The supersonic lifting surface method for isolated surfaces that was originally included in the
Aeroelastic Addition to MSC.Nastran [Rodden, Harder, and Bellinger (1979)] was a version of
the Mach Box method developed for the NASA Space Shuttle. It has been superseded in the software by ZONA51 for multiple interfering surfaces, but has been retained to allow upward compatibility and for users who do not have the ZONA51 option (Aero II).
Mach Box aerodynamics may be used to compute unsteady supersonic aerodynamic forces for a planar, isolated wing at supersonic speeds. The surface (seeFigure 2-4) may have a leading and/or trailing edge crank. There may be one or two adjacent trailing edge control surfaces. The "inboard" edge (side 1-2 on the CAERO3 entry) must be the plane of structural, mass, and aerodynamic symmetry.
The geometry of the planform is specified on the CAERO3 data entry. The two leading edge corners are located by the user. These, along with the flow direction, define the plane of the element. Up to 10 additional points are permitted to specify cranks and controls; these points are dimensional quantities using a coordinate system in the plane of the element and with its origin at point 1.
The recommended minimum number of Mach boxes is 80. The recommended minimum number in the flow direction is seven. The number of boxes in the flow direction is entered on the PAERO3 entry. The number of spanwise boxes is determined within the program as outlined inAerodynamic Theoriessince the total number of boxes depends on the Mach number, m, and will change for different Mach numbers. The number of Mach boxes in the flow direction may be computed as follows:
where:
xmax = maximum chordwise dimension
ymax = maximum spanwise dimension
and Int( ) denotes the integer value of ( ), and NSB is the number of spanwise Mach boxes:
where NBOX0 = initial number of boxes selected
In order to maintain aerodynamic matrices that are the same size for each m,k pair and that are generated at the same physical location, the user must select a set of aerodynamic grid points (also called control points in this case) to be used. On each aerodynamic surface there must be at least three noncolinear control points and usually more than three points are required to represent the modal deflections adequately. It is noteworthy that three noncolinear control points provide sufficient geometric information for rigid body motion of the aerodynamic surface. The CAERO3 entry selects the points by defining their geometric location on the wing. These aerodynamic grid points are located using the coordinate system shown inFigure 2-4. The T3 component of these points is normal to the plane of the element. Additional lists of at least three points are needed for each optional control surface that is included. These aerodynamic grid points are numbered starting with the ID field of the CAERO3 entry, which must be a larger ID number than any GRID, SPOINT, or EPOINT ID in the model. Interpolation from the Mach box centers to determine deflections and slopes at these designated control points is performed with surface spline routines within the program and requires no input from the user.
Figure 2-4. Mach Box Surface
The following restrictions apply to the Mach Box method:
1. The edge 1-2 is taken as a plane of structural, mass, and aerodynamic symmetry; either symmetric or antisymmetric motions can be considered.
2. Both leading edge and hinge line sweepback angles must be greater than or equal to zero. 3. If a leading edge crank is not present, then x5, y5do not have to be input.
4. If a trailing edge crank is not present, then x6, y6do not have to be input.
5. A trailing edge crank cannot be located on the trailing edge of a control surface. It must be located inboard of the inboard surface, outboard of the outboard surface, or exactly at the junction between the two control surfaces.
6. All control surface side edges must be parallel to the flow or swept inward.
7. Points 8, 10, and 12 are used with Points 7, 9, and 11, respectively, to define the control surface edge, and they must be distinct from Points 7, 9, and 11, but with one exception they do not have to lie on the wing trailing edge. The program will calculate new Points 8, 10, and 12 for the wing trailing edge. The exception is that Points 8, 10, or 12 must be located on the trailing edge if the trailing edge is cranked at the side edge of a control surface.
8. When only one control surface is present, it must be control surface one.
9. If the second control surface is not present, then x11, y11and x12, y12are not required as input.
11. No aerodynamic balance for the control surfaces has been included in the program. 12. The number of chordwise boxes used as input (NBOX) to the program should be carefully
selected to provide at least 80 boxes on the wing but NBOX cannot exceed 50. Note that NBOX is the number of chordwise boxes between the most forward point and the most aft point on the lifting surface, as shown inFigure 2-5.
13. If the maximum number of allowable boxes (500 on the main surface, 200 on each control surface) is exceeded, the program will reduce the number of chordwise boxes one at a time until the number of boxes is under the allowable limit.
Figure 2-5. Mach Box Surface Showing Mach Boxes and Diaphragm Region
Strip Theory
Modified Strip Theory can be used for unsteady aerodynamic forces on a high aspect ratio lifting surface, although it is less accurate than the available lifting surface theories. Each strip may have two or three degrees of freedom. Plunge and pitch are always used, and rotation of an aerodynamically balanced control surface is optional. If a control surface is present, either a sealed or an open gap may be used.
The planform (which may have several strips in one macro-element) is specified on a CAERO4 Bulk Data entry. A sample planform is shown inFigure 2-6. The user supplies the two leading edge corner locations and the edge chords as dimensional quantities. Edge chords are assumed to be in the flow direction. All additional geometry (box divisions, hinge locations, etc.) are given in dimensionless units. Multiple CAERO4 entries may be used if there are several surfaces or cranks.
A grid point is assigned to each strip, and is assigned an ID starting with the CAERO4 entry ID and incrementing by one for each strip. The plunge (T3) and pitch (R2) degrees of freedom have the conventional definition. When a control surface is present, the R3 degree of freedom has a nonstandard definition, in the case of the relative control rotation. When interconnecting with the structure, the ordinary (surface or linear) splines can be used for T3 and R2, but a special method (see SPLINE3 data entry) is used for the relative control rotation.
The parameters such as the lift curve slope or the lag function may be varied to account approximately for finite-span effects (three-dimensional flow) and Mach number by AEFACT Bulk Data entry selection from PAERO4. The AEFACT Bulk Data entry format used by Strip Theory is shown in the remarks on the PAERO4 Bulk Data entry. The user may request a Prandtl-Glauert (compressibility) and/or a sweep correction to the value of the lift curve slope. The lag function depends upon the local (i.e., using the chord of the strip) reduced frequency; for incompressible flow, it is the Theodorsen function C(k). An approximate form for this function is given by
Equation 2-1.
in which B0= 0, and may be selected for computing variations on the Theodorsen function that account for compressibility and finite span effects. The choice of parameters bnand B0is left to the user to select values suitable for the requirement. Bisplinghoff, Ashley, and Halfman (1955, pp. 350, 394) give values for various Mach numbers and aspect ratios.
Figure 2-6. Strip Theory Example Lifting Surface