Supersonic Wing Design Method Using an Inverse Problem for Practical Application

Transcription

1 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 29, Orlando, Florida AIAA Supersonic Wing Design Method Using an Inverse Problem for Practical Application Shoji Sakashita 1, Takumi Matsuzawa 2, Kisa Matsushima 3, Kazuhiro Nakahashi 4 Department of Aerospace Engineering, Tohoku University Sendai, JAPAN Supersonic wing design system using an inverse problem realizes high-fidelity geometric control with much smaller number of flow simulations than generic design system. It was used in Japanese SST project NEXST-1 successfully to design the NLF wing. However, there is still need for improvement to apply this method to critical or severe design problems. One of the required improvements is reduction of load in CAD process and another is accuracy in determining geometric correction values. In this research, aiming to reduce the load in CAD process, geometric representation by analytical functions was proposed on the basis of the PARSEC representation. On the other hand, to enhance the accuracy in determining geometric correction values, the basis equation used in inverse problem solver was modified. The redesign of NLF wing of NEXST-1 was conducted to validate the modified design system. Then it is confirmed that the modified one has more ability than original one. Nomenclature x, y, z = cartesian coordinate system of a flowfield x; stream-wise, y; span-wise, z; thickness-wise direction ξ, ψ, ζ = coordinate system for Green s integration M = free stream Mach number γ = specific heat ratio φ = small perturbation velocity potential Δφ = variation of φ f = wing geometry f + = upper wing surface geometry f - = lower wing surface geometry Δf = correction of f = pressure coefficient Δ = difference between target and current I. Introduction Automatic aerodynamic design of aircraft has been one of the major applications of Computational Fluid Dynamics (CFD). There are two traditional computational design techniques. One is the inverse problem technique. The other is numerical optimization which is formed by numerous flow simulations with any of numerical algorithm of searching for the optimum shape. Each method has advantages and drawbacks when compared with the other. From the viewpoint of engineering application, a technique providing high fidelity geometric control and a faster design cycle gives a crucial advantage. In this sense, the inverse problem technique is desirable because it can use much more geometrical control points and it requires a much smaller number of flow simulations than the numerical optimization does. The inverse problem technique recognizes that the designer usually has an idea of the kind of pressure distribution that will lead to the desired performance. A considerable number of inverse design methods have been devised, so far [1, 2]. Among them, the method using integral equations is regarded as one of the most versatile and efficient design methods. In fact, the design 1 Graduate Student, Student member, sakashita@ad.mech.tohoku.ac.jp 2 Graduate Student, (graduated from the Tohoku University in March, 28 and at present works for Honda Ltd.) 3 Associated Professor, Senior Member AIAA, kisam@ad.mech.tohoku.ac.jp 4 Professor, Associate Fellow AIAA 1 of 11 Copyright 29 by the, Inc. All rights reserved.

2 method was used in Japanese SST project NEXST-1 successfully to design the Natural Laminar Flow (NLF) wing. It was originally developed for a wing in transonic flows in 1985 by Dr. Takanashi [3]. As for the integral equation method, later, one of the authors extended his idea to supersonic flow to construct a supersonic inverse problem design system. She has also conducted the NLF wing design [4, 5]. Recently, we have reviewed the design system and recognized that there are still needs for improvement to apply this method to critical or severe design problems. One of the required improvements is reduction of load in CAD process and another is accuracy in determining geometric correction values. In CAD process of this design system, by manual, we need to partly smooth the designed wing surface directly obtained by the inverse solver because it has sometimes C 2 discontinuity and forms an irregular surface. These manual operations take a lot of time and human resource, and which prevent the design system being efficient. In this research, aiming to reduce (or eliminate) the load in CAD process, geometric representation by analytical functions is proposed on the basis of the PARSEC representation. On the other hand, original inverse solver has not enough ability to precisely determine the geometric correction values at leading edge of the wing. The inverse solver is derived from the linear small perturbation theory and thin wing theory, which assume that the second order terms of small perturbation are negligible. However, these assumptions are locally inapplicable for a wing leading edge which is well known as a singular point for the thin wing theory. In the design of NLF wing of NEXST-1, the insufficiency of the accuracy of geometric correction at leading edge caused undesirable peaks of distributions, and which sometimes prevented the convergence. In order to improve the accuracy of geometric correction, in this research, the basis equation used in inverse solver is modified. To validate the capability of modified design system, we conducted the redesign of NLF wing of NEXST-1. II. Design system & Flow solver The design system is iterative. The goal is to determine the wing section geometry which realizes a specified target pressure distribution. Figure 1 shows flowchart of design system using an inverse problem solver and CFD simulation. The detail of design procedure is as follows: (1) Setting the baseline model and target distributions. (2) Grid generation [6, 7] of baseline model. (3) Flow simulation around the baseline model, and get the result distributions. (4) Calculating Δ ( Result ). (5) Getting the geometric correction (Δf) by solving inverse problem using Δ. (6) Updating the geometry by adding geometric correction (Δf) to former geometry f. (7) Smoothing the updated geometry by CAD (8) Combine the updated wing with fuselage. (9) Grid generation of updated model. (1) Flow simulation around the updated model, and getting the result distributions. (11) Iterating (4)-(1), and if the difference between target and current pressure distribution is negligible, the design is complete. Start Baseline model Flow analysis Result Grid Generation CAD Process Update the geometry F = F+ΔF Δ = If No Solve Inverse Problem Target-Result Δ <ε using Δ YES Finish Figure1. Flowchart of design system using an inverse problem This procedure has two primary parts. One is flow analysis. The other is geometric correction by an inverse 2 of 11

3 design part. Both parts are completely independent from each other. The analysis part evaluates the residual. The design part provides the correction values for asymptotic approach to a solution. Thus, the method may be called a residual-correction method. In this research, flowfield was analyzed using the TAS (Tohoku University Aerodynamic Simulation) code for unstructured mesh. Compressible Euler equations were solved by a finite-volume cell-vertex scheme. The numerical flux normal to the control volume boundary was computed using an approximate Riemann solver of Harten-Lax-van-Leer-Einfelds-Wada (HLLEW) [8]. The second-order spatial accuracy was realized by a linear reconstruction of the primitive gas dynamic variables inside the control volume with Venkatakrishnan s limiter [9]. The LU-SGS implicit method for unstructured meshes [1] was used for the time integration. Inverse problem design part is explained in next section. III. Inverse Problem for Supersonic Flows A. Original three dimensional inverse problem solver [11] The formulation starts with the following equations. The flowfield is described in terms of the small disturbance potential, 1 (1) On the wing surface, the flow ought to be tangential to the surface as,,, (2) And pressure is related to the perturbation velocity;, 2,, (3) Where we assume the free stream flow velocity vector is (1,, ). Applying a Prandtl-Glauert transformation for supersonic flow such as,,,, 1, 1 Taking the perturbation of Eq. (1)-(3), and neglect the second and higher order term of the perturbation values, the equations are expressed as (4),,,, (5), 2,, (6) Applying Green s theorem for a hyperbolic equation, and performing some calculus, one obtains the equations to determine. Thickness change (correction) as,,, / (7) where,,,,,,,,, (8),,,,,,,,, (9) Camber change (correction) as,,, (1) where,,,,,,,,, (11),,,,,,,,, (12) The geometrical correction is calculated using Δw s and Δw a as,.,, (13) B. Modification of original inverse solver The original inverse solver is derived from the linear small perturbation theory and thin wing theory, which assume that the second order terms of small perturbation are negligible. However, these assumptions are locally inapplicable for a wing leading edge. In order to enhance the accuracy in determining geometric correction values 3 of 11

4 at leading edge, we modify the approximation form of tangency condition. When a flow can be regarded as inviscid, a flow in the vicinity of the wing surface streams tangentially to the surface shape; that is, the velocity vector on the surface must be perpendicular to the normal of the wing surface plane. The equation of the wing surface plane is z = f (x, y). The normal vector of the plane is,,,,1 (14) The velocity vector of the perturbed flowfield is 1,, (15) Then, the inner product of both becomes zero [11] ; 1 (16) By assuming,,, 1, the basis equation of original inverse problem is obtained (Eq.(2)). At leading edge, however, the value of cannot be negligible. In the modified inverse solver, as tangency condition, we use following approximation form. 1 (17) Eventually, the geometric correction is calculated as,,. 2,,,,,,,, 1 1 2,, (18) The modified inverse solver substitute Eq.(18) for Eq.(13). The subscripts + and. indicate the upper and lower surfaces. From the Eq.(18), the geometric correction values for the upper and lower surfaces are independent each other. Thus the trailing edge might not be closed. To guarantee that every section has a closed trailing edge (T.E), the solution Δf - is modified to satisfy the condition,.,., (19) while the correction values of upper surface is fixed. IV. Parameterization of wing geometry With the inverse design method, a lot of control points can be used to define the design shape at low cost, and therefore high-fidelity shape representation can be realized. However, it makes CAD process to take a lot of time and effort because wing geometry represented by a lot of control points often forms a wavy surface. Then in this research, to eliminate the manual operation and to automate the CAD process, we introduce geometric representation by analytical functions. The parameterization of wing geometry is divided into two parts. First, each sectional airfoil is represented in the polynomials by the modified PARSEC method [14, 15]. Then each coefficient of the polynomials is approximated by the Bezier curves in the span-wise direction in order to smooth the wing surface. Through these two parts, three-dimensional wing geometry is represented by analytic function. A. Modified PARSEC method (airfoil parameterization) The PARSEC method [12, 13], which was originally proposed by Sobieczky, uses 11 parameters to represent an airfoil. The advantage of this method is that geometric features of airfoil are extracted as control point, so this method might be more suitable parameterization for airfoil than other methods. By using this method in the inverse design system, airfoil shape is always C 2 continuous and smooth while maintaining the aerodynamic feature. However, the airfoils designed by inverse solver are more complicated and aerodynamically advanced than the conventional airfoils, so original PARSEC method needs some modifications. In the modified method, as shown in Fig.2, airfoil is divided into the thickness distribution and the camber curve. Then each is respectively represented by the analytical functions. The camber curve is represented by the following polynomial. (2) The coefficients a n are obtained by the least square method, while the thickness distribution is represented as follow; (21) To define the coefficients b n, 6 parameters are used. They are the leading edge radius (R LE ), crest position (X crest, crest ), and its curvature ( xxcrest ), trailing edge location (X TE, TE ), trailing edge angle (β TE ), and location of adding point between leading edge and crest (X add, add ). They are shown in Fig.3. The adding point, which is not used in original method, compensates for the lacks of geometric information between leading edge and crest point. The location of the adding point varies with the geometry of represented airfoil, and which is defined to [14, 15] 4 of 11

5 minimize the gaps between approximated curve and original shape. Eventually, the airfoil is represented by totally 14 parameters, adding the approximated camber curve (7 coefficients) and thickness distribution (7 coefficients). This modified representation technique provides better agreement with original shape than the original PARSEC method does. This technique is sufficient for the relatively simple airfoils like NACA series, however, insufficient for the airfoils such as the result of the optimization which often have complicated geometry. Then, we bridge the gaps between approximated geometry and original one by adding another polynomial function of these gaps. It is expressed as following tenth-order polynomials obtained by the least square method. (22) In this research, we use the linear combination of above three polynomials, Eq.(2) (22), to represent an airfoil. Eventually, we use 24 coefficients for representing one section airfoil. B. Smoothing in the span-wise direction By the airfoil parameterization using modified PARSEC method, the geometry of each airfoil retains continuity, however, in the span-wise direction it is lacking yet. Then we try to smooth the coefficients a n, b n obtained by the airfoil parameterization to ensure the continuity in the span-wise direction. Distributions of them in the span-wise direction are approximated by the Bezier curves. Given control points P, P 1,..., P n, Bezier curve is expressed as follows;, 1 (23) where!!! 1 (24) are known as Bernstein basis polynomials of degree n Original airfoil Camber Thickness Figure 2. Modified PARSEC method Figure 3. PARSEC parameters V. Redesign of NLF wing for NEXST-1 The wing of JAXA s experimental scaled SST NEXST-1 was aerodynamically designed by the original inverse system at 2.. To design high L/D wing, a target pressure is prescribed that its elliptical load distribution minimizes the induced drag and its upper surface distribution can keep the laminar boundary layer as long as possible. To validate the modified inverse design system, we conducted the redesign of NLF wing for NEXST-1. At first, the capability of the design system using wing parameterization was verified. And then, examination was conducted for the modified inverse solver which accuracy in determining geometric correction values was enhanced by handling tangency condition. The wing planform shape, initial geometry, and target distributions are same as those used in the practical design of NEXST-1. Design conditions are also same; 2., the angle of attack α =2. degrees. In order to take the wing-fuselage interaction into consideration, the flow analysis was conducted using wing-fuselage geometry, and the number of grid points is approximately 3 million in 3D space. In the inverse design part, geometric correction were done for a wing alone, and 86 (15% to 1% span stations) 299 (at each span station) control points on the wing surface were used. Throughout the all design process, the location of leading edge points were fixed, and trailing edge points were free. 5 of 11

6 A. Validation of capability of modified design system using wing parameterization In general, an arbitrarily specified distribution does not always correspond to a physically acceptable solution. Sometimes there might be no exact solution. Thus, the desired role of the inverse design method is to find the solution whose distribution is closest to the specified target one. In the present design case, existence of exact solution is guaranteed by the previous NEXST-1 design in In the previous design, however, some particular operation was manually conducted by an expert to obtain exact solution, which needed a lot of time and effort. On the other hand, in present design, all the operations except grid generation are conducted automatically. Thus the purpose of this validation is to obtain the solution converging to the target to a certain extent by the nearly automated design method. distribution and section geometry of the designed wing after 8th iterations at 15, 3, 5, and 7% span stations are shown in Fig.4. At each span stations distribution of designed wing are almost converged. It suggests that the modified design system has capability to design practical wing without manual CAD process. contour visualization of the upper and lower surface of the designed wing is shown in Fig.5. On the upper surface, a flat roof type of distribution along the chord is realized especially at the inner wing. Figure 6 shows section geometry of designed wing and its approximated curve after representation by modified PARSEC method and after span-wise smoothing. It shows that the geometry represented by modified PARSEC method is in good agreement with original designed geometry. On the other hand, there are some discrepancies between original geometry and geometry obtained after span-wise smoothing. However, it s inevitable because designed wing before smoothing has highly irregular surface, as shown in Fig.9. Fortunately, from the fact that the design system works out well, it can be seen that the aerodynamic performance of original wing is maintained after span-wise smoothing. Figures 7 and 8 show the first and second-order derivative of section geometry at 3 and 7% span stations. Each derivative of designed wing, especially second-order, lacks smoothness. After wing parameterization, however, the geometry obtains the continuity in first and second-order. From the result above, it can be seen that the inverse design system using wing parameterization is more efficient and useful than original one Initial Designed Designed geometry Initial geometry Initial Designed Designed geometry Initial geometry (a) 15% span station Initial -.3 Designed Initial geometry.15 Designed geometry (c) 5% span station (b) 3% span station Initial Designed Initial geometry Designed geometry (d) 7% span station Figure 4. distribution and section geometry 6 of 11

7 Figure 5. visualization of designedd wing with contour line (a) 3% span station (b) 7% span station Figure 6. Section geometry after parameterization Designed Airfoil Modified PARSEC after Span Smoothing (BEIER) Designed Airfoil Modified PARSEC -.15 after Span Smoothing (BEIER) Designed Airfoil Modifiedd PARSEC after Span Smoothing (BEIER).15.1 Designed Airfoil Modifiedd PARSEC after Span Smoothing (BEIER) dz/dx.5 dz/dx (a) 3% span station (b) 7% span station Figure 7. First-order derivativee of section geometry 1 7 of 11

8 d2z/dx2 -.5 d2z/dx Designed Airfoil Modified PARSEC after Span Smoothing (BEIER) Designed Airfoil Modified PARSEC after Span Smoothing (BEIER) (a) 3% span station (b) 7% span station Figure 8. Second-order derivative of section geometry.5.5. Designed wing after Span-wise Smoothing. Designed wing after Span-wise Smoothing % span (a) 3% chord location (b) 7% chord location Figure 9. Upper surface coordinates at fixed chord locations in the span-wise direction B. Validation of effect of enhanced inverse solver We proposed modified approximation form of tangency condition to enhance the accuracy in determining the geometric correction values at leading edge. Here, we verify the capability of modified one to design the wing realizing the sudden expansion of the upper surface distribution at leading edge. distribution and section geometry of designed wing are shown in Figs Blue and red lines show the result by original and modified inverse solver respectively. At each span station, distributions on the upper surfaces obtained by modified inverse solver get slightly closer to target than those obtained by original one do. It looks like to depend on the difference in twist angle of section geometry. At the inner wing (15% to 4% span station), the designed wing by modified solver realize the sudden expansion of the upper surface distribution at leading edge, and undesirable suction peaks are suppressed. In contrast, especially at 5% span station, around which there is leading edge kink, undesirable suction peak cannot be suppressed. Thus, the modified inverse solver works out better at the inner wing, however, still needs to be improved in order to suppress the suction peak at the outer wing. % span Designed by original solver Designed by modified solver Figure % span station 8 of 11

9 Designed by original solver Designed by modified solver Figure % span station Designed by original solver Designed by modified solver Figure % span station Designed by original solver Designed by modified solver Figure % span station Designed by original solver Designed by modified solver Figure % span station 9 of 11

10 Designed by original solver Designed by modified solver Figure % span station Designed by original solver Designed by modified solver Figure % span station Designed by original solver Designed by modified solver Figure % span station VI. Conclusion The modifications of supersonic wing design system using the inverse problem were conducted. First, in order to reduce the load in CAD process, geometric representation by analytical functions was proposed on the basis of one of the PARSEC representation. Second, aiming to enhance the accuracy in determining geometric correction values, the basic equations of inverse problem solver was improved. Then the redesign of NLF wing of Japanese SST project NEXST-1 was conducted for the validation. By wing parameterization, the manual load in CAD process is eliminated, and the design system became more efficient and useful. On the other hand, by the modification of the basic equation of tangency condition, designed wing realized distribution closer to the target one. However, undesirable suction peak at leading edge could not completely be suppressed. For the future work, we will conduct the laminar to turbulent flow transition analysis to examine if the present designed wing has wide laminar flow region. 1 of 11

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