24TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES
An Overset Field-Panel Method for Unsteady Transonic Aeroelasticity
P.C. Chen§, X.W. Gao¥, and L. Tang‡ ZONA Technology*, Scottsdale, AZ, www.zonatech.com
Abstract process because it is independent of the The overset field-panel method presented in this structural characteristics, therefore it needs to paper solves the integral equation of the time- be computed only once and can be repeatedly linearized transonic small disturbance equation by used in a structural design loop. an overset field-panel scheme for rapid transonic aeroelastic applications of complex configurations. However, it is generally believed that the A block-tridiagonal approximation technique is unsteady panel methods are not applicable in developed to greatly improve the computational the transonic region because of the lack of the efficiency to solve the large size volume-cell transonic shock effects. On the other hand, the influence coefficient matrix. Using the high-fidelity Computational Fluid Dynamic (CFD) computational Fluid Dynamics solution as the methodology provides accurate transonic steady background flow, the present method shows solutions by solving the Euler’s or Navier- that simple theories based on the small disturbance Stokes’ equations, but it does not generate the approach can yield accurate unsteady transonic AIC matrix and cannot be effectively used for flow predictions. The aerodynamic influence coefficient matrix generated by the present method routine aeroelastic applications nor for an can be repeatedly used in a structural design loop; extensive structural design/optimization. rendering the present method as an ideal tool for Therefore, there is a great demand from the multi-disciplinary optimization. aerospace industry to have an unsteady transonic aerodynamic method with an AIC Introduction matrix generation capability. In fact, such an The unsteady panel methods such as the unsteady transonic aerodynamic method can be Doublet Lattice Method1 (DLM), ZONA62 for developed using the unsteady field-panel subsonic unsteady aerodynamics and ZONA73 method. for supersonic unsteady aerodynamics have been well accepted by the aerospace industry Integral Equations of the Unsteady Field- for many years as the primary tools for routine Panel Methods aeroelastic applications. The unsteady panel Over the past two decades, great progress has methods can handle complex configurations been made in the development of the field- without an extensive model-generation effort, panel method for unsteady transonic flow provides expedient and accurate unsteady computations. In 1985, Voss4 proposed an aerodynamic predictions, and most importantly, AIC-based unsteady field panel method using a generates the Aerodynamic Influence velocity potential approach. Later on, this Coefficient (AIC) matrices that directly relate method was improved by Lu and Voss5 using a the downwash to the unsteady pressure Transonic Doublet Lattice Method (TDLM) to coefficients, i.e., eliminate the wake modeling in the velocity {∆=CAICW} []{} (1) potential approach. The TDLM solves the so- p called Time-Linearized Transonic Small ∆ where Cp is the unsteady pressure jumps Disturbance (TLTSD) equation that reads 222 and W is the downwash due to the ikM∞∞ k M ∂ φφφ++−2 φ + φ =()σ (2) structural oscillations xx yy zzββ22 x∂x v φ The AIC matrix is considered as one of the key where is the perturbed unsteady velocity elements in the industrial aeroelastic design potential, k is the reduced frequency, M∞ is
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2 2 ikM∞ ikM∞ the freestream Mach number, β =−1 M ∞ , − ξ − R 1 β 2 β 2 2 = ()γ + 1 M ∞ G e e is the unsteady σ = φ φ K = R v K ox x . β 2 , and γ is the source kernel. specific heat ratio. and R = ξ 2 +η 2 + ζ 2 , ξ = x − x , Eq. (2) is obtained by linearizing the nonlinear o η = − ζ = − transonic small disturbance equation with yo y , and zo z . respect to the structural oscillating amplitude φ In fact, equation (4) is the integral solution of which is assumed to be small. The term ox in the linear unsteady potential equation and is the the right hand side (RHS) of equation (2) is the equation solved by ZONA6 for the wing-body steady perturbation velocity component along configurations. It should be noted that the the freestream direction, defined here as the TDLM is formulated for the lifting surface “steady background flow,” which contains the only, i.e. the unsteady source integral for steady nonlinear transonic shock effects in bodies is absent. Here, we adopt the ZONA6 transonic flows. This is to say that the solution formulation to handle complex configurations of equation (2) is linearly varying with the such as the wing-body combinations. In the structural oscillating amplitude but it contains ZONA6 paneling scheme, the configuration the nonlinear transonic shock effects embedded surfaces are descritized into many small boxes, in the steady background flow. It should be called the surface boxes, leading to a matrix also noted that that relates the surface singularity strength to equation (2) can lead the normal velocity on each box. to an AIC matrix because of its linear The second and third terms on the RHS of characteristics with equation (2) read 1 ∂ respect to the φ = − ()σ GdV (5) v π ∫∫∫ ∂ v structural oscillating 4 V x amplitude, i.e., the Fig. 1 Integration domain φ = − 1 ∆σ shock ∫∫ vGdS (6) downwash in the of the integral solution 4π RHS of equation (1). shock ∂ In equations (5) and (6), φ is the influence of Assuming ()σ in the RHS of equation (2) v ∂x v the velocity potential from the volume source to be a volume source, the integral solution of φ and shock is the influence of the velocity the TLTSD equation at a point (xo, yo, zo) potential from the shock surface on which ∆σ shown in Fig. 1 consists of three parts: v φ ()x,y,z =++φφφ represents the jump of the volume source ooo s v shock (3) strength across the shock surface. Note that φ φ In equation (3), s represents the influence of shock automatically vanishes if the transonic ∆σ the velocity potential from the surface shock is absent because v = 0. In addition, singularities which can be written as φ in the presence of shock, shock can be ∆ 1 C p ∂φ φ = K − G ds (4) eliminated by performing integration by parts s π ∫∫ ∂ φ 4 s 2 n for the volume integral of v such that ∂φ −εε+ ∆ xxss ∞ ∂ where C p and are the unsteady pressure φ = − 1 + + ()σ ∂n v lim ∫∫ ∫∫∫ v Gdxdydz ε →0 4π −∞ −εε+ ∂x jump and unsteady source distributed on the yz xxss lifting surfaces and bodies, respectively. (7) K is the acceleration potential kernel, gives 1 x s + ε 1 (8) whose detailed expression can be found in φ = lim ∫∫σ G dS + ∫∫∫ σ G dV v π v − ε π v x Ref. 2. ε → 0 4 yz x s 4 V
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An Overset Field-Panel Method for Unsteady Transonic Aerodynamic Influence Coefficient Matrix
where xs represents the shock location and ε and matrices [B] and [D] contain the represents an infinitesimal thickness of the influence coefficients from the volume shock surface. sources to the points on the surface boxes and in the volume cells, respectively. Combining equations (8) and (6) yields 1 Equation (10) can be linked to equation (11) by φ + φ = σ G dV (9) shock v π ∫∫∫ v x σ φ 4 V relating v to through a finite difference Equation (9) can be recast into a matrix operator [T] such that {}σ = []{}φ equation by first defining a volume block v T (12) surrounding the lifting surfaces or bodies and Voss4 suggested that such a finite difference then descritizing the volume block into many operator can be formulated using the Murman’s small volume cells. A typical volume-cell scheme6 that reads. modeling for lifting surfaces and bodies is σµσµσµφµφ=− + + − v()1 i vi i−−11 vi i xi i −− 11 xi shown in Fig. 2. (13) where µi =0 for Mi <1 and µi =1 for Mi ≥ 1, Mi is the local Mach number in the ith volume cell, and i is the index of the volume cells along the freestream direction.
The Murman’s scheme is a conservative finite (a) Lifting Surfaces (b) Bodies difference operator and guarantees the correct Fig. 2 Typical volume cell modeling for lifting surfaces and bodies mathematical shock jumps. It automatically switches from the central differencing in the Unlike the CFD methodology whose volume subsonic flows to the backward differencing in mesh must be extended far away from the the supersonic flows, thereby, introduces the surface mesh, the domain of the volume block directional bias to the integral equations of for the field-panel method needs only to TLTSD for handling the mixed flow problems contain the nonlinear flow region in which the of the transonic flow. The detailed expression σ volume source strength, v , is significant. of the matrix [T] using the Murman’s scheme This is because outside the domain of the can be found in Ref. 7. volume block the solution is dominated by equation (4), thereby the contribution from Substituting equation (12) into equations (10) equation (9) can be ignored. and (11) and combining the resulting equations yields At points located on the surface boxes, the ∆C p (14) normal perturbation velocity reads {}W = []A ∂φ ∆ ! C p ∂n {}! ⋅ ∇φ = {}= [] ∂φ + []{}σ (10) n W A B v where ∂n − []A = []A + [][][]B T E 1 []C (15) At points located in the volume cells, the velocity potential reads and ∆ []E = [][][]I − D T (16) C p {}φ = [] + []{}σ (11) C ∂φ D v Inverting the matrix [A] gives the AIC matrix ∂n ! ! such as the one shown in equation (1). Note where n ⋅ ∇φ represents the normal that the size of the AIC matrix is only the perturbation velocity on the surface boxes, number of surface boxes and the form of the matrices [A] and [C] contain the influence AIC matrix is identical to that of the DLM, coefficients from the surface singularities to ZONA6, and ZONA7. This is a very attractive the points on the surface boxes and in the feature for industrial aeroelastic applications, volume cells, respectively, 3 Chen, Gao, and Tang
because once the AIC matrix is obtained, all when the point (xo, yo, zo) is away from the non- downstream AIC-based aeroelastic solution adjacent sub-blocks. Therefore, the inverse of procedures for flutter, aeroservoelastic and the matrix [E] can be solved approximately by dynamic loads analyses remain unchanged. the following equation: -1 -1 -1 -1 However, the unsteady field-panel method does [E] ≈ [EB] - [EB] [Eε] [EB] (18) -1 have several technical issues that need to be Finally, because [EB] is block-tridiagonal, [EB] resolved before it can be adopted as an can be computed efficiently using a block- industrial tool for expedient unsteady tridiagonal matrix solver. In addition, because aerodynamic computations. the block-tridiagonal matrix solver only needs to hold three block matrices in the computer Block-Tridiagonal Approximation of the [E] memory at a time, a large amount of computer Matrix memory for solving [E]-1 can be saved. The matrix [E] in equation (16) is a complex and fully populated matrix whose size is the Overset Field-Panel Scheme for Complex number of volume cells. To model a general Configurations three-dimensional problem may require more The objective of the overset field-panel scheme than 10,000 volume cells and this number is to minimize the volume-cell generation effort increases rapidly as the complexity of the for complex configurations. For a simple configurations increases. Therefore, to invert lifting surface or body shown in Fig. 2, the (or decompose) such a large matrix is generation of volume cells can be automated; impractical for routine aeroelastic applications. simply specifying the height and number of In the present method, a block-tridiagonal layers of the volume block. However, to approximation technique is employed to generate a set of conformal volume cells over a circumvent this technical issue. complex configuration such as a wing with underwing stores requires an extensive volume- The idea behind the block-tridiagonal cell generation (or grid generation) effort. approximation technique is a simple one. First, the volume cells are grouped into many sub- blocks. For instance, the volume blocks on the top and bottom of the lifting surface can be divided into several sub-blocks and the volume cells within the same sub-block are grouped together. In so doing, the matrix [E] can be written as Fig. 3 Overset field-panel scheme for a wing with an [E] = [EB] + [Eε] (17) underwing missile where [E ] is a block-tridiagonal matrix whose B The overset field-panel scheme allows the tridiagonal blocks contain the influence volume cells to be generated independently on coefficients from the self-block and the each component of the complex configuration. adjacent sub-blocks. For instance, a wing with a underwing missile, [Eε] contains zeros in the tridiagonal blocks a pylon and fins shown in Fig. 3 can be and the influence coefficients from the non- separated into multiple components; wing, adjacent blocks in the off-tridiagonal blocks. pylon, missile body, and fins. On each Next, by comparing the order of magnitude of component, a volume block is defined the coefficients in the matrix [Eε] to the matrix independently which can be automatically [EB], it can be seen that all coefficients of [Eε] divided into volume cells. Therefore, among are small. This is because the integrand in the all volume blocks volume cells of different integral equation shown in equation (5) volume blocks may intersect each other. The contains a 1/R function that decays rapidly overset field-panel scheme constructs the influence coefficient matrices of such an
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An Overset Field-Panel Method for Unsteady Transonic Aerodynamic Influence Coefficient Matrix
overset volume-cell model based on the () And VS is the influence from V to ST, following procedure: T () () VS is the influence from V to SF, is (a) Volume cells in different volume blocks F S V do not influence each other. T the influence from ST to V, () is the (b) Volume cells in the same volume block S V influence only their associated surface F influence from S to V, () is the influence boxes. F VV from V to V (c) All surface boxes influence all volume cells. Equation (19) can be considered as “exact” because of the one-block modeling. The same T-tail configuration can be also modeled by two volume blocks, one for the tail and one for the fin which are depicted by the solid box and (a) One Block Modeling (b) Overset Modeling the dashed box in Fig. 4.b, respectively. Fig. Fig. 4 One-block modeling and overset modeling of a 4.b also shows that the width of the volume T-tail configuration block of the fin is shorter than that of the tail by The above overset field-panel scheme is “2a”. Apparently, the volume cells of the fin formulated with the realization that the are all embedded in those of the tail; rendering interference between volume blocks can be a overset field-panel model. The influence transmitted through the integral equations coefficient matrix equations of this overset shown in equation (3). Unlike the overset CFD field-panel model can be written as: methodology where the interpolation of the ∆C WS p flow solutions in the overset region is required, T ST =+[]A the overset field-panel scheme does not need W ∆C SF p such an interpolation. Therefore there is no SF ()σ need to compute the topology of the ST v BvT 0 vT intersection among cells; greatly simplifying ()σ 0 B SF v (20) the volume-cell generation effort for complex vF vF φ ∆C configurations. To verify this scheme, let us vSTSTCv Cv p TTF=+S T consider a T-tail configuration consisting of a φ ∆ vSFSFpCv Cv C TTF S horizontal tail and a vertical fin. This T-tail F σ configuration can be modeled by a single D 0 ()v vvTT vT rectangular volume block whose volume cells 0 D ()σ vvFF v are conformed to the surfaces of the tail and the vF fin. This one-block modeling is shown in Fig. where the subscript VT and VF denotes the 4.a. and its influence coefficient matrix volume cells of the tail and fin, respectively. equations read: ∆ When “a” = 0, because the domain of the tail C p WS S volume block is identical to that of the fin, it is T = []A T + []B ,B {}()σ W ∆C VS T VS F v v S F p obvious that C = C = C , SF (19) ST V F ST VT ST V = = ∆ C S V C S V C S V , C p F T F F F {}φ = [] ST + []{}()σ v CS V ,CS V Dvv v B = B = B = B and T F ∆C v VT S T VS T V F S F VS F p S F = = DV S DV S DVV , therefore where the subscript S, ST and SF denote the T T F F φ = φ = φ and surface boxes of the tail and of the fin, VT V F V respectively. ()σ = ()σ = ()σ v v v . Thus, the solution The subscript V denotes the volume cells of vT v F v the one volume block of the one-block modeling and the overset field-panel model is identical if “a” = 0.
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When “a” ≠ 0, the overset field-panel modeling the steady shock location in the small becomes an approximation of the one-block amplitude sense; implying that accurate steady modeling. However, because the domain of shock structures can ensure the accuracy of the “a” is far away from the fin, its contribution of unsteady shock structures. the influence to the fin is small and can be To date, many high-fidelity CFD codes for the ignored as long as “a” is not very large. accurate prediction of steady flow over realistic The overset field-panel scheme can also greatly aircraft configurations for transonic flight reduce the computer memory and CPU time for conditions are available. Thus, the steady solving the [E] matrix. As shown in equation background flow of the present method can be 20, because the volume cells in different directly imported from these CFD codes. In volume blocks do not influence each other, the addition, because the CFD mesh is usually [D] matrix becomes block diagonal; leading to much more refined than the volume cells of the a block diagonal [E] matrix that can be inverted field-panel method, to import the CFD steady based on a block-by-block procedure. flow solutions can be easily achieved by a Furthermore, the block-tridiagonal simple interpolation of the CFD steady solution approximation technique can be applied to from the CFD mesh to the volume cells. invert each diagonal block matrix to further Validation of the Unsteady Pressure increase the computational efficiency. Distribution Steady Background Flow From the High- Four test cases are selected to validate the Fidelity CFD Codes unsteady pressure coefficients with the It is well-known that the transonic small experimental data. The steady background disturbance theory may not provide accurate flow of all four test cases are computed by the solutions for strong transonic shock cases CFL3D Navier-Stokes solver9 and interpolated because it cannot correctly model the entropy to the volume cells. gradients from strong shock nor convert the F-5 wing pitching about 50% root chord at M∞ vorticity. However, this is not to say that the =0.9 and k = 0.275 transonic small disturbance theory is not Fig. 5 depicts the field-panel model and the suitable for the prediction of unsteady flows CFL3D surface mesh of a F-5 wing. The field- due to small aeroelastic deformations if the panel model consists of 20x10 surface boxes total unsteady flow is decomposed into a steady and 25x12x12 volume cells whereas the background flow and an unsteady of small 8 CFL3D mesh contains 181x77x71 grid points disturbances. As demonstrated by Liu et al. , for the Navier-Stokes computation. The simplified theories based on the small CFL3D steady pressure coefficients (Cp) at M∞ disturbance approach which can yield accurate =0.9 and angle of attack = 0º (α=0º) are first unsteady flow predictions provided that the compared to the wind-tunnel measurements10 steady background flow on which the unsteady and are shown in Fig. 6 for three span stations disturbance propagate is accurately accounted at y/2b = 51.5%, and = 97.7%. Excellent for. This suggests that, if the steady agreement is obtained except at y/2b = 97.7% background flow in the TLTSD equation is where CFL3D slightly underpredicts the shock externally provided by a high-fidelity CFD strength on the upper surface. steady solution, accurate unsteady flow predictions can be ensured. This is also evident by examining the Murman’s scheme shown in equation 13 where the switching scheme from the central differencing to the backward differencing is based on the local Mach Fig. 5 Field-Panel Model & CFL3D Surface Mesh of numbers of the steady background flow. Thus, F-5 Wing the unsteady shock location is dominated by
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An Overset Field-Panel Method for Unsteady Transonic Aerodynamic Influence Coefficient Matrix
CFL3D surface mesh are depicted in Fig. 13. The field-panel model of the LANN wing consists of 20x14 surface boxes and 24x16x16 volume cells whereas the CFL3D mesh contains a 269x131x71 viscous grid.
Fig. 6 Comparison of Steady Cp between CFL3D & The CFL3D steady Cp at M∞ = 0.822 and α = Wind-Tunnel Results on a F-5 Wing, M∞=0.9 & α=0º 0.6º and at three span stations; y/2b = 47.5%, and 82.5% are compared with the wind-tunnel measured results and are shown in Fig. 14. It can be seen that the steady shock location and strength are well captured by CFL3D except at y/2b = 82.5% where the shock location on the upper surface is slightly underpredicted.
Fig. 7 Unsteady ∆Cp on a F-5 Wing due to a Pitch Fig. 13 Field-Panel Model and CFL3D Surface Mesh Oscillation about 50% chord at M∞=0.9 & k=0.275 of a LANN Wing Using the CFL3D solution as the steady background flow, the unsteady ∆Cp along the same three span stations computed by the present method at M∞ =0.9 and k = 0.275 due to a pitch oscillation about the 50% root chord is shown in Fig. 7. By comparing the present method to the wind-tunnel measurements, it Fig. 14 Steady Cp on a LANN Wing at M∞=0.822 & can be seen that a good correlation is obtained α=0.6º except at y/2b = 97.7% where the present method underpredicts the unsteady shock strength. This is probably caused by the underprediction of the CFL3D steady shock strength y/2b = 97.7%. Also shown in Fig. 7 by the dashed line is the linear results computed by ZONA6 which fails to predict the unsteady shock effects, as expected.
LANN Wing in Pitch Mode About 62% Root Chord at M∞ = 0.822 and k = 0.105 Fig 15. Unsteady ∆Cp on a LANN Wing due to Pitch As discussed earlier, it is generally believed Oscillation about 62% Root Chord, M∞=0.822 & that the small disturbance theories cannot give k=0.105 accurate aerodynamic predictions on the The unsteady ∆Cp obtained by the present supercritical wings in the presence of strong method, wind-tunnel testing, and ZONA6 on shock. To show that this is not the case for the the LANN wing due to a pitch oscillation about unsteady flow predictions using the present 62% root chord at M∞ = 0.822 and k = 0.105 method, the unsteady pressure measurement on are presented in Fig. 15. Good agreement with the LANN wing12 is selected for validation. the wind-tunnel results at y/2b = 47.5% is The LANN wing is a supercritical wing with an obtained by the present method. At y/2b = aspect ration = 7.92 and a 25º swept angle 82.5% the present method underestimates the along ¼ chord whose field-panel model and the unsteady shock location; again, probably due to 7 Chen, Gao, and Tang
the underpredicted steady shock location by one-block modeling. Again, we select the F-5 CFL3D. It should be noted that for this LANN wing at M∞ = 0.9 and k = 0.275 as the test case wing case the ZONA6 results (dashed lines in where the F-5 wing is first separated into two Fig. 15) are totally unacceptable; indicating pieces of lifting surface (denoted as the “root that the linear aerodynamic theories have little section” shown in Fig. 17.a and the “tip applicability to supercritical wings in transonic section” shown in Fig. 17.b) along y/2b = Mach numbers. 62.9%. Next, two volume blocks both have the full span of the F-5 wing; i.e. two identical Validation of the Block-Tridiagonal volume blocks sharing the same three- Approximation Technique dimensional domain, are defined for the root The F-5 Wing at M∞ = 0.9 and k = 0.275 case is selected to validate the block-tridiagonal section and the tip section, respectively. Fig. approximation technique. The one-block field- 17.c depicts that, when joining the root section panel model shown in Fig. 5 is divided into six and the tip section together, an overset field- sub-blocks along the span. Since there are 12 panel model of the F-5 wing is generated. strips of surface boxes, each sub-block evenly According to the overset field-panel scheme, occupies two strips of the surface boxes; these two blocks do not influence each other rendering 25x12x2 volume cells in each sub- and the interference between them is through the influence from the surface boxes. Because the domain of the two volume blocks is identical, there is no approximation of the overset field-panel scheme and identical result between the two-block model and the one- block model is expected.
+ =
Fig. 16 Validation of the Block-Tridiagonal Approximation Technique on a F-5 Wing at M∞=0.9 & k=0.275 (a) Root Section (b) Tip Section (c) Overset Field- block. The result of the six-sub-block model is Panels Fig. 17 Overset Field-Panel Model of a F-5 Wing computed using the block-tridiagonal with Two Volume Blocks approximation technique and compared to that of the one-block model. As seen in Fig. 16, Indeed, as shown in Fig. 18, the result of the nearly identical results between those two sets two-block overset model of the F-5 wing is of results are obtained; showing the validity of nearly identical to that of the one-block model; the block-tridiagonal approximation technique. verifying the mathematical “correctness” of the Note that the six-sub-block model gives an overset field-panel scheme. approximately 50% reduction in CPU time over that of the one-block model. It is believed that more significant computing time can be saved by the block-tridiagonal approximation technique for more complex configurations.
Validation of the Overset Field-Panel Scheme The validation of the overset field-panel scheme can be performed by comparing the results between the overset modeling and the Fig. 18 Validation of the Overset Field-Panel Scheme on a F-5 Wing at M∞ = 0.9 and k = 0.275
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An Overset Field-Panel Method for Unsteady Transonic Aerodynamic Influence Coefficient Matrix
Validation of Flutter Boundary Predictions measurements is seen. Note that the CPU time Three test cases are selected to validate the of computing the AIC matrices of six reduced flutter boundary prediction of the present frequencies at one Mach number for the method with the wind-tunnel measurements. weakened wing is about 3.4 hours on a 2.4 Ghz Again, the steady background flows of all cases computer. Using the AIC matrices of the are computed using the CFL3D Navier-Stokes weakened wing, the flutter computation at four solver. Mach numbers of the solid only takes 71secs.
AGARD 445.6 Wing Flutter Boundary 13 Flutter Boundary of the PAPA Wing The AGARD 445.6 wing has two types of The PAPA (Pitch And Plunge Apparatus) wing stiffness; the weakened wing and the solid has a camber supercritical airfoil with wing; and both wings have the same maximum thickness of 12% and a rectangular aerodynamic geometry; giving an ideal case to Planform with chord of 16 inches and semi- demonstrate the AIC capability of the present span of 32 inches14. The wind-tunnel flutter method. Because the AIC matrix only depends test of the PAPA wing was performed in on the aerodynamic geometry and is NASA/Langley’s Transonic Dynamic Tunnel independent of the structural characteristics, the (TDT). The structural support of the PAPA AIC matrix of the weakened wing can be saved wing provides only two modes; a plunge mode and reused for the solid wing. Four AIC (3.43 Hz) and a pitch mode (5.44 Hz). Because matrices of the weakened wing at M∞ = 0.678, of the simple structural arrangement and the 0.9, 0.95, and 0.98 complex aerodynamics due to the supercritical are first generated, wing characteristics, the objective of the wind- whose matching tunnel test was to provide experimental data for flutter speed the validation of aeroelastic CFD codes. indexes (U ωα µ ) However, it turns out that only a few CFD bs and flutter results of the PAPA wing could be found in the frequencies ()ωωα open literatures. This might be caused by the low-aerodynamic-damping characteristics of are shown in Fig. the PAPA wing which requires a very long 19. It shows that computational time to determine the accurate the transonic flutter flutter boundary using the time-marching dip of a weakened Fig. 19 Flutter Boundary of procedure of the CFD codes15 for transonic wing is well the AGARD 445.6 flutter predictions. However, this low- predicted by the Weakened Wing aerodynamic-damping characteristics is not a present method. Mean-while, the linear results technical issue for the present method for computed by ZONA6 give a large discrepancy flutter predictions since it is formulated in the in the transonic region, as expected. frequency domain whose unsteady Fig. 20 presents aerodynamics can be directly adopted by a the flutter frequency-domain flutter solution procedure boundary of the such as the g-method16. solid wing predicted by the Fig. 21 presents a field-panel model and a present method CFL3D surface mesh of the PAPA wing. The but using the AIC steady background flows are provided by the matrices of the CFL3D Navier-Stokes solver at Mach numbers weakened wing. ranging from 0.5 to 0.85 and at α =1º and -2º. Again, good The CFL3D results show that at α =1º the correlation with transonic shock starts appearing on the upper Fig. 20 Flutter Boundary of surface at M∞ = 0.7 and moves to the wind-tunnel the AGARD 445.6 Solid Wing approximately 40% chord at M∞ = 0.8. At α = 9 Chen, Gao, and Tang
-2º, the transonic shock appears on the lower transonic counterpart of ZONA6/DLM because surface at M∞ = 0.7 and moves to 40% chord at the surface boxes of the field-panel model can M∞ = 0.8. These CFL3D results indicate the adopt those of ZONA6 or DLM. To generate strong transonic effects on the PAPA wing due the volume cells on lifting surfaces or bodies to its supercritical wing characteristic and requires only a few additional input parameters present challenge to the accuracy of the flutter such as the height and number of layers of the predictions. volume block. The overset field-panel scheme can minimize the volume cell generation efforts on complex configurations and automatically transmit the interference between overset blocks. In addition, a block-tridiagonal approximation technique is incorporated in the Fig. 21 Field-Panel Model and CFL3D Surface Mesh present method to greatly improve the of the PAPA Wing computational efficiency of solving the volume-cell influence coefficient matrix that is a complex, fully populated and normally very large size matrix.
The transonic AIC matrix generated by the present method has the same form as that of the linear panel methods and can be readily plugged into any existing AIC-based aeroelastic design processes for rapid flutter, aeroservoelastic, and dynamic loads analyses. Fig. 22 Flutter Boundaries of PAPA Wing, To generate an AIC matrix for a lifting surface α =1º(left) & -2º(right) configuration at one reduced frequency takes The flutter boundaries at α =1º and -2º obtained about a half hour of CPU time. But once by the present method, the wind-tunnel test, generated, it can be repeatedly used in a and the linear method (ZONA6) are presented structural design loop; rendering the present in Fig. 22. It can be seen that the transonic method as an ideal tool for the multi- flutter dips are well predicted by the present disciplinary optimizations. method; verifying that the present method can deal with the strong shock cases on the The present method also shows that simple supercritical wings provided the steady theories based on the small disturbance background flow is accurately predicted by the approach can yield accurate unsteady transonic high-fidelity CFD code such as CFL3D. flow predictions if accurate steady background Results also show that the flutter boundary of flow is given. This is demonstrated by the the PAPA wing in the transonic region is angle- good correlation with the experimental data of of attack dependent. Clearly, this angle-of- the present results using the high-fidelity attack effect on flutter boundary cannot be CFL3D Navier-Stokes solutions on the F-5 accounted for by the linear methods such as wing, the Lessing wing, the LANN wing, the AGARD 445.6 wing, and the PAPA wing. ZONA6. The CPU time of the present method for this case is approximately 3 hours per Mach Acknowlegements number in a 2.4 Ghz computer. The authors would like to thank Professor Shuquan Lu of Nanjing Aeronautical Institute for his Conclusions introduction of the transonic doublet lattice method. An Overset field-panel method has been developed for the transonic AIC matrix References generation and aeroelastic applications. The 1. Rodden, W.P., Giesing, J.P., and Kalam, T.P., present method can be considered as a “New Method for Nonplanar Configurations,”
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An Overset Field-Panel Method for Unsteady Transonic Aerodynamic Influence Coefficient Matrix
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