M. Zhang and A. Rizzi
Royal Insititute of Technology (KTH), Sweden R. Nangia
Nangia Aero Research Associates, Bristol BS8 1QU, UK
Keywords: wing design, airfoils, transonic, wave drag, Euler Abstract
Aerodynamic shape optimization on a 3D lami-nar wing at transonic speed is studied, with three redesigns come out. The inviscid drag is im-proved and its pressure distribution has been re-fined. The wing root bending moment is also reduced a bit. The desired pressure at one cross section is specified and one redesign is obtained by Direct Iterative Surface Curvature method.
1 Introduction
The design of aircraft involves multi-disciplinary approach. The optimization needs to include aerodynamics and structural properties. It helps therefore to be able to parametrize and reduce the order of optimization effort. Within the FP7 NOVEMOR project, a Regional airliner (based on Embraer reference) is being studied.
The ideas is to incorporate a fully parameterized wing including morphing LE / TE. A timely spe-cific goal is to compare with other aircraft (e.g.
Boeing Business Jet based on 737 series). An important aspect for transonic aircraft concerns wave drag estimation and its reduction.
1.1 Goals/challenges for efficient wing design
The authors will establish a computational de-sign framework for a fully parametric virtual air-craft of the reference design with morphing sur-faces. Our specific goal here is to explore the
conceptual design space of the reference model to see if equal or superior aerodynamic perfor-mance can be achieved with respect to a con-ventional regional airplane, e.g., the A-320. The reference design (RD0) is mounted with a pre-liminary designed wingtip device which is about 10% (or 1.5 m) of the main wing semi-span, by reducing the inviscid drag 5%. The reference air-craft is designed as a regional jetliner cruising at transonic speed, with the designed requirements raised by EMBRAER. The MTOW is 58034 kg, with fuselage length 36.86 m, and wing span 34.46 m. It cruises at 11 km, with Mach 0.78, andCL0.47.
However, the winglet on the reference wing has some unfavorable aerodynamics. Figure 1 shows that the pressure distributionCpfrom Eu-ler solution for the spanwise stations near the wingtip inboard and outboard. It can be seen that all of the Cp distributions have a “cross-over” behavior that we should try to avoid.
1.2 Design algorithms
The design is carried out in two-folded. The whole wing is re-designed using the traditional gradient-based optimization method with some reasonable constrains on pitching moment, wing loading and bending moment [7], with the invis-cid drag as objective function that to be min-imized. This is a straightforward method and we trust the numerically computed gradient is correct. To ensure that the gradient indicates the correct search direction which leads
conver-(a)
(b)
Figure 1: The pressure distributionCpfrom Eu-ler solution on the reference wing under cruise condition Mach = 0.78, CL = 0.47, four span-wise stations near the end of the main wing (η= 1)
gence, we should either use the numbering-fixed mesh and virtually deform the nodes of the mesh according to geometry changes, or, as in this pa-per, re-mesh the model every time when the ge-ometry changes to find the gradient, provided that the geometry change is big enough to ef-fect the flow. Figure 3 shows the flow chart of solving the gradient-based optimal problem by re-meshing the model every time. The new ge-ometry is found out by the optimizer in this de-sign loop, while in case that the optimizer finds the local optimum, engineering action will be taken into account based on the solution from the optimizer, to ensure the global optimum is found. For example in this paper, RD1 is from the optimizer while RD1-1 and RD1-2 are from engineering method.
Figure 2: Flow chart of the gradient-based opti-mization procedure
The other approach leads to optimization is the Direct Iterative Surface Curvature (DISC) method, based on the relationship between change in geometry surface curvature ∆κ and the desired change in local pressure distribution
∆Cp:
∆κ=k∆Cp (1)
where the gaink is modified depending on the local Mach-number and curvature. The relation is derived from the normal component of the mo-mentum equation along the streamline [10].
1.3 Flow solver
The flow solver used in this paper is a Fortran code developed by Swedish Defence Research Agency (FOI) [6]. The code is calledEdge, uses
Transonic Wing Design for a Regional Jetliner
Figure 3: The design loop of CDISC approach a n-stage Runge-Kutta time marching procedure to solve the 2D and 3D Euler and Navier-Stokes equations on an unstructured grid. It enables multi-grid method to accelerate convergence.
In this study 3D Euler equation is solved, the forces and moments are obtained by pressure in-tegration. The drag directly from calculation is the inviscid drag.
2 Approach to Design Lifting Surfaces Aircraft design mainly focuses on lifting surfaces design regarding to aerodynamic and structural issues. In this manner the objective turns to parametrize the lifting surfaces by a set of ro-bust parameters which are suitable for optimiza-tion. There are several different parametriza-tion schemes for lifting surfaces. In spite of the 3D planform parameters (wing area Sref, wing span b, taper ratio λ, aspect ratio AR, sweep angle Λ, dihedral angle δ, etc.), the 2D airfoil shape for each wing section is the crucial issue in the process. It relates to the lift, drag, span load distribution, bending moment etc., which are important for aerodynamic and preliminary structural analysis.
The SUface MOdeler 1, sumo, developed by David Eller in KTH, is a tool aimed at rapid creation of aircraft geometries and automatic surface mesh generation. It stores the cross-sectional information (points) as skeletons for the components e.g., wings, fuselages, nacelles, and pylons. sumo together with the TetGen
1http://www.larosterna.com/sumo.html
[16] automatic tetrahedral mesher can provide a high-quality unstructured volume mesh for Eu-ler computation in few seconds. . The surface of the wing can be parametrized withsumo by patches of Bezier surfaces on airfoil stacks which are consisted of points. It allows rapid cre-ation of aircraft geometries and automatic sur-face mesh generation. Figure 4 shows the mech-anism thatsumorepresentation for wing surface creation and how the control points (square) are applied to morph the airfoil leading and trailing edge (LE and TE). Details of how to parametrize the wing including the winglet device are shown in following sections.
Figure 4: Wing surface creation withsumo: air-foil stacks
2.1 Airfoil parametrization
A more powerful parametrization for airfoils is the Class-Shape-Transformation (CST) method pioneered by Kulfan [4] and Sergio [5]. With a small number of design variables it enables a large variety of airfoils to be simulated.
The airfoil is modeled by 4th-order Bernstein polynomial (BPO) CST method, representing the airfoil by a symmetric component (thick-ness) and an asymmetric component (camber) respectively.
The thickness distribution(t) zt is to define the shape function St with zero camber, while the camber shape(c) zc is to define the shape functionScwith zero thickness.
St(x) = X4 i=0
Ati·Si(x) (2) zt=C(x)·St(x) (3)
Sc(x) = X4 i=0
Aci·Si(x) (4) zc=C(x)·Sc(x) (5) Where theSi(x) is component of the 4th-order Bernstein polynomials Si(x) = i!(nn!−i)!xi(1− x)4−i, Atiand Aci are the coefficients added to the shape function that define the thickness and camber shape respectively,i= 0,1,2,3,4. C(x) is the class function, thatC(x) =√
x(1−x) for a shape with round nose and point aft-body. Note that the first term ofAt(orAt0) defines the LE radius (At0=√
2RLE) [4], that we always want to fix or predefine. Similarly, the first term ofAc
(or Ac0) is always 0 (zero LE radius) since the camber line has zero thickness. Both the num-ber ofAt andAccoefficients are reduced to 4 if we want to fix the LE radius of the airfoil. Thus the number of design variables for an untwisted airfoil is 4 + 4 = 8, plus 1 for local twist, we have 9 design variables for one undeformedsection.
The CST method is extended by PoliMI to morphing airfoils[5] . It applies to compliant leading and trailing edges. The Shape and Class Functions of CST are augmented by additional terms.
The approach used in this paper is similar, consistent with the two-stage thickness-camber design stated above. The deformed/morphing leading edge (LE) and trailing edge (TE) are de-fined by deforming the camber line, with thick-ness distribution remaining. Note that the cen-tral region of the wing (wing structural box) is undeformed. The continuity of the camber slope needs to be maintained, when deforming the LE/TE. The nose and tail displacements are notated byZLE and ZT E. The quadratic Bezier curves are used to determine the de-formed LE/TE camber line, see Figure??. Two more design variablesZLEandZT Eare added to eachsectionif we include the morphing strat-egy.
(a)
(b)
Figure 5: Wing surface creation withsumo: air-foil stacks and control points for morphing LE and TE
Transonic Wing Design for a Regional Jetliner
2.2 Geometry representation
The original reference design (RD0) is provided in IGES format. It is converted intosumo[2] ge-ometry by importing the sectional information (points) from IGES, in order to proceed the au-tomated mesh generation within the optimiza-tion process. The deviaoptimiza-tion from both models is really small, the main difference is that the wing from original IGES model has trailing edge gap ∆T Ewith maximum value less than 1%·c, whichcis the local chord along the span. sumo only considers the shape trailing edge, and it is closed by applying minimum change to the air-foil i.e., taking a wedge out of the whole chord by subtracting the∆T E2 from both upper and lower surfaces at the trailing edge that preserving the shape of the airfoil mostly (Figure 6(a)).
The airfoil is parameterized by CST using 4th-order Bernstein polynomials (4th-BPO), which has sufficient accuracy to represent the RD0 ge-ometry in sumo. Figure 6(b) shows the the maximum errors of the cross sections those parametrized by 4th-BPO CST method stated above, which splits the thickness and cambers into two design stages. We can see that the over-all errors are quite smover-all and less than 0.2%, it is safe to use this parametrization method to represent RD0 geometry.
The total number of variables N0 that param-eterizing the whole wing withnsections will be
N0 = 11·n (6)
In this paper, aerodynamic design and struc-tural analysis are done sequentially. We only consider one point design, i.e., the cruise con-dition with LE and TE undeformed. Also the wing box capacity is preserved so the thickness for each cross section is fixed. Only cambers, twists and winglet cant angle are designed for optimized aerodynamics. So the number of vari-ables N1 that parameterizing the whole wing withnsections in the aerodynamic design stage stated here is:
N1 = 5·n (7)
(a) Maximum error for RD0 geometry representation by sumoimported from CAD model
(b) Maximum error for RD0 geometry parametrization using 4th-BPO CST method insumo
Figure 6: Errors in geometry representation along the span
3 Results and Discussion
3.1 Varying the twist and cambers There are two designs RD1 and RD2 are rep-resented in this paper, both of which show sig-nificant drag reduction. The inviscid drag is re-duced from 14.9 counts (RD0) to around 10.5 counts (RD1, RD1-1 and RD1-2), around 30%
reduction in numbers, while the pitching mo-ment stays almost the same, adding a little nose-down pitch. Figure 7 shows the Euler solutions for a 2.7 M nodes mesh for RD0 and its three re-design configurations at transonic speed.
For lowerCL, the RD1 design has lower drag, however with the increasing CL, the advantage in drag eliminates, while the 1 and RD1-2 designs still hold a lower drag at higher CL. This can be explained by the spanload shape in Figure??, RD1-1 and RD1-2 are more towards a triangular loading (as RD0), which has a ben-efit in lower wave drag when shock is strong, while RD1 has more elliptical loading at the outer board, that has a lower induced drag at lowerCLwhen wave drag is not dominated.
Figure 7: Euler solutions for RD0 and its three redesigns at Mach 0.78
Figure 8 shows the pressure distribution from Euler solution under cruise condition for the ref-erence design RD0 and its three redesign cases by varying the twists can cambers on upper and lower surfaces. Figure 11 and 12 represents the pressure distribution and geometry comparisons for reference design (RD0) and its redesigns RD1
and RD1-1, RD1-2 in transonic inviscid flow.
The lambda shock at the root is much weak-ened in RD1 (aroundη= 0.2). At around kink η= 0.4 the mild double shock is replaced with a favorable pressure gradient up to 0.6 airfoil chord followed with a slightly stronger shock.
RD1 design remedies the crossed pressure at the span-wise location after the wingtip (Figure 12 (b) and (c)), but a slightly shock appears at around η = 1, if one sees Figure 3.1 and Fig-ure 12 (a) that the crossed pressFig-ure still exists while unfortunately a shock is raised. That is one of the sources where wave drag comes from.
To avoid the appearance of the shock around η = 1, the local airfoil is twisted up to reduce the local lift (Figure 12 (a), (d) and Figure 3.1), that is the way we get RD1-1. The adverse pres-sure gradient aroundη= 1 in RD1 has been re-fined in RD1-1, however, the ”cross-over” pres-sure still exists, which isinefficent and the pro-duced lift is much less than it is supposed to be.
From the two different designs RD1 and RD1-1 we see that the tip region is a problem, it either produces shock on the upper surface or on the lower side, where adds wave drag. The most efficient way is to not only vary the cambers and twists - making a slightly thiner local airfoil will probably cure the problem. This won’t effect the fuel storage much since the location is around the wingtip and most of the fuel stored in the inboard wing. By reducing the thickness at wing tip section we get RD1-2.
RD1, RD1-1 and RD1-2 are better than the initial baseline RD0, which have more straight and fully-swept isobars along the wing. If we ex-amine carefully their pressure isobars, we can see that on the the upper surface RD1-1 has quite nice looking straight isobars with reduced shock (Figure 3.1), while a tip shock on the lower sur-face is worsen, where in comparison it eliminates in RD1-2 design (Figure 8(b)).
The wing AR is around 10, and its bending moment shows that the root has the highest stress for the planar wing as we know from the-ory. The RBMs (at around 10% semi-span) for the design RD1, RD1-1 and RD1-2 are higher than the initial design RD0, that the wing needs to be designed to hold more loads. This
situa-Transonic Wing Design for a Regional Jetliner
(a)
(b)
Figure 8: Pressure distribution from Euler solutions for RD0 and its three redesigns at Mach 0.78 and CL 0.47 on a) upper, and b) lower surfaces. Marked span-wise stations are (from inboard)η = 0.2, η= 0.4,η= 0.6,η= 0.95, η= 1.05 andη= 1.07
(a)
(b)
(c)
Figure 9: Pressure distribution in sections η=0.2,0.4,0.6, for RD0 and its three re-designs
(a)
(b)
(c)
Figure 10: Airfoil geometry in sections η=0.2,0.4,0.6, for RD0 and its three re-designs
Transonic Wing Design for a Regional Jetliner
(a)
(b)
(c)
Figure 11: Pressure distribution in sections η =0.95,1.05,1.07, for RD0 and its three re-designs
(a)
(b)
(c)
Figure 12: Airfoil geometry in sections η =0.95,1.05,1.07, for RD0 and its three re-designs
tion might be improved by carefully designing the folded angles on the wingtip.
3.2 On-going work for designing the folded angles
The initial design RD0 has a slightly folded an-gle 7 degrees. With carefully designing of the tip folded angles the lift force at the tip can be varied so that the bending moment w.r.t.
the wing root is changed significantly. As the folded angle increasing it shows more triangu-lar wing loadings that elliptical loadings pro-gressively, i.e., the wing tip function decreases but wing root function increases, see Figure 14.
Figure 15 shows the wing sections of those two wings with different folded angles, the sections are shown with wingtip shown as unfolded.
3.3 Preliminary results on target pres-sure
If we specify a target pressure on the wing, for example, the desired fully-swept isobar on the wing which gives minimum drag, then we can use the DISC method to design the wing by solv-ing a inverse design problem. The pressure on the wing is presented by a number of sections, then the problem comes to design a series of air-foil stacks (as mentioned in 4). It is more com-plicated for a 3D wing design problem than the 2D airfoil design. Since the reference wing RD0 is tapered and swept (also twisted), we will de-sign it in two steps:
1. consider the wing without sweep, the lift distribution and span load shape are con-strains when design the airfoil stacks on the wing, the aspect ratioARand taper ratio λwould effect them;
2. the cross section thickness t/c of a swept wing with sweeping angle Λ is equivalent to an unswept wing with cross section thick-ness (t/c)·cos(Λ);
Figure 16 shows the preliminary results us-ing DICS approach to redesign the wus-ing section η= 0.4, with its local lift coefficient preserved.
The work is on-going. The flow inboard would influence the flow outboard.
(a)
(b)
(c)
Figure 13: Euler solutions for RD0 and its three re-designs compared with the planar wing (only thickness) at Mach 0.78, CL = 0.47, a), local lift coefficientCLL, b) span load distribution, c) wing root bending moment at around 10% wing
Transonic Wing Design for a Regional Jetliner
(a)
(b)
Figure 14: Span loads coefficient and local CL from Nangia’s panel code solution [15] on the RD0 geometry, atα= -1, 0, 1 deg , with folded angles (a) 60 degrees; (b) 63 degrees
(a)
(b)
Figure 15: Designed wing sections with wingtip shown as unfolded according to Nangia’s panel code solution [15] on the RD0 geometry with dif-ferent folded angles: (a) folded angle 60 degrees;
(b) folded angle 63 degrees
(a)
(b)
Figure 16: Pressure and geometry in sectionη= 0.4, for RD0 and its re-design RD2 using DISC approach
4 Future Work
This is a on-going work. The DISC approach will be carried out carefully to re-design from inboard to outboard.
5 Bibliography References
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