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An Overset Field-Panel Method for Unsteady Transonic Aeroelasticity

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An Overset Field-Panel Method for Unsteady Transonic Aeroelasticity 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 1 Chen, Gao, and Tang 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 2 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).
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