perhaps of the bottom or Cross members. Results were very repeatable and both multiple slats and cross slat configurations exhibited the same trends.

Tests were not run at angles of attack higher than 18 degrees so that in some cases the actual stall of the fin was never observed. Tuff tests did, however, show that the upper fin members do indeed stall with separated (reverse) flow on the top side of the upper slat. These same tests also showed that bottom slats did not stall at least up to 18 degrees angle of attack.

W

Axial force coefficients, for the most part, increased somewhat as more internal members were added but changed very little with angle of attack. It appears that a x i a l forces, at least for the configurations tested, were dominated by viscous drag. Only the 3-inch by 6-inch double slat fins demonstrated any parabolic tendencies.

+ CROSS A DOUBLE SLAT

1.50 i l 0 SINGLE SLAT I 1 I 'OPEN

ALPHA (deg)

A DOUBLE SLAT 0 SINGLE SLAT

-8.00 -4.00 0.00 4.00 8.00 12.00 18.00 ALPHA (deg)

! 2o:oo

I + CROSS

I I 0.00 u rF-c

-1 004 -8.00 .4.00 000 4.00 8.00 12.00 78.00 2(

ALPHA (deg)

Fig. 5 Grid Fin Aerodynamic Coefficients versus Angle of Attack for the Four 3-Inch by 3-Inch Fins,

'0

W

5

0.00 -- t -1 00 L&

+ CROSS a DOUBLE SLAT 0 SINGLE SLAT r OPEN

4.00 8.00 12.00 18.00 : ALPHA (deg)

_- ,.

+ CROSS A DOUBLE SLAT 0 SINGLE SLAT r OPEN

-8.00 -4.00 0.00 4.00 800 12.00 18.00 2 - l.OC

0.50

I 2 0.00 0

-0.50

-1.O( -1

ALPHA (deg)

+ CROSS A DOUBLE SLAT 0 SINGLE SLAT ! OPEN

-400 000 400 800 1200 1800 i ALPHA (de91

Fig. 6 Grid Fin Aerodynamic Coefficients versus Angle of Attack for the Four 3-Inch by 6-Inch Fins.

For & configurations, the hinge moments were very small and increased only slightly with angle of attack. In some cases--for example, in Fig. 8%- the slight increase was followed by a decrease in CMH, probably as a result of secondary fin element stall. Although one cannot be certain, it appears that most of the hinge moment contribution comes from the fact that at higher angles of attack, the lower slats (unstalled) normal force contribution is in front of the hinge line while

i the upper (stalled) normal force contribution is behind the hinge line. Very little of the hinge moment comes from the fact that on each fin element the center of pressure is in front (= quarter chord) of the hinge line.

3.001

I 2.00

0.00

- 1 . 0 0 1 -8.00 -4.w 0

2 300r-- 00

E l 0 0

5 7 0.00

m .d

.1.001-- -8.00 400 C

0.50 -

z 0.00

-0.50 :r -1.00

THEORY. .- TEST DATA

4.00 8.00 12.00 18.00 : ALPHA ( d q )

THEORY TEST DATA

I 4.00 8.00 12.00 18.00 2C ALPHA (de$ '

THEORY - TEST DATA '

-800 -400 000 400 800 1200 1800 21 ALPHA (deg)

Fig 7 Normal Force, Root Chord Bending Moment, and Hinge Moment Coefficients versus Angle of Attack for the 3-Inch by 3-Inch Single Slat Fin.

Conclusions

Although the wind tunnel tests were not an exhaustive matrix analysis of grid fins, several interesting observations can be made:

1 . Vortex lattice theory is adequate at least

It over the linear range in predicting the load and moment distribution on grid fin configurations. is inadequate for angles of attack over 8 to 10 degrees.

6

4'00r-- 300-

7 00

1 ooc-- i .-- -800 -400 c

THEORY TEST DATA, 1

400 800 1700 1800 2 ALPHA (deg)

3.00- , I

1001 I ~ ,

-800 -400 000 400 800 1200 1800 7000 ALPHA (deg)

ALPHA (deg)

Fig. 8 Normal Force. Root Chord Bending Moment. and Hinge Moment Coefficients versus Angle of Attack for the 3-Inch by 6-Inch Single Slat Fin.

2. Lift and drag tailoring is possible for grid fins by appropriate design of the internal grid structure.

3. For maximum lift and minimum drag, it appears that one should design the internal structure with as many "horizontal" elements as possible. Cross members add integrity to the structure and drag but are not efficient lift producing members.

4. Grid fin configurations demonstrate highly nonlinear CN-alpha curves, at least above 8 t o 10 degrees. It appears that as the complexity of the internal grid work increases that the fluid

particles tend to be "captured" and the fin acts like a tube which channels flow through it. Because of this tendency, stall of the overa l l fin

which results in very efficient lifting surfaces. is delayed to relatively large angles of attack 'J

5. Grid fins, because of their low hinge actuator requirements, may be efficient as lifting surfaces when flow in the normal transverse direction but could be used as '"drag'' brakes if turned to a horizontal position.

6 . Because of their simplicity, grid fins could be used as efficient "trim tabs" attached to fuselages of large vehicles.

References ___ 1. Falkner. V. M.. "The Calculation of

Aerodynamic Loading on Surfaces of Any Shape," R. & M. NO. 1 9 1 0 , British A.R.C.. 1943.

2. Falkner, V. M., "The Accuracy of Calculations Based on Vortex Lattice Theory," Report No. 9 6 2 1 , British A.R.C., 1946.

3 . Falkner, V. M., "The Use of Equivalent Slopes in Vortex Lattice Theory," R. & M. NO. 2293, British A.R.C., 1946.

4 . Rubbert, Paul E., "Theoretical Characteristics of Arbitrary Wings by a Non-Planar Vortex Lattice Method," Document NO. 06-9244, Boeing Company. February 1964.

5. Hedman, Suen G., "Vortex Lattice Method for Calculation of Quasi Steady State Loadings on W Thin Elastic Wings in Subsonic Flow," FFA Report 1 0 5 , Aeronautical Research Institute of Sweden, 1966.

6 . Belotserkovskii, Sergei Mikhailovich, Theory of Thin Wings in Subsonic F k , Plenum Press, New York, 1967.

7. Feifel, W. M., "FORTRAN IV -Program zur Berechnung YO" V -Leitwerken," Mitteilang im Institut fur Aerodynamik und Gasdynamik der Technischen Universitut, Stuttgart, Germany, 1966.

8. Margason, Richard J., and Lamar, John E., "Vortex-Lattice FORTRAN Program for Estimating Subsonic Aerodynamic Characteristics of Complex Planforms," NASA TN E-6142, 1971.

9 . Pistolesi, E. , "Considerations on the Mutual Interference of Airfoil Systems," L.G.L. Report, 1937, pp. 214-219.

10. Byrd, P. F., "Erganzung zu dem Autsatz von N. Scholz, Beitrage zur Theoria der Tragenden Flache," 1ng.-Arch., Bd. XIX, Heft 6 , 1 9 5 1 , pp. 321-323.

11. Abernathy, J. M., and Burkhalter, J. E., "Load Distribution on Multielement Deformed Airfoils with Gap Effects," Journal of Aircraft, Vol. 1 9 , No. 1 0 , October 1982, pp 831-838.

12. Purvis, J. W., and Burkhalter, J. E., "Simplified Solution of the Compressible Lifting Surface Problem," A I M Journal, Vo1. 2 0 , No. 5, May 1 9 8 2 , pp. 589-597.

W