Experimental und numerical investigation of the ...

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A study of the aerodynamic characteristics of two missile configurations has ..... both configuration have in common, that for φ = 45◦ the global extremum of Cl is ...
10.2514/6.2017-3401 AIAA AVIATION Forum 5-9 June 2017, Denver, Colorado 35th AIAA Applied Aerodynamics Conference

Experimental und numerical investigation of the aerodynamic characteristics of a generic transonic missile Christian Schnepf∗ and Erich Sch¨ ulein∗ German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, Bunsenstraße 10, G¨ ottingen, 37083, Germany

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Josef Klevanski† German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, Linder H¨ ohe, Cologne, 51147, Germany

A study of the aerodynamic characteristics of two missile configurations has been performed using the data of an experiment, of a semi-empirical aerodynamic prediction code (Missile Datcom) and a RANS flow solver (DLR-TAU-Code). Considering the design of the configurations one is rather unconventional with large aspect ratio wings, whereas the other one has conventional wings with a small aspect ratio. The latter sustains large lift forces at high angles of attack due to the occurrence of leading edge vortices while the former performs better at low angles of attack. The comparison of the experimental data with the aerodynamic coefficients predicted by TAU shows a good agreement for both configurations in the entire α-range. Also non-linear characteristics are captured well by TAU. Regarding Missile Datcom, for the unconventional design the accuracy of the predicted data is reasonable. For the conventional design the accuracy is less accurate particularly the prediction of the center of pressure.

Nomenclature AW CD CD , b CL Cl Cm cm CD,0 DM lM p∞ q∞ Re SR SS xCG xM RP y+

Wing area, m2 Forbody drag force coefficient (CD = CD,0 − CD,b ) Base drag force coefficient Lift force coefficient Rolling moment coefficient Pitching moment coefficient sectional pitching moment coefficient total drag force coefficient Missile diameter, m Missile length, m Free stream static pressure, Pa Free stream dynamic pressure, Pa Reynolds number Fin span, m Wing span, m Center of gravity on x axis, m Moment reference point on x axis, m Dimensionless wall distance

∗ Research † Research

scientist, German Aerospace Center (DLR), Department of High speed configurations. scientist, German Aerospace Center (DLR), Department of Supersonic and Hypersonic Technology.

1 of 21 American Institute of Aeronautics and Astronautics Copyright © 2017 by Christian Schnepf, Erich Schülein, Josef Klevanski. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

α Λ φ CG FFM MRP RANS SFFM WTSP XCP

Angle of Attack, deg sweep angle, deg Roll angle, deg Center of gravity Free flight missile geometry (blunt rear) Moment reference point Reynolds-averaged Navier-Stokes equation Scaled free flight missile configuration Wind tunnel setup (open rear) Center of pressure

I.

Introduction

ne of the major design goals for subsonic, transonic and supersonic interceptor missiles is high maneuO verability. Good performance in flight maneuvers at high angles of attack is a key in reaching this design goal. Getting reliable aerodynamic data for these maneuvers from aerodynamic prediction tools in Downloaded by Erich Schuelein on October 29, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-3401

1

the preliminary design phase of a missile is very important, but also very challenging. The difficulty is the accurate prediction of the large flow separations and the accompanying complex vortical flow structures that occur on a missile at high angles of attack. Even for state-of-the-art aerodynamic prediction software tools it is still challenging to determine accurately these separations and vortices.2 Hence it is still necessary to validate these software tools with experimental data to ensure their reliability. Since many years a standard industrial tool to determine the aerodynamic coefficients, aerodynamic stability, and control characteristics of a missile in the preliminary design phase is Missile Datcom.3 It is a semi-empirical tool for axisymmetric or elliptical shaped bodies with control surfaces and propulsion systems that generates aerodynamic data sets within seconds. The aerodynamic coefficients are predicted separately and afterwards joined using carryover and synthesis methods.4 Hence also interference effects of the different airframe components are taken into account.5 Although the tool is being continuously improved, its accuracy at high angles of attack and for a complete missile configuration with wings and fins is still an open question. In this regard Abney4 compared experimental data with the predicted data of Missile Datcom for a body-fin configuration for M = 0.8. Rosema5 did a similar comparison for supersonic Mach numbers. Additionally, he investigated a body-wing-fin configuration. In both investigations for the body-fin configuration the predicted trend of the normal forces and the center of pressure with increasing angles of attack compare reasonable well with the experiment. For moderate angles of attack (α < 30◦ ) the predicted data is also quantitatively in a good agreement. In contrast, for the body-wing-fin configuration Rosema5 showed that Missile Datcom overpredicts the downwash effect of the vortices shedding from the wing and their impingement on the fins. This leads to large discrepancies between the experimental data and the predicted data. In recent years also Reynolds-Averaged-Navier-Stokes (RANS) simulations are increasingly applied in the preliminary design phase and throughout the whole development cycle of a missile. With this kind of simulation tool a higher accuracy of the predicted aerodynamic data is achieved. Mostly one- or twoequation turbulence models are used for simulations of high Reynolds number flows. It is well known that with this kind of turbulence modeling the accuracy of the predicted data is decreasing once large separations occur. In Ref. 5 the accuracy of a RANS flow solver was compared with the accuracy of Missile Datcom. In general, for all missile configurations investigated the aerodynamic coefficients were well predicted by the RANS simulations and in a better agreement with the experimental data than the ones of Missile Datcom. Also the nonlinearity of the aerodynamics at high angles of attack was captured. In contrast to Missile Datcom the downwash effect of the wing vortices on the fins was more accurately predicted, resulting in a better agreement with the experiment. Considering different wing configurations, the data agreement was decreasing with a backward extension of the wing. This extension was accompanied with a decreasing distance between the wing trailing edge and the fin leading edge. This modification resulted in a less accurate prediction of the location of the center of pressure. For the smallest distance the maximum discrepancy was about 8% between the experiment and the RANS simulation 5. In Ref. 5 the span SS of the wings was considerably small. The ratio of the span of the wings to the span of the fins SR of the fins was SS /SR = 0.6. Hence, the length scales of coherent flow structures (e.g. vortices) that develop at the wings are small too. It can be assumed that with an increase of SS respectively

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SS /SR , on the one hand the wing-fin interaction is more distinct and on the other hand the impact of these structures on the overall aerodynamics of the missile increases. Since these structures and interactions are particularly difficult to predict, this could degrade the accuracy of a RANS flow solver and Missile Datcom. The present work was carried out in order to investigate how the prediction is influenced by a large ratio of SS /SR . For two body-wing-fin configurations the aerodynamic coefficients were predicted using a RANS flow solver and Missile Datcom. The investigated configurations have a ratio of SS /SR = 1.3 and of 2.25, respectively. These predicted data is compared with experimental data. The comparisons are made for a transonic Mach number of M = 0.85 and a supersonic Mach number of M = 1.2, for different roll angles, and angles of attack.

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II.

Missile configurations LK6D and LK6E

The investigated generic missile configurations LK6D and LK6E are the result of an academic study. The goal of the study was to design a highly maneuverable transonic missile, that should fly at low altitudes. For the aerodynamic design Missile Datcom version 3/99 was used. Not only aerodynamic issues have driven the missile layout, also the integration of necessary components like a warhead, seeker, propulsion and so forth was considered in the design. However, an important aerodynamic design criterion was to get an aerodynamically stable configuration. For that reason a conventional wing-tail-configuration was chosen for the airframe. A further advantage of the wing-tail-configuration is the higher maneuverability at high angles of attack.6 A conventional canard-configuration shows a loss of performance with increasing angles of attack, considering the pitching moment to change the attitude of the missile. The geometries of LK6D and LK6E are shown in Fig. 1 and Fig. 2. For both configurations the design of the missile fuselage is identical. The missile fuselage has a length of lM = 1500 mm and a maximum diameter of DM = 150 mm. The geometry of the fins and the placement of them on the missile fuselage is also identical for both configurations. However, for the wing design different approaches have been chosen. LK6D (Fig. 1) has four swept wings with a high aspect ratio and a small chord with a sweep angle of Λ = 34◦ . In contrast, LK6E has wings with a small aspect ratio and a large chord with a sweep angle of Λ = 49◦ . Accompanied with the large chord, the distance between the wing trailing edge and the fin leading edge is significantly smaller for LK6E in comparison to LK6D. The area of a single-wing of LK6D is AW,D = 0.043 m2 , hence only half the size of a wing of LK6E (AW,E = 0.1 m2 , AW,E /AW,D ≈ 2.3). Despite those differences, for both wing planforms the wing span is larger than the span of the fins (LK6D: SS /SR = 2.25, LK6E: SS /SR = 1.3). For both missiles the wings and the fins are not evenly distributed in circumferential direction. They show an angle of 30◦ to the xy-plane, respectively an angle of 60◦ to the xz-plane (Fig. 1(b) and 2(b)). The reason for this unconventional wing arrangement is the bank-to-turn control method. This method is used for attitude control in the present case. Considering the primary reference plane of the missile (xz-plane), with this unconventional wing design a larger normal force can be achieved in comparison to a conventional x- and +-arrangement of the wings. The position of the center of gravity CG for both generic missiles is at xCG = 0.75 m.

III.

Experimental and numerical setup

The wind tunnel experiments for LK6D and LK6E were conducted in the Transonic Wind Tunnel G¨ ottingen (DNW-TWG). A perforated test section was used to minimize the wind tunnel wall effects. For Mach numbers of M = 0.3, 0.5, 0.7, 0.85, 1.2 at angles of attack of 0◦ ≤ α ≤ 30◦ and for roll angles of φ = 0◦ , 5.625◦ , 11.25◦ , 22.5◦ , 45◦ , 67.5◦ , 90◦ , 180◦ the aerodynamic coefficients were measured with a strain 2 gauge balance. The coefficients were non-dimensionalized using DM for the reference length and πDM /4 for the reference area. A forced boundary layer transition was applied at x/L = 0.037 for both configurations (green dashed line in Fig. 2(a)). The investigated Reynold numbers was 500, 000 for LK6E with respect to the diameter of the wind tunnel model. For LK6D the Reynolds number was reduced in the early stage of the experiment from 500, 000 to 200, 000. This was done to avoid a damage of the thin wings. The wings of LK6D showed a significant bending during the experiment as a consequence of the high wing aspect ratio (Fig. 3)a . aA

reduction of the Reynolds number led to a reduction of the dynamic pressure and hence to a reduction of the wing load

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1500 75

1002 948 903 887 800

60◦

0 35

75

743

75

700 y

87 100

150

x

z 299 y

zoom window in Fig. 6 elliptical edge

(b) Back tated).

view

(90◦

ro-

Figure 1. Geometry of configuration LK6D (dimensions in mm).

1500 598 75

0 20

400

60◦

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(a) Top view.

500 y x

15

5

y

87 100

150

299

zoom window in Fig. 5

z

elliptical edge

(a) Top view.

(b) Back view.

Figure 2. Geometry of configuration LK6E (dimensions in mm).

Figure 3. Overlap of a wind-off (green) and wind-on (red) photograph to illustrate the deformation of the backboard wings of LK6D (M = 0.085, ReD = 200, 000, α = 32◦ ).

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In this study the steady Reynolds-averaged Navier-Stokes (RANS) equations were solved using the finite volume flow solver TAU.7 In general Menter’s shear stress transport (SST) model8 was chosen to closure the system of equation, though some reference simulations were carried out with other turbulence models. For simplification a fully turbulent boundary layer was assumed. This is a reasonable assumption considering the forced transition in the experiment. The simulations with TAU were carried out for two Mach numbers M = 0.85 and M = 1.2. In table 1 the flow properties of the simulations and of the experiments are shown. Furthermore an overview of existing numerical results are given considering angles of attack and roll angles. The dimensions given in Fig. 1 and 2 are for the full scale free flight missile geometries (FFM). This dimensions were used for the academic design study. In the wind tunnel experiment and in the corresponding RANS simulations geometries with a scale of 1 : 3 were investigated. For the RANS simulations, for each of the investigated missile configurations, two geometries exist. One is just the scaled version of the free flight missile geometry (SFFM) with a blunt rear end. The other version is identical with the wind tunnel model and has an open rear end (WTSP). The open rear end was necessary to realize the mounting of the model on the internal balance and sting. Apart from this, the two geometries are identical. The total simulation geometry of the WTSP was supplemented by the sting and the roll apparatus of the experimental setup (Fig. 4). This was done to capture possible interference between the missile and the support. The wind tunnel walls were not part of the computational grid. Instead a farfield boundary condition was used. The moment reference point MRP of the experiment and the simulations was at xM RP = −247 mm (WTSP). φ chimera overlap hinge

sting α u∞ roll apparatus

Figure 4. Computational geometry of the wind tunnel setup.

The unstructured computational grids for the TAU simulations were generated using CENTAURTM . Although the used flow solver is an unstructured solver, the surface grid of the missile shows a structured grid at the nose and at the cylindrical part in front of the wings and downstream of the fins with 160 cells in circumferential direction (Fig. 5(a) and Fig. 5(b)). At the wings and fins the surface mesh consists of triangles with the exception of the leading edges and the trailing edges (Fig. 5(c)-5(e), Fig. 6). This grid topology ensured in Ref. 9, 10 a symmetric flow field for symmetric boundary conditions. A first cell spacing of y + < 1 is achieved on the entire missile surface. In general 40 cell-layers in wall normal direction were used at the fuselage and 37 (33) at the wing (fin) surfaces to resolve the boundary layer with a wall normal stretching of 1.18. The volume mesh consists of tetraeders with a refined grid resolution in the vicinity of the missile. The total mesh size is about 50M nodes (WTSP). To adjust the roll angle φ the missile and sting could rotate relatively to the roll apparatus shown in Fig. 4. In the numerical simulations this was realized by applying the chimera technique. The viscous surfaces of the chimera grids overlapped at the green rectangle in Fig. 4. Table 1. Parameter space of the numerical simulations

LK6E

M [-] 0.85 (1.2)

Re [-] 500, 000

LK6D

0.85 (1.2)

210, 000

α [◦] 0, 5, 10, 15, 20, 25, 30 0, 5, 10, 15, 20, 25, 30

φ [◦] 0, 11.25, 22.5, 45, 67.5, 90 0, 11.25, 22.5, 45, 67.5, 90

p∞ [P a] 47, 000 (29, 000)

q∞ [P a] 23, 500 (29, 000)

19, 400 (12, 400)

9, 800 (12, 500)

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(a) Nose.

(b) Rear with fin

(c) Fin leading edge.

(d) Wing leading edge

(e) Wing junction

Figure 5. Surface mesh of LK6E at the blue marked areas in Fig. 2(a).

(a) Wing junction.

(b) Wing tip

Figure 6. Surface mesh of LK6D at the blue marked areas in Fig. 1(a).

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IV.

Results

In general the aerodynamic data is presented as non-dimensionalized coefficients. The moment coefficient refer to the body-fixed-coordinate system11 . The lift and drag coefficients refer to the wind coordinate system (air-path axis system)11 . Considering the experiments and the RANS simulations, CD refers to the forebody drag coefficient of the missile (CD = CD,0 − CD,b ) and not to the total drag coefficient. The angle of attack α always refers to the angle between the wind axis and the body axis (Fig.4). Analysis of the experimental results

In this section the aerodynamic performance of the two configurations is analyzed based on the experimental data. In Fig. 7 the lift and drag coefficients are shown for both configurations for a roll angle of φ = 0◦ . At low angles of attack both configurations show similar magnitudes and trends of CD . However, the slope dCD /dα changes with the wing planform and is larger for LK6E. Thus the difference in magnitude of CD between both configurations is increasing with increasing angles of attack. Especially for M = 1.2, the difference in CD between both configurations is already significantly at α = 0◦ . In this case, the larger CD -value occurs for LK6D because of the larger wing area facing the wind and the smaller sweep angle of the wings in comparison to LK6D. This has a negative influence on the wave drag. 20

30

Ma=1.2 Ma=0.95 Ma=0.85 Ma=0.7 Ma=0.5

20

LK6D LK6E

10

Ma=1.2 Ma=0.95 Ma=0.85 Ma=0.7 Ma=0.5

25

CL [−]

15

CD [−]

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A.

LK6D LK6E

15 10

5 5 0

0

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α[ ]

20

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10

(a) CD

15

α [◦ ]

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(b) CL

Figure 7. Drag and lift coefficient vs. angles of attack for φ = 0◦ .

A clear distinction between both configurations exists for CL (Fig. 7(b)). A linear characteristic is observable for LK6D for α < 5◦ . In this α-range the slope (dCL /dα) of LK6D is larger than the one for LK6E. This leads to higher CL in this α-range although LK6D has the smaller total wing area. But with a further increase in α a significant reduction of the slope occurs for LK6D, most distinct for M = 0.85. The CFD simulation show that the reason for this reduction is a flow separation occurring over the entire wing. For LK6E a nonlinear CL -characteristic is observable, similar to the one of a delta wing with a relatively high aspect ratio.12 The nonlinearity is mainly due to the vortices developing at the wing leading edges and wing tips. At higher angles of attack they have a beneficial effect on the lift and the slope of the lift. This leads to higher maximum values of CL in comparison to LK6D. However, also for this configuration a gradual decrease of the slope of CL is observable. In contrast to LK6D, this occurs at significantly higher angles of attack (α > 20◦ ). The influence of the different wing planforms on the aerodynamic behavior is even more evident regarding the pitching moment coefficient Cm in Fig. 8(a). For LK6D Cm is negative over the entire range of angle of attack, indicating an aerodynamically stable configuration. After a steady decrease of Cm with increasing α, the angles of attack does not have a major impact on Cm above α ≈ 15◦ . In contrast, for LK6E Cm (α) shows first an increase until a maximum is reached. In the further course Cm decreases gradually over the entire range of angles of attack. The reason of this opposing trends of Cm is the location of the center of

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Center of Pressure x/lM [-]

0.75

Cm [−]

0

−5

−10

Ma=1.2 Ma=0.95 Ma=0.85 Ma=0.7 Ma=0.5

−15

0

5

LK6D LK6E 10

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α [◦ ]

20

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LK6D LK6E

0.70 0.65 0.60 0.55 0.50 0.45 0.40

Ma=1.2

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Center of gravity Moment reference Point

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Ma=0.95 10

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α [◦ ]

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(b) XCP

Figure 8. Pitching moment coefficient and center of pressure (XCP) vs. angles of attack for φ = 0◦ .

pressure XCP (Fig. 8(b)). In case of LK6D XCP is located behind the moment reference point and the aerodynamic forces create a pitch down moment. For LK6E it is the other way around, at least considering XCP for M < 0.95. Furthermore, for LK6D the center of pressure moves to the nose while for LK6E the center of pressure moves to the aft with increasing α. In the latter case this is the reason that Cm decreases after a maximum is reached and even becomes negative for M > 0.85 (Fig. 8(a)).

0

0

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−5

Cm [−]

Cm [−]

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(a) Cm

Ma=0.85

−10

−10

φ = 0.00◦ φ = 11.25◦ φ = 22.50◦ φ = 45.00◦ φ = 67.50◦

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M=0.85 M=1.2 10

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(a) LK6D

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α [◦ ]

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(b) LK6E

Figure 9. Pitching moment coefficient vs. angles of attack for different φ (M = 0.85, M = 1.2).

Figure 8(b) shows clearly for both configurations, that the Mach number effect on the aerodynamic behavior is limited to the Mach number range of M > 0.85. In the subsonic and low transonic flow regime M < 0.85 the Mach number does not have a major impact on the aerodynamic coefficients in Fig. 7 and 8. It can be concluded that for lower Mach numbers the pressure distribution along the missile stays quite similar but it changes significantly approaching supersonic Mach numbers. The general influence of the Mach number on the aerodynamic coefficients is similar between both configurations. The dependence of Cm on the roll angle is shown in Fig. 9. Considering configuration LK6D in Fig. 9(a), for the lower Mach number (M = 0.85) a change in φ does not have a large impact on Cm , except for φ = 67.50◦ . This is especially the case in the α-range of α > 10◦ . With an increase of the Mach number

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1.0

1.0

0.5

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

Cl [−]

Cl [−]

(M = 1.2) also the magnitudes for φ = 45◦ show large discrepancies to the ones of φ < 45◦ . Considering LK6E, it is the other way around. The roll angle has a small impact on Cm for the large Mach number and a large for the lower Mach number. But for both configurations the general trend of Cm (α) does not change with φ.

−1.5 φ = 0.00◦ φ = 11.25◦ φ = 22.50◦ φ = 45.00◦ φ = 67.50◦

−2.0

−3.0 −3.5

0

5

φ = 0.00◦ φ = 11.25◦ φ = 22.50◦ φ = 45.00◦ φ = 67.50◦

−2.0 −2.5 M=0.85 M=1.2 10

15



α[ ]

−3.0 20

25

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30

0

5

M=0.85 M=1.2 10

(a) LK6D

15

α [◦ ]

20

25

30

(b) LK6E

Figure 10. Rolling moment coefficient vs. angles of attack for different φ (M = 0.85, M = 1.2).

In Fig. 10 the rolling moment coefficient (Cl ) is shown for both configurations. A comparison of Fig. 10(a) with Fig. 10(b) shows that in general LK6E can provide larger extremum for Cl (α, φ). In contrast to LK6D, for LK6E Cl increases almost continuously with increasing angle of attack, except for φ = 67.50◦ . However both configuration have in common, that for φ = 45◦ the global extremum of Cl is observable. Another difference between both configuration concerns the sign of Cl . For LK6E it is always negative while for LK6D it changes with α and φ. B.

Validation of the RANS simulations

In this chapter the RANS simulations will be validated with the experimental data. The presented numerical data is the one of the wind tunnel setup geometry (WTSP).

10

15

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦ φ = 90◦

10

CD [−]

15

CD [−]

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−2.5

−1.5

5

0

−5

Experiment LK6D Experiment LK6E 0

5

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α [◦ ]

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TAU LK6D TAU LK6E 20

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦

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−5

(a) M = 0.85

Experiment LK6D Experiment LK6E 0

5

10

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α [◦ ]

TAU LK6D TAU LK6E 20

25

30

(b) M = 1.2

Figure 11. Experimental and numerical drag coefficient for M = 0.85 and 1.2.

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35

20

CL [−]

15

25

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦

20 15

CL [−]

25

10 5

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−5 −5

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦

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−5 35

−5

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5

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(a) M = 0.85

10

15

α [◦ ]

TAU LK6D TAU LK6E 20

25

30

35

(b) M = 1.2

Figure 12. Experimental and numerical lift force coefficient for M = 0.85 and 1.2.

In Fig. 11(a) a comparison of the drag coefficient CD versus α for different roll angles φ is shown. At M = 0.85 the numerical data agrees very well with the experimental data for both configurations. The largest deviation occur for LK6E at high angles of attack, although the prediction is still very good. This accurate prediction of CD is beneficial considering the estimation of thrust and propellant in trade studies. In Fig. 11(b) CD is shown for M = 1.2. For LK6E the agreement between the experimental and numerical data is still very good. However, for LK6D a slight deterioration of the agreement is observable especially at high angles of attack. The maximum percentage deviation is about 11% at α = 30◦ (φ = 67.50◦ ). For both configurations the deviation at α = 0◦ is about 4%. In contrast to LK6D, for LK6E an improvement of the agreement is observable with increasing angles of attack. The roll angle seems to have no large influence on the match of the different data sets. In Fig. 12 the lift force coefficient is shown for the experiment and for the simulations. In case of LK6E, the nonlinear trend of CL (α) is well predicted by TAU. For small φ the TAU data matches also quantitatively well with the experiment data. However, for large φ Tau tends to underpredict CL especially at high angles of attack. At α = 30◦ and φ = 67.5◦ the percentage deviation is about 8.5%. In case of LK6D, the configuration specific trend of CL is captured well by the simulations although the slope of Cl at α < 5◦ is slightly underpredicted by TAU. At high angles of attack TAU provides again a good match with the experimental data (Fig. 12). In contrast to LK6E for LK6D the roll angles has no large impact on the agreement of the data. The accuracy of the prediction is not dependent on the Mach number for both configurations. In Fig. 13(a) the pitching moment coefficient of LK6E is shown for M = 0.85 and 1.2. In general for both Mach numbers, the numerical data matches very well with the experimental data. The agreement worsens slightly with an increase of α for roll angles of 0◦ and 22.5◦ . Whereas for the other φ the agreement is sustained. An explanation for the degradation of the agreement could be the phantom yaw effect2, 13, 14 . This phenomenon is well known for slender bodies and represents an unwanted vortex asymmetry that is influencing the surface pressure distribution and hence the force and moment coefficients. In the experiment for φ = 0◦ a small side force occurred at higher angles of indicating the presence of the phantom yaw effect. In general, the good prediction of CL , CD and Cm for LK6E is reflected in a quite good prediction of the center of pressure. In Fig. 13(b) the center of pressure is shown for the experiment and for the simulation for LK6E. The comparison of both data sets shows a very good agreement for the entire range of angle of attack and for both Mach numbers. Hence TAU is suited to determine accurately the aerodynamic stability of a missile. In case of LK6D, the prediction of Cm is not as accurate as it was for LK6E (Fig. 14(a)). For M = 0.85 the largest mismatches occur at α ≤ 5◦ for all roll angles. Neither the assumption of laminar wings nor the account for a deformation of the wings improved the agreement significantly, because of which the results are not shown. With increasing angles of attack the agreement improves and finally the maximum mismatch is below 13%. However, for M = 1.2 the predicted Cm does not fit the experimental Cm -value over the

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Center of Pressure x/lM [-]

Cm [−]

0

−5 ◦

φ=0 φ = 22.5◦ φ = 45◦ φ = 67.5◦

−10

M=0.85 (exp.) M=1.2 (exp.)

−15

−5

0

5

10

M=0.85 (TAU) M=1.2 (TAU) 15

α [◦ ]

20

25

30

0.55

0.50

0.45

0.40

0.35

35

φ = 0◦ φ = 22.5◦ M=0.85 (exp.) M=1.2 (exp.) 5

(a) Cm,M RP

10

15

M=0.85 (TAU) M=1.2 (TAU) 20

α [◦ ]

25

30

35

(b) Center of pressure

Figure 13. Experimental and numerical predicted pitching moment coefficient and center of pressure for LK6E.

entire range of angle of attack. This is especially the case when the roll angle exceeds 22.50◦ . In Fig. 14(b) for LK6D the center of pressure is shown. Excluding XCP for α = 30◦ and φ = 67.5◦ , the deviation of the predicted XCP from the experimental one is below 5% for all Mach numbers, angles of attack and roll angles. This shows that the pressure distribution along the missile is only slightly different between the experiment and the simulation. However, this difference has an large impact on the predicted Cm . In Ref. 5 the maximum discrepancy between the predicted and the experimental location of the center of pressure was about 8% and thus of the same size like in the current investigation . 5

0.75 M=0.85 (TAU) M=1.2 (TAU)

Center of Pressure x/lM [-]

M=0.85 (exp.) M=1.2 (exp.)

0

Cm [−]

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φ = 45◦ φ = 67.5◦

−5

−10

−15

−20 −5

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦ 0

5

10

15

α [◦ ]

20

25

30

0.70

0.65

0.60

0.55

0.50

35

(a) Cm,M RP

M=0.85 (exp.) M=0.85 (TAU) M=1.2 (exp.) M=1.2 (TAU) φ = 0◦ φ = 45◦ φ = 22.5◦ φ = 67.5◦

5

10

15

20

α [◦ ]

25

30

35

(b) Center of pressure

Figure 14. Experimental and numerical predicted pitching moment coefficient and center of pressure for LK6D.

In Fig. 15 the predicted rolling moment coefficient is compared with the experimental one. The predicted Cl for M = 0.85 shows a good match with the experimental Cl for both configurations, with the exception of the roll angle of φ = 67.50 for LK6D (Fig 15(a)). For this roll angle a major discrepancy exists between the experiment and TAU. TAU is not capturing the change in sign of Cl with increasing angles of attack like it occurs in the experiment. In contrast for M = 1.2, also the simulations shows the change in sign. In general for both Mach numbers, the agreement between the experiment and the simulations is slightly better for LK6E. This section showed that TAU is able to accurately predict the experimental data over a wide range of angles of attack and roll angles. Only at some discrete α and φ the accuracy decreases and larger deviations are observable. Especially this is the case for LK6D. On the other hand also the experimental data could be at some discrete points erroneous due to the Phantom Yaw effect or other unwanted phenomena. Furthermore

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5

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦ φ = 90◦

4 3 2

5

Experiment LK6D Experiment LK6E TAU LK6D TAU LK6E

4 3

Cl [−]

0

1 0

−1

−1

−2

−2

−3 −4 −5

0

5

10

15

20

α [◦ ]

25

30

−3 −5

35

0

5

(a) M = 0.85

10

15

20

α [◦ ]

25

30

35

(b) M = 1.2

Figure 15. Experimental and numerical predicted rolling moment coefficient.

the influence of the deformation was just investigated rudimentarily. May be at some φ e.g. φ = 67.50 the impact is larger leading to the discrepancies between TAU and the experiment.

V.

Detailed analysis of the aerodynamic characteristic based on the numerical simulations

In Fig. 16(a) and Fig.16(b) the pitching moment coefficient Cm and the locations of the center of pressure are shown just for the simulations of LK6D. To quantify the effect of the sting on the aerodynamic coefficients, next to the data of the wind tunnel setup (WTSP) also the data for the scaled free flight missile configuration (SFFM) is plotted for φ = 0◦ , 45◦ and 67.5◦ exemplarily for all roll angles. The difference between the two data sets (WTSP/ SFFM) are quite small. This shows that the sting has a negligible effect on the global aerodynamic of the missile. This conclusion is confirmed by the analysis of other aerodynamic coefficients and the data for M = 1.2 and also applies to the configuration LK6E. 0.64

10

0

−5

−10 φ=0 5



φ = 45 10

15

WTSP SFFM

SFFM; Re=0.5e6

Center of Pressure x/lM [-]

WTSP SFFM

5

Cm [−]

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Experiment LK6D Experiment LK6E TAU LK6D TAU LK6E

2

1

Cl [−]

φ = 0◦ φ = 22.5◦ φ = 45◦ φ = 67.5◦



φ = 67, 5 ◦

20

α[ ]

25

0.62

SFFM; Re=0.5e6

0.60 0.58 0.56 0.54

φ = 0◦ φ = 45◦



30

0.52

35

5

(a) Cm

10

15

20

α [◦ ]

φ = 67, 5◦

25

30

35

(b) XCP

Figure 16. Numerical predicted Cm and the location of center of pressure for LK6D (M = 0.85).

The comparison between LKD and LK6E in section A is based on the data of two different Reynolds numbers. However, Fig. 16(a) shows that for LK6D the Reynolds number has in total a minor effect on Cm (see data for φ = 0◦ and 45◦ ). An increase of ReD entails no large changes in Cm . Only for α = 20◦ and φ = 45◦ the effect is somewhat larger. An analysis of the force coefficients showed that their dependence on

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the Reynolds number is even less (not shown). This conclusion can also be made for M = 1.2. Ma 0,85

6

4

4

2

2

0

0

−2

−2

−4 −6 −8

M Hull Nose W Lee

−10 −12 −14 −16

0

5

−6 −8

W Luff Ru Lee Ru Luff

−10 −12 WTSP

10

−4

15



α[ ]

20

−14

SFFM 25

−16

30

0

5

(a) M = 0.85

10

15

α [◦ ]

20

25

30

(b) M = 1.2

Figure 17. Sectional pitching moment coefficient for LK6D for φ = 0◦ (M=total missile, Hull: cylindrical fuselage of the missile + nose, W Lee: lee side wing, W Luff: windward wing, Ru Lee: lee side fin, Ru Luff: windward fin).

8

8

6

6

4

4

2

2

cm [−]

cm [−]

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8

6

Cm [−]

Cm [−]

8

0 −2

−2 M Hull Nose W Lee

−4 −6 −8

0

0

5

W Luff Ru Lee Ru Luff WTSP 10

M Hull Nose W Lee

−4

15



α[ ]

20

−6

SFFM 25

−8

30

0

5

(a) M = 0.85

W Luff Ru Lee Ru Luff

10

15

α [◦ ]

20

25

30

(b) M = 1.2

Figure 18. Sectional pitching moment coefficient for LK6E for φ = 0◦ (M: total missile, Hull: cylindrical fuselage of the missile + nose, W Lee: lee side wing, W Luff: windward wing, Ru Lee: lee side fin, Ru Luff: windward fin).

In Fig. 17 and 18 the pitching moment coefficients cm are shown for different components of the missile configurations. For M = 0.85 the sectional pitching moment coefficient cm is plotted for the geometry SFFM and for the geometry WTSP (17(a) and 18(a)). The data of those two geometries agree very well which confirms that the sting has only a minor impact on the flow field of the missile. A comparison of Fig. 17(b) with Fig. 17(a) shows that for LK6D the general trend of cm (α) of each missile section does not change with Mach number. Only the magnitude of cm does vary partially significantly with M . In contrast for LK6E, the sectional cm of the fin on the lee side shows a change in sign with an increase of M . At M = 0.85 this fin induces a positive moment (pitch down moment) at high angles of attack and a negative at M = 1.2. Considering the other fin and the fins of LK6D this change in sign is a unique feature at this φ. A similar trend also is observable for the wings of LK6E, especially to the one on lee side. Only for M = 0.85 a change in sign for cm is observable with increasing α. Apart from the wings and the fins, the trends of the sectional cm with increasing α and M is identical for both configurations. Hence those missile sections are responsible for the characteristic trend of Cm of the two configurations with increasing α and M in Fig. 13(a).

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An interaction of a vortex with the flow at the lee side fin is responsible for the change in sign of cm with increasing Mach number that occurs for this component (Fig. 19(a)). The vortex originates from the leading edge of the windward wing. For M = 0.85 this vortex induces low cp -magnitudes on the lower side of the fin. They are even smaller like the ones on the upper side of the fin. For M = 1.2 this interaction is weaker and on the lower side of the fin the pressure is still higher like on the upper side. Hence the efficiency of the lee side fin is just reduced in comparison to the windward fin (Fig. 19(b)). In comparison to LK6D, the fins of LK6E show in general a lower efficiency. This is due to the distance between the wing trailing edge and the fin leading edge. In case of LK6D this distance is larger and hence the impact of the wake of the wings on the fins is smaller. Furthermore no large coherent vortices originate from the wings of LK6D which could influence the flow field like in case of LK6E. In general this paragraph showed, that the wake of the wings can have an significant influence on the efficiency of the fins. The impact of this wake clearly depends on geometric properties. cp,S : cp,S :

low cp

high cp low cp

high cp

low cp

low cp

high cp

high cp (a) M = 0.85

(b) M = 1.2

Figure 19. Visualization of the flow field with streamlines and cp -distributions for LK6E at α = 25◦ and φ = 0◦ (cp,S : cp -distribution at the surface).

Hull Nose

W Lee W Luff

Ru Lee Ru Luff

M=0.85 M=1.2

1.0 0.8 0.6 0.4 0.2 0.0

5

10

15



α[ ]

20

25

1.2

Center of Pressure x/lM [-]

1.2

Center of Pressure x/lM [-]

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cp : cp :

30

Hull Nose

W Lee W Luff

Ru Lee Ru Luff

M=0.85 M=1.2

1.0 0.8 0.6 0.4 0.2 0.0

5

10

(a) LK6D

15

α [◦ ]

20

25

30

(b) LK6E

Figure 20. Sectional center of pressure for LK6D and LK6E for φ = 0◦ (M: total missile, Hull: cylindrical fuselage of the missile + nose, W Lee: lee side wing, W Luff: windward wing, Ru Lee: lee side fin, Ru Luff: windward fin).

In Fig. 20 the location of the center of pressure for different missile sections is shown. The effect of the wing planform on the location of the center of pressure of the wings is clearly distinguishable. Additionally

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at lower angles of attack, the wing planform has a strong impact on the center of pressure of the cylindrical fuselage of the missile. These effects are the main reason that the location and the trend of the center of pressure of LK6D is completely different to them of LK6E in Fig. 8(b). In contrast to LK6D, for LK6E the location of center of pressure of the wings shows over the entire range of angle of attack a dependence on the Mach number, especially the wing on lee side (Fig. 20). Furthermore, the shift of XCP of the wings with increasing α is significantly larger for LK6E than for LK6D. In case of LK6D, at high angles of attack even a stagnation of the shift of XCP is observable. This is due to the occurrence of flow separations at the wings. For M = 0.85 the flow separates at the entire wing already between 2.5◦ < α < 5◦ (Fig. 21). This large separation explains the decrease of the slope of CL in this range of angles of attack (Fig. 7(a)). With an increase of the Mach number up to M = 1.2, the onset of stall at the root of the wing is shifted to higher angles of attack. Furthermore the range of angle of attack becomes larger within the separation is spreaded over the entire wing (5◦ < α < 20◦ ). In Fig. 7(b) the expansion of this range with increasing Mach number is reflected in a more gentle decrease of the slope of CL .

(a) α = 2.5◦

(b) α = 5◦

Figure 21. Visualization of the flow field with streamlines, skin friction lines and cp -distributions for LK6D for M = 0.85 and φ = 0◦ .

In Fig. 22 the skin friction line pattern and the cp -distribution at the wings for LK6E are shown. Each figure shows at the upper half a top view at the wing and fin on lee side and at the lower half a top view at the wing and fin on windward (e.g. Fig. 22(a)). For this wing planform the occurrence of a local separation in the vicinity of the wing leading edge at a certain angle of attack is characteristic (e.g. Fig. 22(a)). This separation extends along the entire wing leading edge. Beneath the separation the cp -values are relatively low which is beneficial considering the lift force. At first the size of this local separation extends in downstream direction with increasing angle of attack. Above α ≥ 15◦ the skin friction line pattern changes. No longer a reattachment line is observable that originates at the wing root in the vicinity of the wing leading edges (compare Fig. 22(a) with 22(g)). In Fig. 23 a visualization of the flow field in the vicinity of the wings is shown. For the low angles of attack it can be clearly seen that boundary layer that separates at the leading edges forms so called leading edge vortices, with a more distinct one on windward (e.g. Fig. 23(c)). Based on the streamlines and the cp -distribution, these vortices are better to identify for the lower Mach number. However, for both Mach numbers the vortices run along the wing tip and leave distinct footprints on the cp -distribution of the wing. With an increase of α the footprints extend to the wing trailing edges due to an increase of the vortices. The occurrence of these vortices is responsible for the non-linear characteristic of CL with increasing angle of attack (Fig. 7(b)). The cp -distribution and the streamlines indicate that the flow development at the windward wing and at the wing on the lee side is slightly different. At the windward wing the flow is influenced by the fuselage and of course by the lee side wing. The flow at the lee side wing itself is additionally influenced by the lee vortices which originate at the nose of the missile. This is especially relevant at high angles of attack (Fig. 23(g) and 23(h)). Furthermore for M = 0.85, the streamlines and the skin friction line pattern indicate a breakdown of the leading edge vortices at high angles of attack (α ≥ 15◦ ). This is the reason for the change in skin friction line pattern with increasing angle of attack respectively the absence of the reattachment line in the vicinity of the leading edges. The vortex breakdown phenomenon occurs at first on the wing at lee side (Fig. 22(e) and Fig. 23(e)). With further increase of α it occurs also on the windward wing (α ≈ 25◦ ,Fig. 19(a)). The vortex breakdown has a negative effect on the cp -distribution. The influence of the vortex footprint is reduced in size and strength which is reflected in Fig. 12(a) by a decrease of the slope of CL in the corresponding range

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(a) α = 5◦ , M = 0.85

(b) α = 5◦ , M = 1.2

(c) α = 10◦ , M = 0.85

(d) α = 10◦ , M = 1.2

(e) α = 20◦ , M = 0.85

(f) α = 20◦ , M = 1.2

(g) α = 25◦ , M = 0.85

(h) α = 25◦ , M = 1.2

(i) α = 30◦ , M = 0.85

(j) α = 30◦ , M = 1.2

Figure 22. Top view: Skin friction lines and cp -distributions at the upper side of the wing for LK6E for φ = 0◦ .

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(a) α = 10◦ , M = 0.85

(b) α = 10◦ , M = 1.2

(c) α = 15◦ , M = 0.85

(d) α = 15◦ , M = 1.2

(e) α = 20◦ , M = 0.85

(f) α = 25◦ , M = 1.2

(g) α = 30◦ , M = 0.85

(h) α = 30◦ , M = 1.2

Figure 23. Visualization of the development of the vortices at the wings with increasing angle of attack for LK6E. Next to streamlines and cp -distribution at the surface the cp -distribution in cross sections is shown.

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of angles of attack (α > 15◦ ). For M = 1.2 this phenomena cannot be determined. Instead two primary vortices can be clearly distinguished at the wing at higher angles of attack (e.g. Fig. 23(h)). Downstream of the leading edge they merge to one primary vortex . After the merge they show a significant lift off from the wing. Analogous to the vortex breakdown this has a negative effect on the cp -distribution and hence on CL . The last paragraphs showed that the wing planform does significantly influence the flow field at the missile. With a change of the planform also a significant change of the characteristic flow structures can occur. Accompanied with this, also new phenomena arise what has to be considered in the design of a missile. Comparison of the experimental data with the data of TAU and Missile Datcom

In this chapter the data of Missile Datcom is compared with the experimental data and with the results of the TAU simulations. The comparison is made for the geometry SFFM. After the configuration had been designed with Missile Datcom for a Reynolds number of ReF F ≈2, 500, 000 the derived geometry was scaled in order to make a comparison with the experiment and with the RANS simulations for a Reynolds number of 500, 000. A comparison of the Missile Datcom data showed that the change in Re has no impact on the aerodynamic coefficients. Analogous to the TAU simulations also for the simulations with Missile Datcom a fully turbulent boundary layer was assumed. In Fig. 24 the lift force coefficient is shown. For LK6E the agreement between the data of Missile Datcom and the one of the experiment is quite accurate. Even at high angles of attack the difference between the data sets is relatively small. Although the non-linear trend of CL (α) is not captured precisely. Especially the decrease in slope (∂CL /∂α) starting from α ≈ 20 is not present. But in general the prediction is sufficient enough for trade studies. In contrast, for the more unconventional configuration LK6D the same comparison shows not such a good agreement over the entire α-range. At low angles of attack the differences between the data sets are also rather small but they increase significantly with increasing angles of attack. The reason is the different prediction of the slope of CL . Missile Datcom is not capturing accurately enough the distinct decrease of the slope like it occurs in the experiment at α = 5◦ . It seems that Missile Datcom does not sufficiently determine the effect of the flow separation at the wings on CL . Horton15 observed differences of similar magnitudes in his investigation of an unconventional missile configuration. 30

30

25

25

20

20

15

15

CL [−]

CL [−]

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A.

10 5

5

0

MDatcom; M=0.85 MDatcom; M=1.2 Exp.; M=0.85

−5 −10

10

0

5

10

15



α[ ]

20

0

Exp.; M=1.2 TAU; M=0.85 TAU; M=1.2 25

30

MDatcom; M=0.85 MDatcom; M=1.2 TAU; M=0.85

−5 −10

35

0

5

(a) LK6D

10

15

α [◦ ]

20

TAU; M=1.2 Exp.; M=0.85 Exp.; M=1.2 25

30

35

(b) LK6E

Figure 24. Comparison of the Lift force coefficients.

In Fig. 25 a comparison of the pitching moment coefficient is presented. For LK6D, the predicted Cm by Missile Datcom decreases gradually with increasing angles of attack which is similar to trend of the experimental Cm . However, in detail the trends of Cm do not agree. Especially the slope dCm /dα at small angles is predicted to small and the saturation of Cm at high angles of attack is not captured at all by Missile Datcom. In general, the magnitudes of the prediction are not as accurate as the ones of TAU, particularly for the lower Mach number. However, the difference in magnitude is significantly larger for LK6E. Because of opposing Cm trends, the difference is increasing with increasing angle of attack. Taking the good agreement of the normal forces into account, this indicates the load distribution at the missile is predicted not accurately

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15

MDatcom; M=0.85 MDatcom; M=1.2 Exp.; M=0.85

10

MDatcom; M=0.85 MDatcom; M=1.2 TAU; M=0.85

30

0

Cm [−]

Cm [−]

5

40

Exp.; M=1.2 TAU; M=0.85 TAU; M=1.2

−5 −10 −15

TAU; M=1.2 Exp.; M=0.85 Exp.; M=1.2

20

10

0

−20 −25

0

5

10

15



α[ ]

20

25

30

−10

35

0

5

10

α [◦ ]

20

25

30

35

(b) LK6E

Figure 25. Comparison of the pitching moment coefficient.

enough by Missile Datcom. In Fig. 26 the location of the center of pressure is shown. For LK6D the predicted location of XCP matches well with the one of the experiment. The better agreement is achieved for the lower Mach number. But also for the higher Mach number the maximum discrepancy is below 8%. However, for LK6E the prediction of XCP is not very accurate. Neither the trend nor the magnitudes matches. At high angles of attack the discrepancies are about 20%. In comparison to TAU and the experiment the predicted location of XCP shows a forward shift. This could be due to the leading edge vortices and their effect on the load distribution. Perhaps the impact of the vortex footprints, which extent at certain angles of attack all the way to the wing trailing edges, is underestimated in Missile Datcom. Rosema5 observed in his investigation a similar trend for the prediction of XCP. 0.7

Center of Pressure x/lM [-]

0.7

Center of Pressure x/lM [-]

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(a) LK6D

15

0.6

0.5

0.4

0.3

0.2

MDatcom; M=0.85 MDatcom; M=1.2 Exp.; M=0.85 5

10

15

Exp.; M=1.2 TAU; M=0.85 TAU; M=1.2 20

α [◦ ]

25

30

0.6

0.5

0.4

0.3

0.2

35

MDatcom; M=0.85 MDatcom; M=1.2 TAU; M=0.85 5

10

(a) LK6D

15

TAU; M=1.2 Exp.; M=0.85 Exp.; M=1.2 20

α [◦ ]

25

30

35

(b) LK6E

Figure 26. Comparison of the location of the center of pressure.

VI.

Conclusion

In this study the aerodynamic characteristics of two missile configurations with different wing planforms were investigated. Configuration LK6D has four backward swept wings with a high aspect ratio and a small chord. In contrast, the conventional configuration LK6E has four wings with a small aspect ratio and a large chord. The wings of LK6E have a backward swept leading edge and a unswept trailing edge. The analyzed

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database consists of aerodynamic coefficients measured in a wind tunnel experiment and of aerodynamic coefficients predicted by Missile Datcom and the RANS flow solver TAU. The experiments were carried out for Mach numbers of 0.3 < M < 1.2. However, the comparison of the different data sources is done for M = 0.85 and 1.2. In this Mach number range the effect of the Mach number on the aerodynamic coefficients was most distinct. The following conclusions may be drawn from the investigation considering the comparisons of the different data sets:

Downloaded by Erich Schuelein on October 29, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-3401

• The RANS flow solver TAU very accurately predicts the aerodynamic force coefficients for both configurations. Even the non-linear trends are captured by TAU. For LK6E also the location of the center of pressure is predicted accurately. This is relevant considering the determination of the aerodynamic stability of a missile. For LK6D the prediction of XCP is somewhat poorer. However the maximum discrepancy is still small (5%). • As expected Missile Datcom did not show the same accurate prediction of the aerodynamic coefficients. For LK6E the prediction of CL was reasonably good in the entire α-range. In contrast, for LK6D this was only the case for small angles of attack. The agreement is degrading with increasing α. For the pitching moment coefficient larger differences were observed for both configurations at the entire α-range. Hence, also for the location of center of pressure quite large differences occurred between Missile Datcom and the experiment (Maximum deviation: 10%). Considering the aerodynamic characteristic of the different configurations the following conclusions are drawn: • Considering aerodynamic stability, LK6D shows a better behavior due to a center of pressure that is located abaft the center of gravity. • Both configurations show opposing trends of XCP with increasing Mach number but similar trends with increasing angle of attack. • LK6D shows larger lift forces at low angles of attack. But a flow separation that occurs over the entire wing leads to a poor performance at high angles of attack. The predicted sectional aerodynamic coefficients by TAU do not indicate the occurrence of a significant interaction of the wakes of the wings with the fins. • Considering the lift force LK6E performs better at high angles of attack. This is due to the occurrence of leading edge vortices which have a beneficial effect on the pressure distribution and on the separation at the wings. However at certain conditions these vortices also can effect the performance of the fins. Although LK6E shows higher lift forces than LK6D also for this configuration the slope dCL /dα does decrease at very high angles of attack due to a breakdown of the leading edge vortices.

References 1 Eugene,

L. F., “Tactical missile design,” AIAA Education Series, AIAA, 2001, pp. 1–18. R. M., Forsythe, J. R., Morton, S. A., and Squires, K. D., “Computational challenges in high angle of attack flow prediction,” Progress in Aerospace Sciences, Vol. 39, No. 5, 2003, pp. 369–384. 3 Rosema, C., Doyle, J., and Blake, W. B., “MISSILE DATA COMPENDIUM (DATCOM) User Manual 2014 Revision,” Tech. rep., DTIC Document, 2014. 4 Abney, E. J. and McDaniel, M. A., “High angle of attack aerodynamic predictions using missile datcom,” AIAA, Vol. 5086, 2005, pp. 2005. 5 Rosema, C. C., Abney, E., Westmoreland, S., and Moore, H., “A Comparison of Predictive Methodologies for Missile Configurations with Strakes,” 33rd AIAA Applied Aerodynamics Conference, 2015, p. 2588. 6 Levin, D., “Vortex flaps canard configuration for improved maneuverability,” Journal of aircraft, Vol. 34, No. 5, 1997, pp. 693–695. 7 Schwamborn, D., Gerhold, T., and Heinrich, R., “The DLR TAU-Code: recent applications in research and industry,” ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006 , Delft University of Technology; European Community on Computational Methods in Applied Sciences (ECCOMAS), 2006. 8 Menter, F. R., “Zonal two equation k-turbulence models for aerodynamic flows,” AIAA paper , Vol. 2906, 1993, pp. 1993. 9 Schnepf, C., Wysocki, O., and Sch¨ ulein, E., “Numerical investigation on the development of the Phantom Yaw Effect on a maneuvering missile,” 32nd AIAA Applied Aerodynamics Conference. AIAA, Vol. 3097, 2014, p. 2014. 2 Cummings,

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