An Approximate Method to Calculate Nonlinear Rolling Moment Due to Differential Fin Deflection

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 09-1 January 01, Nashville, Tennessee AIAA An Approximate Method to alculate Nonlinear Rolling Moment Due to Differential Fin Deflection F. G. MOORE * and L. Y. MOORE Aeroprediction, Inc., King George, VA 485 Abstract A new nonlinear method has been developed to predict the roll moments due to differential deflection of fins (often referred to as roll driving moment). The method utilizes the nonlinear aerodynamics currently available in the 009 version of the Aeroprediction ode (AP09) for the horizontal wing alone and wing-body aerodynamics. Specialized treatment is given to the leeward and windward plane fins as well as engineering approximations to handle the physics associated with fin-to-fin interference, wing-tail interference, a fin in close proximity to a large wing upstream, and fins located on a boattail. omparison of the new method to seven body tail configurations, five wing-body-tail cases with tail control and six wing-body-tail cases with canard control showed the new method followed the general trends of the experimental data for most cases. Results were quite good for most cases with the poorest agreement with experiment being for canard control cases where the tail fins were very large. Results from several of the cases where comparisons of the new theory to experiment have been made are presented in this paper. Nomenclature AOA = Angle of attack SB, SBT = Slender body, Slender-body Theory A REF = Reference area (maximum cross-sectional area of body, if a body is present, or planform area of wing, if wing alone)(ft ) A w, A f = Wing or fin planform area (ft ) AR = Aspect ratio = b /A w b w, b c = Wing or canard span (not including body)(ft) l = Roll moment due to differential fin deflection, l/(qa ref d ref ), u m = Unmodified and modified moment, respectively pd V = Roll damping moment coefficient ( ( )) P δ = Roll moment due to differential fin deflection derivative, N = Normal force coefficient = Negative canard shed vortex normal force coefficient on tail N T ( V ) ( ) W N N δ (per deg) W, α = Normal force coefficient or normal force coefficient derivative of wing alone, = Normal-force coefficient of wing or tail in presence of body NW ( B ) NB( W ) N α = Normal-force coefficient derivative (per radian) due to angle of attack P = Pressure coefficient P P 1/ ρ V c, c r, c t = Local chord, root and tip chord, respectively d ref = Reference body diameter (ft) F S, F = Factors used to multiply leeward and windward plane roll moment FIF = Fin-to-Fin Interference factor * President; drfgmoore@hotmail.com; Associate Fellow of AIAA omputer Scientist 1 opyright 01 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Go

2 F(α), F V, F B = Factors used to allow decay of the wing-tail interference with angle of attack and to represent the loss in roll moment from a tail located close to a wing or on a boattail, respectively K W(B), k W(B) = Ratio of normal-force coefficient of wing or tail in presence of body to that of wing or tail alone at δ = 0 deg or α = 0 deg, respectively M, M S = Mach number where FIF attains its value of 1.0 or Mach number where FIF starts to increase from its SBT value, respectively M = Freestream Mach number p = Roll rate (rad/sec) P L, P = Local and freestream pressure, respectively (lb/ft ) Q L, Q = Local or freestream dynamic pressure, respectively (lb/ft ) r B, r W = Base radius and body radius at wing, respectively (ft) r r = Reference body radius (ft) r/s = Local body radius/(body radius + wing semispan) V, V L = Freestream and local velocity, respectively (ft/sec) x,y,z = oordinate system, x along body axis, y out right wing, z up X G = Distance to center of gravity (ft) X P /d = enter of pressure (calibers from some reference point) X LE, X AFT = Distance from nose tip to wing or tail leading edge or afterbody, respectively (in feet or cal from reference point that can be specified) y P = Distance from body centerline to wing centroid of presented area α = Angle of attack (deg) Λ LE = Sweepback angle of wing or canard (deg) γ = Ratio of specific heats of air δ = Wing, canard or tail deflection (deg) δ* = Boundary layer displacement thickness (ft) φ = Velocity potential Φ = Roll position of missile fins (Φ = 0 deg corresponds to fins in the plus (+) orientation. Φ = 45 deg corresponds to fins rolled to the cross ( ) orientation). θ = Local angle between tangent to body surface and freestream velocity (deg) θ b = Boattail angle (deg) μ = Mach angle (deg) 1,,3,4 = Fin numbering system with fins 1 and 3 being the leeward and windward fins and fins and 4 being the horizontal plane fins I. Introduction Many fin stabilized rockets and mortars differentially deflect a portion of (leading or trailing edge) or all of their fins in order to generate a small amount of roll. By differential deflection of the tail fins is meant that a pair of wings are deflected in opposite direction as opposed to being deflected in the same direction for either pitch or yaw control. The differential deflection of the fins is generally designed so as to give a clockwise roll when viewing the weapon from the rear or from the gunner s perspective. The roll moment generated is generally referred to as the roll driving moment. The low level of roll for differentially deflected fins is designed to be high enough to minimize weapon dispersion due to manufacturing asymmetries of the weapon or launch asymmetries from the rocket motor. The fin deflection is also designed so that the roll rate will not be close to the natural frequency of the weapon in order to minimize the possibility of roll-yaw lockin. Most unguided weapons will deflect all or a portion (either leading or trailing edge) of the tail fins by a slight amount. Usually a degree or less of deflection for each fin is adequate. Guided weapons also differentially deflect fins to provide roll control. The fin deflection can be from either the canards or tails. If the canards are the control surfaces, additional physics must be accounted for due to the canard shed vortices impacting the tail surfaces. There have been several attempts in the past to predict roll moments of configurations having differential fin deflection. The earliest attempt was by Adams 1 for slender cruciform wings. Adams 1 used slender body theory to predict the roll moment for both planar and cruciform wings. Adams and Dugan extended it to slender wing-body configurations a year later. The results of Refs 1 and are very important with respect to wing-wing interference effects. Nielson 3 extended the Ref analysis from and 4 fins to 6 and 8 fins. Nicolaides and Bolz 4 used basically strip theory to bring Mach number into the roll moment computations since the slender body theory of Refs 1-3 are Mach number independent. Oberkampf 5 extended the method of Ref 4 from linear pressure distributions on the fin to a cubic theory. He applied the method to the Army-Navy Finner configuration at supersonic speeds and for angle of attack (AOA) zero and got good agreement between theory and experiment for each and. Oberkampf 5 also had some success in predicting induced roll up to about 0 deg AOA, as well as predicting roll damping for a δ P

3 limited set of conditions. The Ref 4 and 5 methods were limited to supersonic flow due to the method they used to predict the pressure distribution on the airfoil surface. Oberkampf 6 extended his method to the transonic flow region by using a strip theory for local fin pressure distributions. The local fin distributions were approximated by functions Oberkampf defined based on observing experimental pressure distributions on fins. Both Refs 5 and 6 assumed no body-wing interference and both were limited to only one set of lifting surfaces. Prakash and Khurana 7 computed the roll damping and roll driving moments at zero AOA by assuming an elliptic spanwise load distribution on the fin, with each chordwise segment assuming a -D pressure distribution. Reference 7 showed reasonable agreement with experiment for a limited number of cases at supersonic speeds. Miller and Burkhalter 8 conducted experiments and used lifting surface theory to predict fin-fin interference effects at supersonic speeds for cruciform missiles. The Ref 8 method showed reasonably good comparisons to experiment for AOA to about 10 deg and at various roll angles. Landers, et al 9 compared several approximate, as well as exact numerical codes to experimental data for a canard controlled missile. For roll moment coefficient, no code gave completely accurate results; however, the full Navier Stokes code in general gave the best results as may be expected. AOA up to 15 deg and Mach numbers 3-6 were considered. Wienacht and Sturek 10-1 used full Navier Stokes equations to predict the roll driving moment on a configuration that had beveled leading and trailing edge fins to provide roll. They showed good predictions compared to data at low AOA and Mach numbers To summarize, of all the theoretical methods used to calculate nonlinear roll moment due to wing deflection, there is no general approximate method available. The full Navier Stokes methods of Refs 9-1 are beyond the scope of approximate methods. The method of Ref 6 is nonlinear but too limited in scope (transonic Mach numbers and one set of lifting surfaces). It is therefore the objective of the present work to develop a new approximate method to compute nonlinear roll moment due to differential wing deflection. The method should be robust enough so it can be applied to most tactical weapons under realistic flight conditions. This means Mach numbers 0 to about 6, angles of attack up to about 30 deg, up to two sets of lifting surfaces, and with 4, 6 or 8 tail fins and with or 4 canards allowed. II. Physical Phenomena In Nonlinear Roll Driving Moment Prediction Before an approximate method can be developed to predict the nonlinear roll driving moment l or δ, the physical phenomena involved in the nonlinear moment must be understood. Figure 1 attempts to illustrate the axis system we will be using along with typical nomenclature. In viewing Fig 1, it is seen that based on the right handed x-y-z axis system used in the Aeroprediction ode (AP), positive roll is counterclockwise as viewed from the rear. This means that with respect to the present axis system, most weapons will roll in the negative direction since they roll in the clockwise direction as viewed from the rear or gunners position. Figure attempts to illustrate some of the nonlinear aerodynamics phenomena that must be modeled. In viewing Fig A, one can in general have a body-tail or canard-body-tail configuration at low, medium, or high AOA. The fins can have small or large span depending on the airframe design. If the Mach number is supersonic, the bow shock can intersect the windward plane canard or tail if the combination of AOA and Mach number is high Figure 1. Axis system used in aeroprediction code enough. The windward plane fin would not only be in a and positive roll for a cruciform fin weapon. high compressibility region, but may have additional aerodynamics due to the intersection of the shock with the fin. The leeward plane fin would experience less effectiveness due to being in a reduced pressure region. The horizontal plane fins of Fig A would experience nonlinearities due to AOA in both the wing alone and wing-body interference term due to AOA and control deflection. To get a positive roll as shown in Fig 1, would require fin to be deflected leading edge up and fin 4 leading edge down. Fin 1 would have the leading edge in the negative y direction or to the left whereas fin 3 would have the leading edge deflected to the right or in the positive y direction. If the roll is in the clockwise, or negative direction with respect to our axis system, (Fig A), then all the deflections discussed above for a positive roll would be in the opposite direction. Fins and 4 of Fig 1 would have a nonlinear load at a combined α ± δ in addition to loads generated by roll and fin to fin interference (Fig B). Fins 1 and 3 would have loads due to ±δ only plus any 3

4 loads due to roll, fin to fin interference, as well as additional loads from being in the compressible windward or low density leeward plane (Fig ). Figure 3 (taken from Ref ) attempts to illustrate the wing loads due to differential deflection as well as fin to fin interference. The wing loads due to deflection have already been discussed. The loads due to fin to fin interference come from both roll and deflection. In roll, the more fins that are present, the more interference exists from one fin to another. This type of fin to fin interference means that 4 fins are not twice as effective as two fins in terms of either roll damping or in roll driving moment. The fin to fin interference from a wing deflected actually causes a negative pressure distribution on an adjacent wing, negating some of the fin effectiveness due to deflection, as seen in Fig 3. The final physical phenomena one needs to mathematically model for roll driving moment is canard shed vortices if the configuration is a canard or wing-body-tail. If the canards are deflected, the vortices will be stronger than if they are at AOA only. If the configuration is at zero roll and only the planar fins deflected, only two canard shed vortices need be considered. If all four fins are deflected, the horizontal fins will have stronger vortices (due to being at α ± δ) than the windward or leeward plane fins which will have vortex strength primarily from fin deflection. If the vehicle is at Φ = 45 deg roll, all four forward lifting surfaces will generate vortices, regardless of whether the fins are deflected or not. The net result of the vortices from the forward lifting surfaces impacting the tail surfaces is to reduce (and in some cases change the direction of) the roll driving moment. In addition to canard or wing shed vortices impacting on the tail surfaces, an additional loss on the tail surfaces can occur if the forward lifting surfaces are large and close to the tails. The tail surfaces are in the wake of the wings, even if the wings are not deflected, and thus the tails are not as effective if they are used for roll control. Also, if the tail surfaces are on a long boattail, significant loss of tail effectiveness for roll control occurs due to a large boundary layer displacement thickness. One physical phenomena we have not discussed is induced roll. That is because the Aeroprediction ode, by definition, is a planar code designed to provide aerodynamics for conceptual design or for particle ballistic or trim (3 degree of freedom) trajectory models. As such, the AP is limited to roll positions of Φ = 0 deg (fins in plus ) or Φ = 45 deg (fins in cross ) roll orientation. In principal, if the fins are equally deflected, no yaw is present, the model is perfectly machined, and no flow angularity exists in the wind tunnel, the induced roll should be zero. Table 1 summarizes the physical phenomena that must be mathematically modeled in order to predict nonlinear roll driving moment. The bow shock Figure. Some of the physical phenomena that occur on a rolling missile. Figure 3. oefficient of rolling moment effectiveness for cruciform wing-body combinations with differential incidence of the horizontal surfaces (taken from Ref.). 4

5 Table 1. Physical phenomena for a wing-body-tail with differential wing deflection. intersecting a fin will only occur at higher Mach number and AOA Nonlinear wing alone and wing body loads due to AOA and or control deflection conditions. Thus from a Windward plane compressibility and leeward plane separation practical standpoint, this Bow shock intersecting windward plane fin physical phenomena is Fin to fin interference the least important of Wing-tail interference those listed in Table 1. Wake effects of a large forward fin on the tail The next section of the Loss of tail effectiveness when located on a long boattail paper will deal with the mathematical model for each of these physical phenomena listed in Table 1, except for the explicit treatment of the bow shock intersecting a wing or tail surface. III. Mathematical Model To Define Nonlinear Roll Moment Due To Differential Fin Deflection This section of the paper will define the mathematical model used to compute the roll moment changes due to the physical phenomena of Table 1. A. Nonlinear Wing Alone and Wing-Body Loads Due to AOA and or ontrol Deflection This section of the paper will discuss the calculation of the normal force and roll moment coefficient of fins and 4 of Fig 1. Fins 1 and 3, which are in the leeward and windward plane respectively, will be discussed in the next section where we will discuss leeward plane separation and windward plane compressibility effects. We will analyze the roll moment at a roll orientation of Φ = 0 deg for convenience. However, the same analysis could be applied to any roll orientation, particularly Φ = 45 deg. One of the key parameters needed in predicting nonlinear roll moment on a fin is the normal force on the fin in the presence of the body. The method used is described in detail in hapter 5 of Ref 13, with refinements of Ref 13 given in Refs 14 and 15. The wing-body normal force coefficient is given by 13 [ ]( ) W N = K ( ) W( B) sinα + k W B W( B) sinδ N (1) α Each of the terms K W(B), k W(B) and ( N α ) are nonlinear in AOA or control deflection and Mach number. The W nonlinearities were based on several large wind tunnel data bases in addition to other missile data used in validating and refining the values of the nonlinear parameters, particularly k W(B), as a function of M, α. As an example of the nonlinearities, consider the wing-body interference term K W(B). K W(B) is typically close to the slender body theory value at zero AOA and at all Mach numbers. However, at low Mach numbers, as α increases, K W(B) will typically decrease to a value of 1.0 at high AOA. On the other hand, as Mach number increases K W(B) will approach one at fairly low AOA s. k W(B) was defined 13 empirically based on comparing the Aeroprediction ode to experimental data for several missile configurations that had control deflection at both Φ = 0 and 45 deg. The approach to compute k W(B) was based primarily on the fact that the primary data base 16 used for defining the K W(B) had small control surfaces. Also, only a limited number of deflections were done in the Ref 16 tests. As a result, k W(B) had to be refined using other missile data bases. The wing alone normal force coefficient of Eq (1) is estimated 13 using a 4 th order equation in total AOA (where total AOA is α + δ ), where the constants were determined using several wing alone data bases The local slope, was then computed at the given value of α W (α + δ). ( N α ) W The roll moment for a single fin, or 4 of Fig 1, is:,4 [ N N ] ( y )( ) ( ) = () P where y P is the spanwise center of pressure. Note that using our sign convention, fin is leading edge up or positive and fin 4 is leading edge down or negative. Thus, both normal force terms in Eq () will generate positive roll moments since there is a negative sign. The spanwise center of pressure is available in the AP09. omparing the AP09 values of in front of ( N W(B) ) 4 y P for wings of different planforms, an approximation for y P is W(B) W(B) 4 5

6 y P 0.95 ya + rw = (3) d ret where for trapezoided wings, b 3 c = t 1 c t y A (4) 4 cr cr y P applies to all the fins 1-4 of Fig 1 whereas the wing-body normal force coefficient will be different for wings 1-4 as the wings are in different locations and will be deflected differentially. B. Windward Plane ompressibility and Leeward Plane Separation Effects Fins and 4 were subject to body upwash along with the nonlinear wing normal force due to AOA and control deflection effects. However, fins 1 and 3, the leeward and windward plane fins respectively, are not appreciably affected by body upwash, but are subject to changes in load due to either being in a compressible flow field or in the separated flow region of the leeward plane where the body is at AOA. Thus, the roll moment for fins 1 and 3 is where y P is given by Eqs (3) and (4) and ( from the N α ) W nonlinear wing alone data bases as discussed earlier Note that δ 1 is deflected to the left, giving a negative times a negative, so that both terms of Eq (5) will give a positive roll moment for positive wing deflections. It should be noted that to get clockwise roll, which is the normal roll direction of weapons, requires a negative control deflection of all fins using the axis system of Fig 1. Also note that Eq (5) has two factors F S and F, for separation and compressibility effects respectively (see Fig 4A). These two factors will now be defined. We will consider F S first. To compute F S, we will assume that separation begins at [ ( ) δ F + F ( ) ] 1, 3 = yp N W 1 S N δ α α W 3 (5) Figure 4. Separated flow and compressibility factors for roll moment effectiveness of leeward and windward plane fins. the base of the body and progresses to the nose tip over a 50 deg AOA range. In numerical experiments investigating the effectiveness of a leeward plane fin of r/s = 0. to 0.5, Fig 4B was derived. Fig 4A indicates the leeward plane fin loses its effectiveness as AOA increases. The effectiveness is lost slowly up to about 0 deg AOA, but then the rate of loss increases until it is assumed the leeward plane fin is completely ineffective in producing roll at 50 deg AOA for r B /r ref = 1.0. Note that as the diameter of the base decreases, the effectiveness of the fin 1 is lost even faster as AOA increases, assuming fin 1 lies on the boattail. If fin 1 lies ahead of the boattail, the Fig 4B curve for r B /r ref = 1.0 will be applied to fin 1 to compute the factor F S. The windward plane fin will be assumed to be in a compressible flow region (see Fig 4A). At high Mach number and AOA, the fin could be intersected by the bow shock, generating an additional roll moment term which will be discussed in the next section. We will attempt to calculate the additional roll moment on the windward plane fin due to compressibility effects. Typically, compressibility effects start around M = 0.5. In deriving a simple approximation to compressibility effects, Newtonian Impact Theory (NIT) will be utilized. Experience has shown NIT can be utilized as low as M = 1.5 with good results. Thus: 6

7 F F F = 1.0 = = ; M ( F ) 1 + ( F 1) M ; 0 M 1.5 ( F ) ; M (6) where ( F ) 1 is the value for the compressibility factor computed by NIT. Newtonian Theory States = sin θ (7) P where θ is defined as the angle between the velocity vector and a local tangent to the body or wing surface. Since we are using Eq (7) for the wing deflection δ, the angle θ can be shown to be or since Φ = 180 for the windward plane, then ( sin δcosα sin αcosφcosδ) cosλle sin θ = (8) θ = sin 1 [( sin δcosα + sin αcosδ) cosλ ] LE (9) Also, the local dynamic pressure ratio to that of the freestream can be approximated by (see Ref 0): Q Q L γ M cos α γ M α [ 1+ γm sin θ] = cos (10) for M 1.5. For M < 1.5, the value of Q L /Q at M = 1.5 is used. Here θ of Eq (10) is defined by Eq (9). Thus = QL 1 (11) Q ( F ) and Eq (6) defines the compressibility factor for fin 3 and Eq (5) can now be used to compute the roll moment for fins 1 and 3 of Fig 1. Figure 4 summarizes the method to compute the compressibility factor F for fin 3.. Bow Shock Intersecting Windward Plane Fin As the combination of Mach number and AOA increase, the bow shock will intersect the windward plane fin. The farther forward on the body the fin is located (or the closer to the nose the fin is located), the lower the value of Mach number and AOA increase needed for the shock to intersect the fins. If the nose is blunt or truncated, the bow shock angle will start out at 90 deg. At zero AOA, the bow shock will curve and eventually approach the Mach angle. 7 1 µ = sin 1 (1) M As AOA increases, the shock in the windward plane approaches the body whereas the bow shock in the leeward plane gets farther from the body. Initially, since the bow shock intersecting the lifting surface occurs primarily at high M and α, we will neglect the additional roll moments generated by this physical phenomena and instead focus on the compressibility effects on the windward plane fin. The windward plane compressibility effects occur over most Mach numbers (M 0.5) and AOA s. D. Fin To Fin Interference Effects Adams 1 and Adams and Dugan applied Slender Body Theory (SBT) to Wings Alone 1 and Wing Body onfigurations to calculate both roll damping moment and rolling moment due to differential wing deflection. It is

8 believed the SBT results of Refs 1 and should be reviewed as the SBT will be the approach used for fin to fin interference prediction. The linearized small perturbation equation is given by (excluding transonic flow): ( M ) φ + φ + φ = 0 1 XX YY ZZ (13) where the subscripts xx, yy, and zz indicate the second partial derivative of the velocity potential, φ. Adams rationalized that if φ XX is small and M not too large, then the first term of Eq (13) can be neglected compared to the velocity gradients in the y and z direction. Equation (13) then becomes φ + φ 0 (14) YY ZZ = which is the conventional La Place equation. Dugan and Adams solved Eq (14) for planar wings, cruciform wings, planar wing-bodies and cruciform wing bodies for both roll damping moment and roll moments due to differential wing deflection. Without going through all the theoretical details of Refs 1 and, the results will be summarized here. Adams derived a value of roll damping from cruciform wings as 1.6 that for planar wings. The fact the value of 1.6 for cruciform wings is not.0 times that for a planar wing is a form of fin to fin interference effects. Nielsen 3 extended the SBT of Adams to an infinite number of fins. In particular he showed the roll damping for 6 and 8 fin cases was.0 and.3 times the roll damping of a planar set of fins. To summarize SBT fin to fin interference effects for roll damping we have: ( ) = 1.37( ) P 3f P f ( ) = 1.6( ) P 4f P f ( ).0( ) = ( ) P 6f 1.3 P f P 4f ( ).3( ) = 1.4( ) P 8f P f P 4f The above values for fin to fin interference effects on roll damping are already utilized in the AP09. In addition to the fin to fin interference effects, Adams and Dugan also gave values of as a function of r/s using SBT. P The SBT values of roll moment due to differential wing deflections from Ref are defined for planar wings, for cruciform wings when only a pair of fins are differentially deflected, and for cruciform wings with all four fins differentially deflected. In particular, they showed for a cruciform set of wings with only one pair of fins differentially deflected If all 4 fins are differentially deflected then ( ) = f ( ) δ f (15) (16) ( ) = 1.5 4f ( ) δ f (17) Notice the fin to fin interference for cruciform fins result in a loss of 4 percent of δ for 4 fins compared to planar fins. Adams and Dugan also defined the rolling effectiveness of a plane wing as P f = 1.70 (18A) and for a cruciform wing as P 4f = 1.59 (18B) Thus 8

9 P 4f P f 1.59 = = (19) Hence, even though the roll damping of a cruciform wing is only 6 percent greater than that of a planar wing and the roll moment due to differential fin deflection of a cruciform fins is only 5 percent greater than that for a plane wing, the rolling effectiveness of cruciform fins is only 6 percent less than that for planar wings. While not defined in either Refs 1-3, it will be assumed the 6 percent reduction in rolling effectiveness will apply to 3, 6 and 8 fin configurations as well. In other words The factor of Eqs (16), (17) and (0) includes the loss of Reference also defined the body interference on δ using SBT. Figure 5 shows the results of Ref plotted in a form that is a function of δ where no body is present and as a function of r/s. The value shown as the AP09 of Fig 5 is actually the same value used for for body interference as well. In examining Fig 5, once the value of δ ( ) = ( 0.94)( 1.37)( ) 1.9( ) δ 3f ( ) ( )( = δ ) f ( f = 1.9 / 1.54 = 0.85 ) 4f 4 f ( ) = ( 0.94)(.0)( ) 1.88( ) δ 6f ( )( )( = δ ) f ( ) f = = 1.16 δ 4f 4 f ( ) = ( 0.94)(.3)( ).16( ) δ 8f = δ f f ( ) = 8f ( 0.94)( 1.4)( ) = f ( ) δ 4f δ for the fins is computed, the value of δ where the body is present can be computed. Note there is little effect of the body on up to values of r/s = 0.4. δ Also note there is little difference between or 4 fin cases as well. Defining the values of fin interference factor (FIF) as Eqs (16), (17) and (0) and the SBF from Fig 5, Table summarizes the results of SBT when applied to roll moment due to differential fin deflection. The factors of Table are to multiply roll moments computed due to differential fin deflection of two fins for various numbers of fins. For example 8/8 signifies 8 fins deflected and the roll moment of the 8 fins is.16 times that of fins. Before we leave the section on fin to fin interference, the assumptions associated with SBT need to be evaluated with respect to their impact on expected accuracy of results. Table 3 lists the SBT assumptions. δ due to fin-to-fin interference. 9 (0A) The first assumption is believed to not be as important for roll moment due to fin deflection as it is for axial or normal force. The AP09 uses second order perturbation methods primarily for axial force computation and first order perturbation methods for normal force computation of the body. Furthermore, the values of wing-body normal force and interference effects use SBT, but also perturbation theory as well as large wind tunnel data bases for computation of wing alone and wing-body interference effects. As a result, assumption 1 has been minimized to a great extent by the way N W(B) and ( N W ) α are calculated in the AP09. It is believed the second assumption of Table 3 was taken care of in the AP74 (see Ref 13, page ) and has continued in all follow-on versions of the AP. In essence, what was done for sweptback trailing edge fins was (0B) (0) Figure 5. Effect of body radius on differential incidence of the horizontal surfaces.

10 Table. Fin interference factor (FIF) and slender body factors (SBF) for various number of fins. FIF for No. fins/no. fins deflected r/s SBF/ ( lδ ) / 4/ 4/4 6/6 8/8 r= Table 3. Most significant slender body theory assumptions. Body is slender (perturbations in freestream due to presence of body are small) Fins are triangular in shape with no sweep on trailing edge Fins are low aspect ratio (slender) Mach number not too large (flow is adiabatic and isentropic so shocks are weak) to reduce the wing-body and body-wing interference factors by the ratio of the chord of the sweptback trailing edge fin to the no sweepback wing. This simple engineering approximation has worked quite well. Assumptions 3 and 4 of Table 3 are more troublesome. Experience has shown that SBT can be applied to wings of aspect ratio up to about one (and at low AOA) with reasonable accuracy. The higher the aspect ratio, the more inaccurate SBT becomes. It is expected that some of the error in using SBT for large aspect ratio fins is reduced by forming ratios of SBT results for to 4 fin cases or 6 to 4 fin cases, etc. Experience has also shown that as Mach number increases above about, the isentropic flow, or low Mach number assumption, becomes more and more questionable. Again, using SBT as a ratio of 4 fin to fin results should reduce the error somewhat. Many cases of practical interest combine assumptions 3 and 4. That is, the weapons have moderate to high aspect ratio wings flying at Mach numbers much greater than.0 and at AOA. It is therefore expected that the FIF and possibly SBF of Table will need to be modified empirically to minimize the SBT assumptions 3 and 4. Based on comparing theoretical results using the Table factors to data, the following modifications to the Table factors for high aspect ratio and Mach number appear to be needed. It was found for small fins (A f /A r 0.5) minimal fin to fin interference was present and FIF = 1.0 at all Mach numbers. For most fins, it was found the SBT values of Eqs (16), (17) and (0) were applicable up to a certain Mach number where the FIF started approaching 1.0 (no fin to fin interference). Designating M S as Mach number where FIF starts increasing from say its 4 fin value of 0.76 and M as the Mach number where FIF = 1.0, we have: M M S = 1. = 1.8 ; ; FIF = 0.76 FIF = 1.0 (1) In addition to an exception to Eq (1) for small fins, where FIF = 1.0 at all Mach numbers, it was found that when we have large fins (r/s 0.5 and A f /A ref 3.0), the values of Eq (1) were increased. In other words, the SBT value of 0.76 was valid for higher Mach numbers. An empirical equation for M S was derived and is given by: where i = 0 if δ < 5 and i = 1 if δ 5 ( δ 5) i A f r /s MS = and M = 1.5 MS Equation () says that for large fins, fin to fin interference remains present at a much higher Mach number than for medium size fins. Not enough 6 and 8 fin cases exist to draw conclusions on FIF as a function of Mach number, as we did for cruciform cases. Thus, for the time being, it will be assumed () 10

11 ( ) 6f = 1.16( ) 4f ( ) ( ) = f 4f (3) In other words, the modified FIF s for cruciform fins of Eqs (1) and () will be assumed to apply to 6 and 8 fins as well. E. Wing-Tail Interference The wing-tail interference term N T(V) is also calculated in the Aeroprediction ode for both Φ = 0 and 45 deg. At Φ = 0 deg, two wing shed vortices are considered whereas at Φ = 45 deg, four wing shed vortices are used in the computation process. is nonlinear in both Mach number and AOA. The details of the nonlinear N T(V) methodology for N T(V) are given in Ref 13, pages In general N T(V) is a negative normal force term when the configuration is at a positive AOA and control deflection. The negative N T(V) tends to decrease static margin but also will decrease roll damping and roll moment due to a control deflection if the deflection is from the forward lifting surfaces. If the aft fins are used for roll, then the forward fin shed vortices will tend to cancel out on the rear fins with respect to roll moment. That is because one forward fin shed vortex will tend to increase the roll moment on the tail whereas the other forward fin shed will need to vortex will decrease the roll moment by an equal amount, assuming no yaw is present. Thus [ ] N T(V) be used in computing roll moment due to differential fin deflection, where the δ indicates differential fin deflection of the forward lifting surfaces. In using the values available for nonlinear wing-tail interference in the AP09 for calculating differential roll moments, it was found that the values near zero AOA and some value of δ worked quite well. However, as AOA increased, it was found the values of N T(V) became asymmetric for many cases when α + δ switched signs. While this behavior is typical and correct for calculating aerodynamics in pitch, experimental data suggested symmetric values of l due to as AOA increased. As a result, an approximation was arrived at to represent the N T(V) nonlinear nature of N T(V) in a symmetrical sense, while using all the AP09 methodology near zero AOA. Using this approach we have: ( ) = ( ) ( α) = ( ) [ α F α T V) α±δ α for 0 α 0 ( ) = ( ) ( α ) α α±δ for 0 < α 40 N N N T(V) T(V) 3 N T V = 0, α > 40 δ = α N T(V) δ = α α δ = N T V 0 0 ]; α in degs [ 0] F. Loss of Tail Efficiency Due to Being in Wake of Large Wings in lose Proximity to Tail When a tail surface that is used for providing roll control is in close proximity to a larger span lifting surface upstream, some loss of dynamic pressure exists at the tail due to the wing wake. This loss of dynamic pressure can cause a reduction in control effectiveness. Results of several cases suggests an approximate equation to account for this loss in control effectiveness is: F V (4) = (5A) M U where F V = 1 for wing/canard control 3 X TW = for tail control (5B) 3 11

12 and X TW is the distance in calibers between the trailing edge of the forward lifting surface and the leading edge of the tail. That is X TW ( X ) ( X ) LE T ( + ) r LE W rw = (5) d and X TW 3.0 cal. If X TW > 3.0 cal, F V = 1.0 and no loss in tail control effectiveness is assumed. G. Loss of Tail Effectiveness Due to Being Located On a Long Boattail The last physical phenomena we will try to account for in deriving a robust, approximate, nonlinear model to predict roll moments due to differential fin deflection is the case when the control surfaces are located on a long boattail. Practical examples of weapons that have their tail fins on long boattails include mortars and low drag bombs. The nonlinear roll and pitch damping methods developed for the AP09 0 required special attention for configurations with long boattails. The fin span was reduced by approximating the boundary layer displacement thickness (BLDT) and a special model was developed for nonlinear roll damping on configurations with tail surfaces on long boattails. Utilizing the experience of Ref 0, the BLDT for a long boattail case can be approximated by δ * X X aft = 0.10 θb (6) dr where θ b is the boattail angle and (X X aft )/d r is the distance in calibers of the mean geometric center of the tail to the start of the boattail. In actuality, the actual BLDT is generally larger than Eq (6) suggests since there is some BLDT from the nose of the body to the start of the boattail. The loss in normal force on the tail due to BLDT is a combination of both the reduced fin span and reduced aspect ratio. For low aspect ratio fins, SBT states = πar N α (7) so that the N α is directly proportional to the aspect ratio. Eq (7) is a good approximation for low AOA up to aspect ratio of about 1.0. Most low drag bombs and mortars fit this aspect ratio range. Since the moment arm is increased somewhat by the BLDT, the loss in normal force is compensated somewhat by the increased moment arm. An approximation for roll moment loss due to BLDT and aspect ratio reduction is: X Xaft = F = M U B (8) U 3d r Equation (8) says that for a 3 caliber distance (X X aft ), one can lose 5 percent of the roll moment due to the fins being on a long boattail. Most mortars have X X aft actually larger than 3 calibers. H. Summary Mathematical Model for Roll Moment Of Differentially Deflected Fins The roll moment for differentially deflected fins will include most of the physical phenomena discussed previously. The overall equation for nonlinear roll moment is: + rw y = dr [( ) ] F ( ) NW rt +.95y d r 3 α= 0 t i W {[( N ) ( N ) ] [ N ] FS } W 1 α= 0 FIF ( N ) ( ) α F α FV FB 0 0 W T δ = W(B) α+δ W(B) α δ cosφ (9) 1

13 where i = 1 for wing/canard control = 0 for tail control y, of Eq (9) is defined by Eq (3); F by Eqs (6) (10); F S by Fig 4B; FIF by Eqs (16), (17), (0) (3); W y t F(α) by Eq (4); F V by Eq (5); F B by Eq (8). Of course, N W(B) and N W come from the AP09 for a given wing planform shape and given freestream conditions. Notice that cos Φ is included in Eq (9) because if the AP09 aerodynamics are at the roll position of Φ = 45 deg (wings in x orientation), is for 4 wings and must be multiplied by to convert to the Φ = 0 deg plane. If only two fins are differentially deflected versus four, the value of roll moment computed by Eq (9) is divided by, since Eq (9) was derived for a cruciform fin configuration. If 3, 6 or 8 fins are used for roll, the FIF of Eq (9) accounts for other than four fins differentially deflected. IV. Results and Discussion The mathematical model represented by Eqs (9), (4), (6)-(10), (16), (17), (0)-(3), (4), (5) and (8) was implemented into the AP09/11 and will be released as the AP09/1. A total of 18 unlimited distribution cases were considered. This includes 7 body-tail cases, 5 wing-body-tail cases with tail control, and 6 wing-body-tail cases with canard controls. One of the wing-body-tail cases includes canard body as well as two different tail sizes. Finally, three limited distribution cases were considered as part of the validation process, but these cases were not presented in Ref due to the distribution statement. In general, most results were quite encouraging. Results of the new theory compared to experimental data will now be presented for several of the 18 unlimited distribution cases. Reference should be reviewed for additional comparisons of the theory to experiment. The first case considered is the Standard Army Navy Finner (ANF) that has been studied extensively in the past. Figure 6 compares the predictions of roll moment predicted by the AP09 compared to experimental data as a function of AOA at M = 0. and.5, the only Mach numbers where data was given in Ref 3. Also, as a function of Mach number δ is given compared to both experimental data and two other approximate theories near zero AOA and for a fin deflection of 10 deg. As seen in the Fig 6, the new nonlinear theory does a pretty good job of predicting the roll moment as a function of AOA, although for M = 0., data was not available above 15 deg. Reference 3 stated that the roll moment could not be extracted from the methods they were using due to nonlinearities in the data at M =.. Also, it is comforting to see the excellent agreement to data for the low AOA comparisons as a function of Mach number, although no data was available below M = 1.3. The second body-tail configuration considered is the Navy Low Drab Bomb configuration shown in Fig 7. Results of A. Body-Tail ases δ are shown at M = 0.8 for AOA up to 5 deg and at low AOA for Mach number 0.6 to.0. The comparison at M = 0.8 of theory to experiment as a function of AOA is quite good. omparison of with experiment as a function of Mach number is quite good at transonic speeds. However, δ both the Ref 4 and 5 data was taken from the old supersonic tunnel at the former White Oak Laboratory. Based on the authors personal experience in testing in that tunnel, model blockage effects were practically impossible to 13 N W B Figure 6. omparison of theory and experiment for roll moment due to differential fin deflection for army navy finner configuration (all fins deflected indicated amount).

14 Figure 7. omparison of theory and experiment for roll moment due to differential deflection of fins (δ= deg) of Navy low drag bomb. eliminate at transonic Mach numbers. Hence, the fact the theory is higher than the data around M = 1.5 may not be completely realistic, as the data could have been impacted by wind tunnel blockage effects. The third body tail configuration where roll moment data was found was for the Small-aliber-Smart-Munition (SSM) round (see Fig 8). The SSM round is a very low drag configuration designed for high Mach number launch. It had 6 tail fins, all deflected 0.1 deg. Figure 10 also shows the AP09 approximation for the SSM round where wing area, span, and centroid of presented area were held constant. Data were given for AOA up to ±10 deg. The AP09 gives identical roll moment data for AOA plus or minus. Also the data for M =.5 was nearly identical for AOA plus or minus. However, the data at Mach 3.0 was significantly different for AOA plus and minus, and the reason was not stated in Ref 6. One would expect the roll moment to be about the same for AOA plus or minus. The AP09 code compares to the M =.5 data quite well up to the 10 deg AOA and it compares quite well to the negative AOA data for M = 3.0. Since l is directly proportional to and since decreases with N W(B) N W(B) Mach number, the negative AOA data for M = 3, which is slightly lower than that at M =.5, is more believable. In general, agreement of the AP09 to experiment for the body tail cases of Figs 6-8 is fair to good. The only easy configuration to model was the ANF with the remaining cases requirement engineering judgment on how to model them in the AP09 as well as geometry approximations in many cases. Anytime a configuration geometry is modified in order to fit the geometry requirements of the AP09 code, reduced accuracy generally occurs. B. Tail ontrol Wing-Body-Tail onfigurations Three different wing-body-tail (WBT) configurations that utilize tail control will be considered in the validation of the new methodology of Eq (9). Since the tails are used for control, the wing-tail interference is considered negligible for roll moment since it occurs equally on the left and right side of the weapon, thus cancelling out. 14 Figure 8. omparison of theory and experiment for roll moment due to differential deflection of fins (δ=0.1 deg). Figure 9. omparison of theory and experiment for roll moment due to differential fin deflection of a tail controlled missile.

15 The first case is one of the examples found in the user guide sent out to all new purchases of the AP09 and is referred to as UG Example 4. The configuration is shown in Fig 9 along with roll moment as a function of AOA up to AOA of 5 deg for Mach numbers of 3.95 and Results are given for Φ = 0 where only the horizontal fins and 4 (see Fig 1) are deflected differentially 0 deg and for Φ = 45 deg where all 4 tail fins are differentially deflected 0 deg. deg). Good agreement of theory and experiment 7 is seen up to 15 to 0 deg AOA at both Mach numbers. The roll Φ = 45 deg results are particularly impressive. The second tail controlled WBT configuration is shown in Fig 10. The Fig 10 configuration body fits within the scope of allowable geometries for the AP09, however the fin shape was not trapezoidal so the shape was modified using the automated fin planform module of the code where fin area, fin centroid of presented area and leading edge sweep are held constant. The modified configuration is also shown in Fig 10. For this case, the vertical fins 1 and 3 were deflected 10 deg and the roll moment due to the differential deflection was given for Mach 1.6 up to 10 deg AOA and up to 0 deg AOA for M =.0 and.5. The AP09 gives excellent prediction of roll moment compared to data 8 at all Mach numbers, even through the allowed fin input Figure 10. omparison of theory and experiment for roll moment due to differential fin deflection of tail fins or wing-body-tail configuration(φ=0, vertical fins deflected 10 Figure 11. omparison of theory and experiment for roll moment due to differentially deflected tails of a wing-body-tail configuration (Φ=0, all 4 tails deflected 10 deg). 15

16 geometry had to be modified to fit the code requirements. The last tail control case shown is a somewhat complicated configuration (see Fig 11) in that a nose spike and several external appendages were present (presumably for wiring, etc.). None of these geometry perturbations were modeled in the AP09 as the clean AP09 representation of Fig 11 indicates. All four tails were deflected 10 deg and results are given at M =.4,.96, 3.95 and Excellent agreement of theory and experiment 9 is seen at all four Mach numbers up to 0 deg AOA, where the data base ended. To summarize the comparisons of theory and experiment for the three tail controlled WBT configurations of Figs 9-11, it is fair to say that overall good agreement is seen. It is also interesting to note that the AP09/1 new theory for roll moment prediction due to differential deflection of the fins was actually better for the tail controlled WBT cases than for the wing body cases. It is suspected that this statement may not be true if the wing body cases were more conventional (such as ANF) and did not require significant geometry modifications to fit within the required inputs for the code.. anard ontrolled anard-body-tail onfigurations Three canard controlled WBT configurations will be shown in the validation of the new AP09 roll driving moment prediction capability. The canard controlled class of configuration is the most difficult of all the configurations to accurately predict the roll driving moment because here we must consider the wing-tail interference term. The wing-tail interference term is in general a term which has the opposite sign to the roll driving moment of the canards. This is because the vortex shed from the canards rotates in a counterclockwise fashion, creating a download on the tail fin behind the canard, and thus negating some or all of the canard load. In comparing the AP09 new roll driving moment model to experiment, it is also believed that the vortices shed from the canards may also contribute a positive contribution to the vertical tail, particularly for large tails and short span canards. The vortex shed from the canards will have a download on the horizontal tail, creating a roll moment in the opposite direction to the canard term. However, this same canard shed vortex, if located close to the vertical tail, can contribute to the roll driving moment in a positive way at low AOA, since the velocity of the vortex on the vertical tail contributes to roll moment in the same direction as the canards. The positive component of roll driving moment is presently not accounted for in the mathematical model of Eq (9). This positive term to roll driving moment may be considered for refinement of the AP09 and the Eq (9) mathematical model at a future date. The first canard controlled configuration to be considered is shown in Fig 1. Bert 34 conducted wind tunnel tests of a configuration that had small canards over a broad range of Mach numbers at low AOA. The nice thing about the Ref 30 data was they not only had main balance data collected, but had separate balances for the canards as well as the tails. That way, one could distinguish between the contribution of the canards and the contribution in the opposing direction of the tails. Figure 1 shows comparisons of the AP09 predictions for the canards, tails and total roll moment separately so as to see how the theory Figure 1. omparison of theory and experiment for roll moment due to differentially deflected canards of a canardbody-tail configuration (Φ=0, all 4 canards deflected -5 deg). 16