PowerPoint Presentation*
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Aerodynamic theories
Supersonic Reference
Journal of Aircraft, Vol. 28, No. 9, September 1991, pp. 598-605
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Flutter solutions can use all theories
Static and aerodynamic gust solutions are restricted to the subsonic Doublet-Lattice method with body interference and the supersonic ZONA51 option
Reduced frequency is a key parameter
Where
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Doublet Lattice Method
DLM can be used for interfering lifting surfaces in subsonic flow
All lifting surfaces are assumed to lie nearly parallel to the flow because small-disturbance, linear aerodynamic theory is used
Each interfering surface is divided into small trapezoidal lifting elements ("boxes")
The boxes are arranged to form strips that are parallel to the free-stream
Fold lines and hinge lines must lie on the box boundaries
Symmetry options are available to enable reduced problem size
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Double Lattice Method
Unknown pressure on the box represented by a line of doublets at the box quarter chord
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The linearized condition of tangential flow may be expressed as:
Where
is the deflection mode of the surface measured normal to the
surface. By prescribing
and equating this equation with the integral
equation, an expression for the unknown pressure amplitude can be established.
The coordinate system is
Then
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Aerodynamic theories
Modeling Guidelines
Aerodynamic grid point identification starts with element identification given on the CAERO1 entry and is incremented by unity for each box generated by the entry
Aerodynamic grid point identification numbers are treated independently from structural grid, scalar and extra point ID’s so that duplicate and/or overlapping ID’s are permitted across the structural and aerodynamic models.
Aerodynamic grid point identification numbers cannot overlap
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Aerodynamic theories
Modeling Guidelines
Boxes are identified with k-set degrees of freedom with two DOFs per box
One panel represents a flat plate with a trapezoidal planform. The inboard and outboard edges are parallel to the streamwise direction
The trapezoidal lifting elements (i.e., the "boxes") should maintain an aspect ratio of less than 3 in N5KA; the aspect ratio in N5KQ should be less than 6. It is possible, depending on the configuration, that higher aspect ratios can be used in both N5KA and N5KQ. Convergence studies are recommended when higher values are selected.
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Boxes should be concentrated near downwash discontinuities, e.g., leading edge, trailing edge, and hinge lines
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Re...
n
S
K
w x, y, z( ) = ddx h x, y, z( )+ i kl h x, y, z( )
wx,y,z
Fig. 2 Idealization of lifting surface using trapezoidal boxes.
Fig. 1 Lifting surface coordinate system.
Introduction
T HE doublet-lattice method (DLM)1 is a !nite element method for the solution of the oscillatory subsonic pres-
sure– normalwash integral equation for multiple interfering surfaces
N 1
w(x, s) = K(x, j; s, s )p(j , s ) dj ds (1)O EE8p n=1 Sn
where (x, s) are the orthogonal coordinates on the nth surface Sn such that the undisturbed stream is parallel to the x axis, and K is the complex acceleration potential kernel for oscil- latory subsonic "ow. The original DLM algorithm was pre- sented at the same time as the lifting line element method (LLEM) of Landahl and Stark.2 Although numerous compari- sons2,3 with experiments were shown at the time, the complete details of the LLEM were never published. Stark has written a note on the LLEM, and this was included in Ref. 4 as Ap- pendix A.
A re!nement to the expressions for the kernel given by Ro- demich in Ref. 5, and Landahl6 was presented by Rodden et al.7 in the form
2 4K = (K T /r 1 K T * /r )exp(2ivx /U ) (2)1 1 2 2
to analyze nonplanar interference correctly. K1 and K2 are the planar and nonplanar parts of the kernel numerator, respec- tively,
T = cos(g 2 g ) (3)1 r s
T * = (z cos g 2 y sin g )(z cos g 2 y sin g ) (4)2 0 r 0 r 0 s 0 s
2 2 2r = z 1 y (5)0 0
The coordinate system is illustrated in Fig. 1. The description of K1 and K2 as the planar and nonplanar parts of the kernel numerator is a convenience because both are obviously non- planar in general. The re!nement in Eq. (2) is in the second term; this was found to be necessary so that the DLM could predict the interference between a nearly planar wing and hor- izontal tail.7 The re!nement retained the original primary ap- proximation,1 i.e., that the incremental oscillatory normalwash factors are obtained by integrating the difference between the oscillatory and steady kernels over the length of the bound vortex assuming a quadratic (parabolic) variation in the nu- merator of the difference. The total normalwash factor is then the sum of the incremental oscillatory normalwash factor and
the steady normalwash factor obtained from the expressions for a horseshoe vortex, e.g., the vortex-lattice method (VLM) of Hedman.8 In this way, the DLM converges to the VLM at zero reduced frequency, and the error in the parabolic approx- imation of the kernel numerator difference is small at low re- duced frequencies but increases with reduced frequency.
Extensive experience with the VLM and DLM has led to guidelines for the idealization of lifting surfaces into !nite el- ement models. It is assumed that each surface can be approx- imated by segments of planes. The surface is divided into small trapezoidal panels (boxes) in a manner such that the boxes are arranged in strips parallel to the freestream (Fig. 2) and surface edges and fold lines lie on box boundaries. Boxes should be concentrated near wing edges and hinge lines or any other place where downwash is discontinuous and pressures have large gradients. (The usual practice is not to concentrate boxes near hinge lines because viscous effects, which dominate trail- ing-edge control-surface aerodynamics, reduce hinge moments from potential theory results, so that more chordwise boxes tend to overpredict hinge moments.) The chord lengths of ad- jacent boxes in the streamwise direction should only change gradually. If a surface lies in (or nearly in) the plane of another surface, the spanwise divisions of the downstream surface should lie along the spanwise divisions of the upstream sur- face. Strips at the intersection of lifting surfaces should have approximately equal widths.
The foregoing qualitative modeling recommendations have been quanti!ed (the guidelines have been summarized by Rod- den and Johnson9) as follows. The aspect ratio of the boxes should be less than three. The chord length of the boxes should be less than 0.08 times the minimum velocity divided by the maximum frequency (in Hz) of interest, i.e., Dx < 0.08U / f (this is a requirement for approximately 12 boxes per minimum wave length; however, no less than four boxes per chord should be used). The limitation of box aspect ratio to three is a consequence of the DLM assumption of a parabolic variation in the incremental kernel numerator. The variation of the real part of the incremental kernel numerator along the box quarter- chord is shown in Fig. 3 for an aspect ratio of 2.
Aeroservoelastic analyses frequently deal with high frequen- cies in control-system components, and new design criteria for short wavelength gusts, e.g., the tuned discrete gust requires a minimum gradient distance of 30 ft be analyzed,10,11 also re- quire that higher reduced frequencies be considered. The com- bination of the box chord length limitation and the box aspect ratio limitation can result in a requirement for a large number of aerodynamic boxes. A higher-order (higher than quadratic) approximation to the numerator of the incremental oscillatory kernel will increase the limit on box aspect ratio for accurate
RODDEN,TAYLOR,ANDMCINTOSH 721Fig.2Idealizationofliftingsurfaceusingtrapezoidalboxes.Fig.1Liftingsurfacecoordinatesystem.