ABSTRACT

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Aeroelastic analyses of a wing/tip-store configuration are performed using a computational methodology that couples transonic small-disturbance theory and an interactive boundary layer approach. Flutter and limit-cycle oscillation responses are computed and compared with solutions obtained for three locations of tip store mass. Unmatched and matched analyses are performed to find limit-cycle oscillations at different Mach numbers and altitudes. For some of the configurations considered, non-unique states of limit-cycle oscillation are observed over certain parameter ranges. Generally, the presence of viscosity increases the onset speed of limit-cycle oscillation and quenches oscillation growth, so that their amplitude becomes insensitive or decays with further increase in flight speed. The quenching mechanism is found to be separation of the boundary layer at the trailing edge of the wing. However, in a limited number of cases, the presence of viscosity is found to lead to increased amplitudes of limit-cycle oscillation.

NOMENCLATURE a speed of sound, ft/sec c chord, ft cg, ea center of gravity and elastic axis, ft from

leading edge Cf skin friction coefficient Cp pressure coefficient CL lift coefficient i, j, k computational index coordinates l span, ft M Mach number ρ density, slugs/ft3

t time, typically nondimensional (based on freestream velocity and wing chord)

U velocity, ft/sec Q dynamic pressure, lb/ft2 x, y, z physical coordinates (streamwise, spanwise,

vertical), feet

* Principal Research Aerospace Engineer; AIAA Associate Fellow; narendra.khot@wpafb.af.mil † Principal Research Aerospace Engineer; AIAA Associate Fellow; philip.beran@wpafb.af.mil ‡ Senior Aerospace Engineer; AIAA Senior Member § NRC Senior Research Associate; AIAA Fellow

ζ structural damping coefficient

INFLUENCE OF TIP STORE MASS LOCATION ON WING LIMIT-CYCLE OSCILLATION

N.S. Khot* P.S. Beran† J.V. Zweber‡ F.E. Eastep§ MultiDisciplinary Technologies Center, Air Vehicles Directorate, AFRL

AFRL/VASD, 2210 Eighth Street, Building 146 Wright Patterson AFB, OH 45433-7531

UNITED STATES

Subscripts ∞ freestream condition s, w store or wing property, respectively

INTRODUCTION High-performance fighter aircraft with external stores are required to operate with high manoeuvrability in the transonic flight regime. In this regime, the potential exists for encountering transonic nonlinear flutter, known as limit-cycle oscillation (LCO). LCO is a limited amplitude, self-sustaining oscillation produced by an aerodynamic-structural interaction, which for the cases of interest, is exasperated by the occurrence of shock waves on the surface of the wing and/or stores. LCO results in an undesirable airframe vibration and limits the performance of the flight vehicle.

The main goal of the current work is to investigate the influence of the location of tip-store mass on the LCO and flutter response. A finite-element model of the Goland+ wing, with an additional store mass, is used for this study. Flutter onset velocity, LCO response, modal contributions to LCO response, tip deflections, and other quantities, are computed and compared for three mass locations at different flight speeds and Mach numbers. Unmatched and matched analyses are conducted to establish the stability boundaries and range of flight conditions over which LCO occurs.

LCO typically occurs near linear flutter boundaries in the nonlinear, transonic regime (Mach number ranging between 0.8 and 1.1), suggesting that classical flutter predictions using linear aerodynamic theories can be applied to the identification of lightly damped modes that may nonlinearly participate in LCO. Indeed, using traditional approaches, Denegri1 had limited success in relating observed store-induced LCO to “hump’’ (or “soft’’ crossing) modes found in velocity-damping diagrams. However, in many cases, the linear approach is inadequate in predicting response characteristics of vehicle configurations in the transonic regime.

The transonic regime differs from the linear, subsonic regime by the appearance of shocks. These structures may strongly interact with vehicle boundary layers, with the possible consequences of flow separation or significant shock movement. In a coordinated manner, we examine the ability of aeroelastic models of varying

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1731

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

fidelity to predict accurately LCO onset and amplitude. Models based on linear analysis, transonic small-disturbance theory (TSDT), and TSDT with interactive boundary layer are considered. Through this approach, we discern: (1) the limitations of linear theory for LCO prediction vis-à-vis the simplest nonlinear theory capable of producing weak shocks; (2) the ability of TSDT to predict store-induced LCO in inviscid flow, and (3) the effects of viscosity on store-induced LCO. This work provides the increased understanding of the LCO phenomenon depending on Mach numbers and velocities.

Two computational methodologies are employed in this investigation: the MSC/NASTRAN aeroelastic analysis program and the TSDT-based NASA/LaRC CAPTSDv computational aeroelasticity algorithm for inviscid and viscous flow. MSC/NASTRAN is used in the development and analysis of structural models, and the prediction of linear aeroelastic response of the wing/store configurations. The CAPTSDv algorithm is used to carry out nonlinear aeroelastic analyses for inviscid flow and viscous flow (through interactive boundary layer coupling).

PROBLEM FORMULATION Geometry and Structure The wing we study is derived from the “heavy” version of the original Goland wing. Like the original, the heavy wing is structurally represented by a beam, but with additional non-structural mass2. This latest version, referred to as the Goland+ wing, is a heavy wing modeled with a box structure to enable a variety of store attachment options3. The Goland+ wing is rectangular (lw = 20 ft, cw = 6 ft) and cantilevered from an infinite midplane. The airfoil section is assumed to be constant over the spanwise extent of the wing and is chosen to be that of a symmetric, 4% thick, parabolic-arc airfoil. The wing-tip store is mounted flush to the wing tip (see Figure 1), although in this paper, the store is not modeled aerodynamically. In a previous study3, for which the sectional shape of the rectangular store (upstream offset, coff = 3 ft, ls = 1 ft, and cs = 10 ft) is also a parabolic arc (50% thicker than the wing), the influence of store aerodynamics on LCO onset is found to be insignificant.

The geometry of the wing structure is simple. The origin is at the mid-height of the root of the leading edge spar. The three spars are un-swept and placed at 0, 2 and 4 ft along the positive x-coordinate. The eleven ribs are evenly spaced on 2 ft centers along the positive y-coordinate. The shear elements are defined by the intersections of the spars and ribs, with 10 elements per spar and 2 elements per rib. The mass properties of this wing are modeled by placing lumped masses with no rotational inertia at each grid point. The lumped masses

are sized to match the mass properties (total mass, cg, and inertia) of the heavy Goland wing. Further details of this model are described in a previous paper3.

Store Mass and Linkage The store configuration examined in this work is that of a tip store. This store structure is modelled as a series of rigid bar elements that result in a 10 ft long rigid bar. The resultant bar is centred 0.5 ft outboard of the wing tip and 2 ft aft of the wing leading edge. The store is then rigidly connected to the six wing tip grid points. The mass properties of the tip store are chosen to match the properties of one section of the wing: a mass of 22.496 slugs and a rotational inertial of 50.3396 slug-ft2.

During this study, the position of the store mass is fixed at the lateral and vertical centers of the store (i.e., y = 20.5 ft and z = 0 ft) and varied in the streamwise direction. The three store-mass locations considered in this investigation are 0, 1, and 2 ft upstream of the elastic axis (towards the wing leading edge), denoted herein as the 0 ft, 1 ft, and 2 ft store-mass locations, respectively.

The CAPTSDv Computational Methodology CAPTSD solves the three-dimensional, transonic, small-disturbance, potential-flow equations for partial and complete aircraft configurations4,5. In this investigation, a single version of the CAPTSDv algorithm is used to perform both inviscid and viscous aeroelastic analysis. Viscous analysis is carried out through an interactive, integral boundary layer coupling strategy described by Howlett6, Edwards7, and Edwards8. The aeroelastic responses of selected surfaces are simulated by linearly time integrating a generalized modal representation of the structure splined to the aerodynamic grid (see Modal Analysis section) and coupling this procedure with the flowfield analysis scheme. In executing the CAPTSDv procedure, structural damping (ζ) is neglected.

RESULTS Physical Conditions The aeroelastic analysis is performed using two different kinds of methods for representing atmospheric conditions: unmatched and matched analysis. With unmatched analysis, freestream density is assigned a constant value equivalent to sea-level conditions (0.0023771 slugs/ft3). Then velocity and dynamic pressure are consistently varied according to the relationship Q∞ = ½ρ∞U∞

2, while treating Mach number as an independent parameter. With matched analysis, freestream density and speed of sound are assigned values corresponding to altitudes of either 20,000 or 30,000 ft. Then, velocity and dynamic pressure are linked according to the previous formula and

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consistently varied through the definition of the Mach number: U∞ = M∞ a∞.

Summary of Grid Construction The view of computational grid used for all CAPTSDv calculations reported in this paper is shown in Figure 2. Owing to the geometry of the wing/store configuration and the mid-plane formulation of the surface boundary condition used in CAPTSDv, this grid is rectilinear. Clustering of grid points is enforced along the edges of the geometry and normal to the wing and store surfaces. Further details of this grid were in a previous paper3.

Modal Analysis Modes of the structural model are computed with MSC/NASTRAN and then splined to aerodynamic surface grids (specified at z = 0) with the infinite plate spline, as implemented by Harder and Demaris9. The modes are scaled to yield generalized masses of magnitude 1. The results given in this paper are obtained by retaining the 4 modes of lowest frequency in the aeroelastic analysis and by excluding in-plane modes. The vibration frequencies of the first four retained modes of the clean wing (i.e., excluding tip-store mass) are computed to be 1.97, 4.05, 9.65, and 13.4 Hz, respectively. The vibration frequencies of the first four retained modes of the wing with the store mass at the 2 ft location are computed to be 1.69, 2.96, 9.13, and 10.8 Hz, respectively. The vibration frequencies of the first four retained modes of the wing with the store mass at the 1 ft location are computed to be 1.69, 3.29, 9.15, 11.1, respectively. The vibration frequencies of the first four retained modes of the wing with the store mass at the 0 ft location are computed to be 1.67, 3.57, 8.85, 11.7, respectively. Computed mode shapes are similar to that shown in a previous paper3.

Numerical Results Five different sets of aeroelastic analyses were performed during this investigation:

A. MSC/NASTRAN flutter calculations at different Mach numbers for all store-mass locations.

B. Unmatched, stability analysis with inviscid CAPTSDv at different Mach numbers for all store-mass locations; LCO analysis for Mach 0.91, 0.92, 0.93, and 0.94 for all store-mass locations (sea level).

C. Matched, inviscid CAPTSDv LCO analysis for different Mach numbers and 1 and 2 ft store-mass locations at altitudes of 20,000 ft and 30,000 ft.

D. Unmatched, viscous CAPTSDv LCO analysis at Mach 0.92 for all store-mass locations (sea level).

E. Matched, viscous CAPTSDv LCO analysis for different Mach numbers and 1 and 2 ft store-mass locations at 30,000 ft altitude.

Results were generated (for all three store-mass locations) at Mach numbers and altitudes other than that shown above, but are not presented in this paper for want of space. It may be mentioned here that for the 0 ft store-mass location, for which many results are not presented, particularly for matched analysis, the response at different Mach numbers was typical LCO. However, the computed lift coefficient and tip deflections were of such large magnitudes that the validity of these dynamic responses were not regarded by the authors as being quantitatively accurate.

Set A The flutter velocities obtained by using P-K method for the three Mass locations for different Mach numbers are shown in Figure 3. It is observed that for any specific Mach number the flutter velocity increases as the tip mass is moved towards the leading edge. For the 2 ft store-mass location no flutter velocity was found for Mach numbers equal to or greater than 0.89. The V-g diagrams indicated that damping parameter remained negative for all modes at all velocities. At Mach 0.88 the flutter velocities for 0, 1, and 2 ft store-mass locations are 381.5, 442.0, 691.5 ft/sec, respectively. Moving the mass location from 0 ft to 2 ft (i.e., towards the wing leading edge) increased the flutter velocity by 80% for Mach 0.88.

Set B Boundaries of flutter and LCO onset are computed with CAPTSDv for the store mass located at all three offset positions. The flow is assumed to be inviscid but nonlinear. For Mach numbers equal to or less than 0.90, flutter is observed. However, at selected Mach numbers between 0.90 and 0.95 (0.905, 0.91, 0.92, 0.93, and 0.94), LCO solutions are observed. Flutter and LCO boundaries for the three locations of the mass are compared in Figure 4. At flight speeds above the aeroelastic stability boundaries, equilibrium solutions are no longer dynamically stable and diverge in an oscillatory fashion (i.e., flutter) for Mach numbers less than or equal to 0.9. The nonlinear version of CAPTSDv at Mach 0.9 predicts flutter speeds of 409.5, 559.5 and 877.0 ft/sec for the 0, 1, and 2 store-mass locations, respectively. CAPTSDv clearly confirms that the forward movement of the store mass has a stabilizing effect on the aeroelastic system for Mach numbers at or below 0.905.

The LCO onset velocities between Mach 0.9 and 0.94 decrease as the tip mass is moved towards the leading edge. For Mach numbers 0.91, 0.92, 0.93 and 0.94, the LCO onset velocities in ft/sec for the (0, 1, 2) ft store-mass locations are found to be (528.0, 427.0, 429.0,

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583.0) ft/sec, (483.0, 409.0, 405.0, 575.0) ft/sec, and (436.0, 381.0, 382.0, 543.0) ft/sec, respectively. The peak LCO lift coefficients for these same Mach numbers and store-mass locations, are (0.23, 0.16, 0.13, 0.062), (0.22, 0.16, 0.10, 0.070) and (0.20, 0.16, 0.17, 0.074), respectively. The LCO calculations are carried out to flight speeds of 650 ft/sec. Generally, LCO amplitudes increase with increasing velocity, and for sufficiently large velocities, computed oscillations become so large that the assumptions of TSDT may break down. The peak LCO lift coefficients at 650 ft/sec for the above-mentioned Mach numbers and mass locations are (0.26, 0.77, 0.66, 0.20), (0.92, 0.85, 0.67, 0.31) and (1.01, 0.88, 0.71, 0.34), respectively. This data indicates that as the store mass is moved towards the wing leading edge, the peak lift coefficients during LCO increase.

In Figure 5, for Mach numbers 0.91 and 0.93 peak lift coefficients are compared for 1 and 2 ft store-mass locations in order to show the effect of mass location on the system stability. All responses are calculated for initial disturbance of 0.01 in the first-mode amplitude. For Mach 0.91, the stability curves are multi-valued for certain velocities.

For Mach 0.91, in the case of 1 ft offset, the peak lift coefficient increases monotonically from 0.22 at 483 ft/sec (point A1) to 0.37 at 592 ft/sec (point B1). At a speed of 594 ft/sec, the peak lift coefficient jumps to 0.91 (point C1). After point C1, up to point D1, there is gradual increase of the lift coefficient. In order to investigate the LCO response for velocities less than 594 ft/sec, additional CAPTSDv runs are made with the initial condition selected to be the LCO state at point C1. With this approach the curve C1-E1 is obtained. This is a stable branch of the curve, indicating two different solutions depending on initial condition for velocities 500 ft/sec (point E1) through 594 ft/sec (point C1). The existence of a curve connecting points B1 and E1 is speculated that represents an unstable branch (i.e., physically unrealizable). At 594 ft/sec, where the jump occurs from low magnitude of peak lift coefficient to a high value, the time history of lift coefficient is shown in Figure 6. After a low-amplitude LCO is nearly established at a time of 1500, the lift coefficient slowly increases from about 0.4 to about 0.9.

At 490 ft/sec (point F1), where the initial condition used is the LCO state at C1, LCO amplitude decreases to 0.20 from 0.91. See Figure 7. The fraction 1.3/1.2 and 7.7/4.7 at points A1 and D1 are the ratio of amplitudes of the first mode to the second mode. At point A1, where the velocity is 483 ft/sec, the contribution of the second mode is 92% of the first mode, while at point D1 the contribution of the second mode reduces to 61%. For velocities less than 483

ft/sec, the initial response damps out. For velocities greater than 650 ft/sec, peak lift coefficients increase with increasing flight speed.

For the 2 ft store-mass location at Mach 0.91, the peak lift coefficient increases from 0.20 at U = 436 ft/sec (point A2) to 0.35 at U = 494 ft/sec (point B2). At 496 ft/sec, the peak lift coefficient jumps to 0.96 (point C2). From point C2 up to D2 (650 ft/sec), the peak lift coefficient gradually increases to 1.01. In order to investigate the LCO response at velocities less than 496 ft/sec, CAPTSDv runs are made using as an initial disturbance the LCO state at point C2. At 400 ft/sec, the response damps out after a short time for this initial condition. The dotted curve shown in the figure connecting points B2 to E2 is the speculated unstable branch. The time history for U = 498 ft/sec, where jump in the peak lift coefficient is observed, is shown in Figure 8. For U = 436 ft/sec (point A2), the ratio of the amplitude of the first mode to that of the second is 1.1/1.4, indicating that both modes significantly participate in the LCO response. At point D2, the amplitudes of the first mode and the second mode are 8.2 and 6.1, respectively. For velocities less than 436 ft/sec, the transient aeroelastic response vanishes after a short time.

The response curves at Mach 0.93 for the 1 and 2 store-mass positions indicate gradual increase in the LCO peak lift coefficients without any jumps (although LCO amplitude varies rapidly near each onset point). The contributions from the two modes are indicated at the beginning and end of each curve. Transient responses are observed to vanish for both mass positions at flight speeds below LCO onset.

Set C In this case, the LCO response is investigated for 1 and 2 ft store-mass positions at altitudes of 20,000 and 30,000 ft using matched, inviscid analysis. The peak lift coefficients for the two mass locations at the elevation of 20,000 ft are first compared in Figure 9. The LCO response is bounded between Mach numbers 0.906 and 0.944 for the 2 ft store-mass position. However, for the 1 ft position, dynamic responses of large magnitude persist to Mach numbers as low as Mach 0.75. For both cases, peak lift coefficients decrease as the Mach number is increased. On the right side of the curves for higher Mach numbers the response was damped after a short time for both mass locations. The contributions of the first and second mode to the LCO response are indicated at the end of the curves.

Under matched conditions (elevation of 20,000 ft), MSC/NASTRAN predicts flutter onset for the store mass at the 1 ft position to be Mach 0.740. This is in excellent agreement with the observed development of

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sustained dynamic responses at about Mach 0.75 predicted by CAPTSDv.

Figure 10 indicates the peak lift coefficient distribution for the two mass locations at 30,000 ft elevation. For the 2 ft store-mass position, LCO response is bounded between Mach numbers 0.906 and 0.944. The contribution of the first and second modes to the LCO response for the two extreme Mach numbers are indicated on the curve. In the case of the 1 ft store-mass position, two sets of sustained dynamic responses are found. One set of LCO states is bounded between Mach 0.907 and 0.944, another set of dynamic responses exists between about Mach 0.85 and 0.90. From Mach 0.9 to 0.907, transient responses damp out. At Mach 0.899 and 0.9, LCOs are observed that are dominated by the first mode. At Mach 0.90, the ratio of the amplitudes of the first and the second modes is 8.7/0.21 (while the peak lift coefficient is 0.17). When Mach is reduced to about 0.89, a non-simple LCO response is observed. Such dynamic responses are observed for Mach numbers down to Mach 0.85.

Under matched conditions (elevation of 30,000ft), MSC/NASTRAN predicts flutter onset for the store mass at the 1 ft position to be Mach 0846. This is in excellent agreement with the observed development of sustained dynamic responses at about Mach 0.85 predicted by CAPTSDv. However, it should be noted that for the store mass at the 2 ft position, linear analysis with MSC/NASTRAN does not predict flutter, while nonlinear analysis predicts onset of sustained dynamic responses at about Mach 0.906. Thus, not surprisingly, in cases where dynamic stability is lost in the transonic regime, linear analysis does not appear to be adequate for determining the flutter or LCO onset speed.

Set D LCO solutions are computed for viscous flow assuming unmatched analysis at Mach 0.92 and compared to those obtained for inviscid flow in Figure 11 for all store-mass locations. In this figure, it is generally observed that as flight speed increases, LCO amplitude increases. However, in all cases shown, the growth becomes relatively flat beyond certain speeds, and for two of the cases shown, there is a slight diminishment of amplitude around U = 650 ft/sec. The effects of increased store-mass position (i.e., movement towards the wing leading edge) are reduced LCO onset speed and increased LCO amplitude (for fixed flight speed). As will be described further in later sections, the LCO amplitude grows to an amplitude much larger for inviscid flow than viscous flow. Furthermore, it should be noted that for the viscous solutions, computed dynamic responses lose their simple LCO character at a sufficiently large flight speed, gaining increased

frequency content. The reason for this change in the response character has not yet been investigated.

Set E Finally, viscous solutions arising from matched analyses (for an altitude of 30,000 ft) are compared to inviscid solutions in Figure 12 for the 1 and 2 ft store-mass locations. The resulting solution space is quite complicated, showing two distinct regions of dynamic response for the 1 ft store-mass location. Starting first with the 2 ft store-mass location, it is observed that the dynamic responses available when the flow is viscous occur over a range of Mach numbers (0.906 to about 0.93) which is smaller than that seen for inviscid flow. Beyond about Mach 0.925, these responses become irregular, as was described above for the Set D calculations, but diminish in amplitude with increasing Mach number, in the fashion of the “atypical” LCO observed by Denegri1.

When the store-mass location is moved to 1 ft, a new set of dynamic responses become available between about Mach 0.85 and 0.9 for both inviscid and viscous flow. These branches of dynamic responses are primarily characterized by temporal oscillation in mode 1 (see Set B discussion), and are separated from another pair of branches (for inviscid and viscous flow) that occur at Mach numbers exceeding about 0.906. This latter set of branches are characterized by strong coupling between modes 1 and 2. Between the two different sets of aeroelastic responses, there is a “void” region (Mach 0.9 to 0.906) for which inviscid and viscous analyses are linearly stable. In contrast, the LCO branches that occur below about Mach 0.9 erupt from a reverse Hopf bifurcation in such a way that the analyses are not linearly stable below Mach 0.9 until Mach number becomes small enough to re-stabilize the systems.

One feature distinguishes the viscous flow branch from the inviscid one for the dynamic responses emanating from the bifurcation points near Mach 0.9. In the case of viscous flow, a “tongue” of the solution branch extends back over the bifurcation point to at least Mach 0.91. Points on this tongue are found by using as initial conditions solutions from converged LCO simulations. For Mach 0.91, two LCO states are observed: a large-amplitude state, and a small-amplitude state. This is the first three-dimensional, aeroelastic computational model, which the authors are aware of, that represents a physically realizable condition (viscous and matched) permitting multiple LCO states.

Further discussion of the viscous flow branch, and how certain points on this branch are computed is given below in the Effects of Viscosity on LCO Dynamics section.

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Dependence of LCO Amplitude on Flight Speed The variation of LCO amplitude with respect to changes in flight speed at Mach 0.92 is now shown for four categories of analysis: CAPTSDv inviscid analysis of wing with store mass; CAPTSDv inviscid analysis of wing with store modeled aerodynamically, CAPTSDv viscous analysis of wing with store mass, and Euler analysis of wing with store mass10. The position of the store mass is offset 1.75 ft upstream of the wing elastic axis. Results are compared in Figure 13 to show the effects of varying the level of modeling fidelity within the context of transonic small-disturbance theory. Assuming inviscid flow, it is observed that modeling the store aerodynamically with transonic small-disturbance theory has little impact on the onset or computed amplitude of LCO, at least near the point of LCO onset. (The reader should note that when LCOs of larger amplitude are simulated with modeling of store aerodynamics, the CAPTSDv algorithm becomes unstable owing to a large pressure spike that develops in the juncture region of the wing leading edge and the store3). The primary effect of viscosity is to diminish LCO amplitude, with the relative diminishment increasing with flight speed. Beyond a flight speed of about 450 ft/sec, LCO amplitude is observed to be rather insensitive to further increases in flight speed. Also, the large increase in LCO amplitude observed at U = 430 ft/sec for inviscid flow conditions is not observed when the flow is viscous, possibly extending the speed range over which the assumption of small disturbances is arguably satisfied. With the Euler equations, LCO onset occurs at a flight speed about 25% higher than that predicted by CAPTSDv. While this difference is large, and merits further examination, it is encouraging that the LCO predicted with the Euler equations consists of a coupling of the 1st and 2nd modes like that observed with CAPTSDv (Snyder et al. (2003)). Thus, CAPTSDv appears to predict properly the qualitative aeroelastic response of the wing/store configuration, assuming inviscid flow.

Dependence of LCO Frequency on Flight Speed LCO frequencies are compared for the CAPTSDv and Euler models, assuming store aerodynamics are negligible, and found to be in reasonably good agreement. Computed frequencies are summarized in Figure 14 over the range of computed LCOs and for single points for which equilibrium solutions are stable. Frequencies for stable cases are recorded during the decay of the aeroelastic response. For each of the models, LCO onset occurs at a flight speed at which frequency is locally decreasing. However, at larger flight speeds, frequency is generally observed to increase at the same rate for all three models. One exception to this consistent behavior is observed for the CAPTSDv inviscid analysis between flight speeds of

410 and 450 ft/sec. In this range, frequency is observed to increase rapidly, as does LCO amplitude (Figure 13). As described above, it is possible that with this jump in LCO amplitude, the CAPTSDv model becomes invalid. The Euler prediction of LCO frequency is only about 15% lower than that of CAPTSDv (inviscid).

Wing-Tip Conditions The history of the wing tip once the aeroelastic system achieves LCO of moderate amplitude is shown in Figure 15 for Mach 0.92, an elevation of 30K ft, and a store-mass position of 1ft. Viscous flow conditions are assumed. As shown, the maximum angle of attack is 3.3 degrees. The tip deflections at the elastic axis (EA), leading edge (LE) and trailing edge (TE) are 0.132 ft, 0.05ft and 0.36 ft, respectively. These response curves follow the same pattern of development as that of the lift coefficient. It should be noted that the oscillation is of a very regular character, and that tip conditions, which are the most extreme, are well within what would be expected to be valid for small-disturbance theory.

Effects of Viscosity on LCO Dynamics As described in Figure 12, the presence of viscosity both complicates and simplifies the LCO dynamics in the transonic regime. Using matched analysis (30,000 ft) with the store position at 1 ft, two sets branches of sustained dynamic oscillations are computed: Branch I, lying between about Mach 0.85 and 0.90, and Branch II, lying between about Mach 0.906 and 0.93. For Branch II, the impact of viscosity is to diminish LCO amplitude; peak lift coefficients are about 0.2, versus much higher values for inviscid flow (which are likely to be quantitatively inaccurate in the context of small disturbances). For Branch I, large-amplitude dynamic responses are obtained for both inviscid and viscous flow, but in the case of viscous flow, the branch is “stretched” out over Branch II in a manner that is not evident in the inviscid calculations. In this section, we examine the dynamics of the flowfield, as represented by the pressure coefficient on the surface of the wing, to better understand the changes in the flowfield brought about by viscosity.

First, attention is given to the solution on Branch II at Mach 0.2, and the corresponding solution on the branch for inviscid flow. As shown in contour plots of surface pressure coefficient, the presence of viscosity (1) moves the shock on the wing surface towards the trailing edge, and (2) leads to larger variations in pressure over outboard wing sections11. Contour plots are given in Figure 16 for inviscid flow and Figure 17 for viscous flow. Each figure provides snapshots of the wing upper surface at two instantaneous states during LCO, the first given by a forward position of the shock and the second given by an aft position of the shock. In Figure 16 (left), the shock is primarily located near the trailing

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edge, detaching somewhat at stations beyond about 25% of span (it should be noted that over part of the LCO cycle, the wing upper surface is in compression, and these states are not shown), while in Figure 16 (right), the shock is at the trailing edge over a greater portion of the span with an associated enlarged region of reduced pressure over the wing.

The role of viscosity in reducing LCO amplitude is clarified in Figure 18 through examination of the skin friction coefficient on the wing surface. Skin friction is shown for the same two instants in time selected for examination of pressure coefficient in Figure 17 (Branch II, Mach 0.92). Examination of Cf at those instants reveals that when the moving shock is near the trailing edge, significant shock-induced separation of the boundary layer occurs. Thus, while LCO is observed for both inviscid and viscous flow, boundary-layer separation appears to restrict the rearward movement of the shock (cf. Figure 16) and limit the growth of oscillation amplitude. Thus, for this configuration and flight condition, the loss of dynamic stability is not brought about by trailing-edge separation (viscosity delays onset; cf. Figure 12), but the boundary-layer phenomenon induced by the shock has a major impact on the development of the response. Trailing -edge separation has also been identified as a quenching mechanism for LCO in a separate study in which a full-aircraft configuration was studied12.

Now, attention is turned to Branch I. This branch is characterized by large-amplitude, non-simply periodic responses dominated by mode 1 participation. See Figure 12. In contrast to what is seen for Branch II, the potential effect of viscosity for this class of solutions is the increase of LCO amplitude. The extent to which viscosity affects LCO behavior is determined by initial conditions. For an initial condition of small amplitude (i.e., baseline), an LCO of small amplitude (peak lift coefficient of about 0.15) is obtained that is very similar to the LCO computed assuming inviscid flow. However, a second LCO state, of much larger amplitude, is computed with viscous analysis by using the LCO state at a smaller Mach number (Mach 0.89) as an initial condition. The inviscid solution is compared to the large-amplitude viscous solution in Figures 19 and 20 using the distributions of pressure coefficients computed at the wing surface at selected instants of time. In these figures, it is seen that on the large-amplitude portion of Branch I (viscous), the shock travels further downstream than in the case of viscous flow. Examination of the skin friction coefficient indicates that when the shock is near the trailing edge, the boundary layer separates (Figure 21). Further study of the effects of viscosity is warranted to better understand how viscosity can both magnify and diminish LCO amplitude.

SUMMARY AND CONCLUSIONS A detailed investigation was carried out for the LCO response of the Goland+ wing for three positions of tip-store mass using transonic small-disturbance theory. For unmatched analysis, LCO was observed between Mach numbers 0.91 and 0.94. The LCO amplitudes increased with increasing velocity, and for sufficiently large velocities, oscillations become so large that TSDT likely became quantitatively inaccurate. At lower Mach numbers, the forms of bifurcations were subcritical, such that LCO amplitude jumped abruptly beyond onset. The time history of lift coefficient at the transition velocity indicated the initial establishment of a low-amplitude LCO that slowly increased in response to large-amplitude LCO.

For matched analysis, the LCO response was bounded between the Mach numbers 0.91 and 0.94 for the store-mass location of 2 ft. However, when the store mass was moved downstream to the 1 ft location, a non-simple form of LCO was observed with the first mode dominating the response. The effect of viscosity in unmatched analysis was to reduce the LCO amplitude. For matched analysis, the effects of viscosity were the decrease of LCO amplitude and the narrowing of the band of Mach numbers over which LCO is observed. The impact of viscosity was primarily felt through the development of boundary-layer separations triggered by shocks moving over the upper and lower surfaces of the wing.

As shown for both inviscid and viscous flow, non-unique LCO states existed for particular combinations of flight speed, altitude, and position of store mass. For these conditions, the choice of large-time behavior (i.e., the selection of LCO states) was determined by initial conditions.

The observed behaviors that were observed were partially validated through linear analysis and through, in a separate study, aeroelastic simulation using the Euler equations. Onsets of dynamic response that occurred in the linear regime (Mach numbers less than about 0.85-0.9) were well predicted by both MSC/NASTRAN and CAPTSDv. LCO states observed at larger Mach numbers were also observed with the Euler analysis. While the modal characters of the responses were in good agreement, quantitative differences in modal amplitudes did exist. Future work should concentrate on the validity of the LCO states predicted by CAPTSDv using viscous analysis.

ACKNOWLEDGEMENTS This work was sponsored by the Air Force Office of Scientific Research under Laboratory Task 99VA01COR, monitored by Dr. Dean Mook.

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1. Denegri, C. M., “Limit Cycle Oscillation Flight Test

Results of a Fighter with External Stores,” Journal of Aircraft, Volume 37, Number 5, September-October 2000, pp. 761-769.

2. Eastep, F. E., “Transonic Flutter Analysis of a Rectangular Wing with Conventional Airfoil Sections,” AIAA Journal, Volume 18, Number 10, October 1980, pp. 1159-1164.

3. Beran, P. S., Khot, N. S., Eastep, F. E., Snyder, R. D., Zweber, J. V., Huttsell, L. J., and Scott, J. N., “The Dependence of Store-Induced Limit-Cycle Oscillation Predictions on Modelling Fidelity,” RTO Applied Vehicle Technology Panel Symposium on Reduction of Military Vehicle Acquisition Time and Cost Through Advanced Modeling and Virtual Product Simulation, Paris France, April 2002, Paper #44.

4. Batina, J. T., “Efficient Algorithm for Solution of the Unsteady Transonic Small-Disturbance Equation,” Journal of Aircraft, Volume 25, July 1988, pp. 598-605.

5. Batina, J. T., “Unsteady Transonic Algorithm Improvements for Realistic Aircraft Applications,” Journal of Aircraft, Volume 26, February 1989, pp. 131-139.

6. Howlett, J. T., “Efficient Self-Consistent Viscous Inviscid Solution for Unsteady Transonic Flow,” Journal of Aircraft, Volume 24, November 1987, pp. 737-744.

7. Edwards, J. W., “Transonic Shock Oscillations Calculated with a New Interactive Boundary Layer Coupling Method,” AIAA Paper 93-0777, January 1993.

8. Edwards, J. W., “Calculated Viscous and Scale Effects on Transonic Aeroelasticity,” AGARD-R-822, Numerical Unsteady Aerodynamic and Aeroelastic Simulation, March 1998, pp. 1-1 – 1-11.

9. Harder, J. T., and Desmaris, R. N., “Interpolation Using Surface Splines,” Journal of Aircraft, Volume 9, Number 2, 1972.

10. Snyder, R.D., Scott, J.N., Khot, N.S., Beran, P.S., and Zweber, J.V., “Predictions of Store-Induced Limit-Cycle Oscillations Using Euler and Navier-Stokes Fluid Dynamics,” 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA 2003-1727, April 2003.

11. Beran, P.S., Khot, N.S., Eastep, F.E., Strganac, T.W., and Zweber, J.V., “Effects of Viscosity on

Store-Induced Limit-Cycle Oscillation,” 2003 ITEA Aircraft-Stores Compatibility Symposium XIII, Destin FL, February 2003.

12. Melville, R., “Aeroelastic Instability of Tactical Aircraft in Nonlinear Flow Regimes,” 32nd AIAA Fluid Dynamics Conference, AIAA 2002-2970, June 2002.

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x ls

ycoff

U∞

cs

ea

lw

cw

Figure 1: Schematic of Goland+ Wing planform geometry (including store).

x/cwy/

c w-1 0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

Figure 2: CAPTSDv grid along wing mid-plane: grid lines (red) and wing boundary (green)

Mach

Velocity(Ft/Sec)

0.7 0.75 0.8 0.85 0.9 0.95300

350

400

450

500

550

600

650

700

750

800

850

900

Zero Ft OffsetOne Ft OffsetTwo Ft Offset

Flutter point notobserved aboveMach 0.88

Figure 3: Comparison of MSC/NASTRAN flutter

boundaries for three store-mass locations.

Mach

Velocity(Ft/Sec)

0.7 0.75 0.8 0.85 0.9 0.95300

350

400

450

500

550

600

650

700

750

800

850

900 Zero Ft Offset (Flutter)One Ft Offset (Flutter)Two Ft Offset (Flutter)Zero Ft Offset (LCO)One Ft Offset (LCO)Two Ft Offset (LCO)

Figure 4: Comparison of CAPTSDv stability

boundaries (inviscid flow) for three store-mass locations.

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Velocity (Ft/Sec)

PeakLiftCoefficient

350 400 450 500 550 600 650 700

0.25

0.5

0.75

1

1.25 One Ft Offset - Mach 0.91One Ft Offset - Mach 0.93Two Ft Offset - Mach 0.91Two Ft Offset - Mach 0.93

8.2/6.1

7.7/4.7

6.7/4.5

7.8/3.6

1.3/1.21.1/1.4

0.30/0.47

0.74/1.2

Branch I

Unstable Branch(speculated)

Branch IID1

E2

A2

B1

C1

E1

A1

F1

E2

C2

D2

B2

Figure 5: Variation of peak lift coefficient during sustained dynamic response for two Mach numbers and

store-mass locations (participation of modes 1 and 2 reported for selected LCO states as #1/#2).

Time (Nondimensional)

LiftCoefficient

1000 2000 3000 4000 5000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 6: Time response of lift coefficient at a flight speed of 594 ft/sec and Mach 0.91 starting from

baseline initial condition (converging to limit cycle C1).

Time (Nondimensional)

LiftCoefficient

1000 2000 3000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 7: Time response of lift coefficient at a flight speed of 490 ft/sec and Mach 0.91 starting from limit

cycle E1 (converging to limit cycle F1).

Time (Nondimensional)

LiftCoefficient

1000 2000 3000 4000 5000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 8: Time response of lift coefficient at a flight speed of 490 ft/sec and Mach 0.91 starting from

baseline initial condition (converging to limit cycle C2).

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Mach

PeakLiftCoefficient

0.9 0.91 0.92 0.93 0.94 0.95

0.25

0.5

0.75

1

1.25

One Ft Offest (20K)Two Ft Offset (20K)

8.4/4.4

2.0/0.73

9.0/6.0

2.3/1.7

Dynamic response diesout near Mach 0.75

Figure 9: Variation of peak lift coefficient with Mach number at an altitude of 20K ft (matched results) for

two store-mass locations (participation of modes 1 and 2 reported for selected LCO states as #1/#2).

Mach

PeakLiftCoefficient

0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95

0.25

0.5

0.75

1

1.25 One Ft Offest (30K)Two Ft Offset (30K)

C2: 1.5/1.1

C1: 0.91/0.62

A2: 6.7/6.1

A1: 8.7/0.21

B1: 5.8/4.7

Dynamic responsewith higher frequencycontent

Figure 10: Variation of peak lift coefficient with Mach number at an altitude of 30K ft (matched results) for

two store-mass locations (participation of modes 1 and 2 reported for selected LCO states as #1/#2).

Velocity (Ft/Sec)

PeakLiftCoefficient

350 400 450 500 550 600 650

0.25

0.5

0.75

1

1.25 Zero Ft Offset - InviscidZero Ft Offset - ViscousOne Ft Offset - InviscidOne Ft Offset - ViscousTwo Ft Offset - InviscidTwo Ft Offset - Viscous

Non-Simple LCO

2.4/1.6

1.8/1.3

1.3/1.0

7.7/4.7

8.2/6.1

0.74/1.20.3/0.47

0.15/0.260.17/0.26

0.17/0.22

Figure 11: Variation of peak lift coefficient with flight speed for unmatched analysis of inviscid and viscous

flow at Mach 0.92 and three store-mass locations (participation of modes 1 and 2 reported for selected

LCO states as #1/#2).

Mach

PeakLiftCoefficient

0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95

0.25

0.5

0.75

1

1.25

1.5One Ft Offest - InviscidOne Ft Offset - ViscousTwo Ft Offset - InviscidTwo Ft Offset - Viscous

Non-simple LCO

Unstable Branch(Speculated)

40/1.9

8.2/2.7

0.91/0.61

1.5/1.1

1.3/0.88

1.2/0.94

Figure 12: Variation of peak lift coefficient with Mach (matched analysis for 30,000 ft altitude) for inviscid

and viscous flow and two store-mass locations (participation of modes 1 and 2 reported for selected

LCO states as #1/#2).

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Velocity, U

PeakLiftCoefficient

300 350 400 450 500 550 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Inviscid (G1)Inviscid (G2)Viscous (G3)ENS3DAE

Figure 13: Variation of LCO amplitudes with flight speed for inviscid (CAPTSDv/G1 and

ENS3DAE/Euler) and viscous (CAPTSDv/G3) flow at Mach 0.92 (analysis G2 includes tip-store

aerodynamics)10.

Velocity, U

Frequency(Hz)

300 350 400 450 500 550 6002.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

CAPTSDv-InviscidCAPTSDv-ViscousENS3DAE-Euler

Figure 14: Variation of LCO frequencies with flight speed for inviscid (CAPTSDv and Euler) and viscous

(CAPTSDv) flow at Mach 0.9210.

Time

AOA

TipDeflection(feet)

20 21 22 23 24 25-4

-3

-2

-1

0

1

2

3

4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

AOATip Def. at EATip Def. at TETip Def. at LE

Figure 15: Typical LCO response of wing tip in terms of angle-of-attack (AOA in degrees) and deflection at the leading edge (LE), elastic axis (EA), and trailing edge (TE): matched viscous analysis at Mach 0.92 (30K elevation

for 1 ft store-mass position).

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x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.500.430.360.290.210.140.070.00-0.07-0.14-0.21-0.29-0.36-0.43-0.50

Cp

Figure 16: Distributions of pressure coefficient on wing upper surface during LCO cycle for Mach 0.92 (inviscid

matched analysis at 30K elevation for 1 ft store-mass position): (left) shock forward; (right) shock aft.

x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.500.430.360.290.210.140.070.00-0.07-0.14-0.21-0.29-0.36-0.43-0.50

Cp

Figure 17: Distributions of pressure coefficient on wing upper surface during LCO cycle for Mach 0.92 (viscous

matched analysis at 30K elevation for 1 ft store-mass position): (left) shock forward; (right) shock aft.

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x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.00420.00390.00360.00330.00300.00270.00240.00210.00180.00150.00120.00090.00060.00030.0000

Cf

Figure 18: Distributions of skin friction coefficient on wing upper surface during LCO cycle for Mach 0.92 (viscous

matched analysis at 30K elevation for 1 ft store-mass position): (left) shock forward; (right) shock aft.

x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.500.430.360.290.210.140.070.00-0.07-0.14-0.21-0.29-0.36-0.43-0.50

Cp

Figure 19: Distributions of pressure coefficient on wing upper surface during LCO cycle for Mach 0.899 (inviscid

matched analysis at 30K elevation for 1 ft store-mass position): (left) shock forward; (right) shock aft.

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x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.500.430.360.290.210.140.070.00-0.07-0.14-0.21-0.29-0.36-0.43-0.50

Cp

Figure 20: Distributions of pressure coefficient on wing upper surface during LCO cycle for Mach 0.899 (viscous matched analysis at 30K elevation for 1 ft store-mass position – top branch): (left) shock forward; (right) shock aft.

x/cw

0 0.5 10

1

2

3

x/cw

y/c w

0 0.5 10

1

2

3

0.00420.00390.00360.00330.00300.00270.00240.00210.00180.00150.00120.00090.00060.00030.0000

Cf

Figure 21: Distributions of skin friction coefficient on wing upper surface during LCO cycle for Mach 0.899

(viscous matched analysis at 30K elevation for 1 ft store-mass position – top branch): (left) shock forward; (right) shock aft.

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